Geometric Analysis Seminar

Organizers: Pengfei Guan, Joshua Flynn and Jérôme Vétois


Tuesday, February 6, 14:35-15:35, Burnside Hall 1205
Fang Hong (McGill University)   
Title: On normal derivative of Dirichlet eigenfunctions
Abstract: Normal derivative gives a lot of useful information of Dirichlet eigenfunctions and eigenvalues on Riemannian manifolds. We will introduce Hadamard's formula for eigenvalues and local relationship between normal derivative and curvature of boundary. Then we will review Ozawa's and Tao's results, giving more detailed estimates to normal derivatives. 
Tuesday, February 20, 14:35-15:35, Burnside Hall 1205
Cale Rankin (University of Toronto)   
Title: A geometric approach to the Ma–Trudinger–Wang estimates
Abstract: For a long time the regularity of optimal transport with general costs was a fundamental open problem. Whilst this question had been answered in particular cases by Delanoë, Caffarelli, and Urbas (separately) via apriori estimates for Monge–Ampère equations, it remained unknown how to deal with general costs. The extension of these estimates and the regularity of the optimal transport maps was proved in a groundbreaking work by Ma, Trudinger, and Wang who introduced a mysterious fourth order condition on the cost. Kim and McCann realised this condition amounts to a curvature condition in a particular pseudo-Riemannian geometry and, with Warren, realised after a conformal rescaling the graph of optimal transport maps are maximal surfaces in this geometry. In this talk, which describes joint work with Brendle, Léger, and McCann we show how the original MTW estimates may be realized via estimates for maximal surfaces in pseudo-Riemannian geometry.
Tuesday, March 12, 14:35-15:35, Burnside Hall 1205
Pengfei Guan (McGill University)   
Title: Anisotropic Gauss curvature flows and Lp Minkowski problem
Abstract: The classical Minkowski problem is a problem of area measures. Lutwak introduced Lp version of the Minkowski problem. A basic question is under what conditions one may produce a weak solution. The problem can be reframed as a variational problem for the associated entropies with volume constraint. We use Andrews anisotropic Gauss curvature flows provide good path to achieve the goal. The talk is based on a recent joint work with Karoly Boroczky.
Tuesday, March 19, 14:35-15:35, Burnside Hall 1205
Hyun Chul Jang (Caltech)   
Title: Instability of minimal entropy rigidity for product spaces of rank-one symmetric spaces
Abstract: The volume entropy is a geometric invariant defined as the volume growth of geodesic balls in the universal cover equipped with the pull-back metric. The minimal entropy and the corresponding rigidity for some special cases have been established in the literature. In this talk, I will review the minimal entropy rigidity results and discuss a couple of new results concerning product spaces of rank-one symmetric spaces. Firstly, I will present the uniqueness of the spherical Plateau solution, which is closely related to the minimal entropy rigidity. Secondly, I will demonstrate the geometric instability of the minimal entropy rigidity for product spaces with a counterexample. This observation differs from the recent results of Song for locally symmetric spaces.
Tuesday, March 26, 14:35-15:35, Burnside Hall 1205
Joshua Flynn (McGill University)   
Title: Conformal Boundary Operators, Extension Theorems and Trace Inequalities.
Abstract: The Caffarelli-Silvestre extension theorem allows one to obtain the fractional Laplacian of fractional order in (0,1) via a Dirichlet-to-Neumann map which maps solutions to a weighted Dirichlet problem on Euclidean halfspace to functions on its boundary. Dirichlet's principle gives a resulting sharp Sobolev trace inequality. Given that the fractional Laplacian may be defined for higher fractional orders and analogous fractional operators are naturally defined on conformal or CR infinities of certain geometries, it is natural to extend Caffarelli-Silvestre's results to the higher order and for general geometries. In this talk, we discuss relevant recent results obtained in collaboration with G. Lu and Q. Yang.
Tuesday, April 2, 14:35-15:35, Burnside Hall 1205
Min Chen (McGill University)   
Title:Inhomogeneous Gauss Curvature flow.
Abstract:We consider the flow of convex hypersurfaces in Euclidean space R^{n+1} under the in-homogeneous speed functions of Gauss curvature. We establish the existence and convergence of the flow to a limit which is the round sphere (after rescaling) under appropriate conditions of the speed functions for $\alpha>\frac{1}{n+1}$. This generalized the celebrated results on Gauss curvature flow by Andrews-Guan-Ni and Brendle-Choi-Daskalopoulos. This is joint work with Prof.Pengfei Guan and Jiuzhou Huang.

Friday, April 5, 13:30-14:30, Burnside Hall 1104, joint with the Analysis Seminar
Yi Wang (Johns Hopkins University)
Tuesday, April 9, 14:35-15:35, Burnside Hall 1205
Sébastien Picard (University of British Columbia)   
Title: G2 flows and parabolic complex Monge-Ampere equations
Abstract: We will discuss how parabolic complex Monge-Ampere equations arise in the context of G2 geometry. Estimates for these general Monge-Ampere flows can be used to provide examples of the G2 flow which exist for all time and converge to a torsion-free G2 structure on a compact manifold. This talk will survey joint works with X.-W. Zhang and C. Suan.

Tuesday, April 23, 14:35-15:35, Burnside Hall 1205
Xiangwen Zhang (University of California, Irvine)
Title: A geometric flow in symplectic geometry
Abstract: Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. In this talk, we will discuss the progress on the so-called Type IIA flows, introduced in a joint work with Fei, Phong and Picard, aiming to study the Type IIA equations from the flux compactifications of superstrings.

Tuesday, April 30, 14:35-15:35, Burnside Hall 1205
Samuel Zeitler (McGill University) 

Tuesday, May 7, 14:35-15:35, Burnside Hall 1205
Mariel Sáez Trumper (Pontificia Universidad Católica de Chile) 
Tuesday, May 21, 10:00-10:55, Burnside Hall 1205
Meijun Zhu (The University of Oklaoma) 
Title:On the extension operators - sharp inequalities and the applications
Abstract: Define the general extension operator $E_{alpha,beta}f(x^2,x_n)=\int_{\partial R^n_+}x_n^\beta f(y)/(|x'-y|^2+x_n^2)^((n-\alpha)/2))dy$ In this talk, I shall recall how we arrive at this general form in the study of the extension operators, what kind of sharp estimates we have obtained so far in the last ten years. I shall also report some recent results on the application of these estimates, in particular in the study of Carleman type inequalities in Hardy space.
Tuesday, May 21, 11:00-11:55, Burnside Hall 1205
Jyotshana Prajapat (University of Mumbai)   
Tuesday, May 28, 14:35-15:35, Burnside Hall 1205
Liokumovich (University of Toronto)   

FALL 2023

Tuesday, October 3, 14:30-15:30, Burnside Hall 1104
Marcin Sroka (McGill University)  
Title: Second order estimate for Monge-Ampere type equations on Riemannian manifolds with additional structure
Abstract: We will discuss the challenges and differences in obtaining second order estimates for some second order, elliptic equations (of Monge-Ampere type) arising naturally on Riemannian manifolds endowed with additional structure (like complex or quaternionic one). This type of equations emerged for example in the course of proving Calabi’s volume prescribing conjecture or its analogue in hypercomplex case – Alesker-Verbitsky conjecture.
Tuesday, October 17, 14:35-15:35, Burnside Hall 1104
Dylan Cant (McGill University) 
Title: Floer theory in convex-at-infinity symplectic manifolds
Abstract: "Floer theory" is a sort of elliptic Morse homology constructed by counting solutions to a certain inhomogeneous perturbation of the J-holomorphic curve equation. The nonlinear PDE is now called "Floer's equation." As discovered by Gromov and Floer, if one works in a symplectic manifold which "tames" the almost complex structure J and uses inhomogeneous perturbations coming from certain Hamiltonian vector fields one has a priori W1,2 estimates (for suitable choice of Riemannian metric). When the symplectic manifold is non-compact, the variety of invariants which can be constructed using Floer theory is still an open field of research. A well-studied class of noncompact manifolds are those which are symplectically "convex" at infinity, in the sense of Eliashberg-Gromov. I will present some recent work which proves a maximum principle for solutions to Floer's equation in convex-at-infinity symplectic manifolds.
Tuesday, October 24, 14:35-15:35, Burnside Hall 1104
Min Chen (McGill University)  
Title:Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere
Abstract: The Alexandrov-Fenchel inequalities in the Euclidean space are inequalities involving quermassintegrals of different orders and are classical topics in convex geometry and differential geometry. Brendle-Guan-Li proposed a conjecture on the corresponding inequalities for quermassintegrals in the sphere. In this talk, we introduce a new progress to this Conjecture.
Tuesday, October 31, 14:35-15:35, Burnside Hall 1104
Amir Moradifam(University of California Riverside)  
Title: The Sphere Covering Inequality and Its Applications
Abstract: We show that the total area of two distinct Gaussian curvature 1 surfaces with the same conformal factor on the boundary, which are also conformal to the Euclidean unit disk, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total areas as the Sphere Covering Inequality. This inequality and it’s generalizations are applied to a number of open problems related to Moser-Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results. In particular we confirm the best constant of a Moser-Truidinger type inequality conjectured by A. Chang and P. Yang in 1987. This is a joint work Changfeng Gui.
Tuesday, November 14, 14:35-15:35, Burnside Hall 1104
Jacob Reznikov (McGill University)  
Title: Isoperimetric inequality using curvature flows of conformal vector fields
Abstract: We describe a flow approach to the Isoperimetric inequality first derived by Guan-Li in 2013 and then improved by Guan-Li-Wang in 2018, and its generalization by Li-Pan in 2023. We describe the key algebraic properties of a special type of conformal vector field which allows us to define a curvature flow on embedded hypersurfaces. We then derive the evolution equations of geometric properties along this flow and finish by proving its convergence.
Tuesday, November 28, 14:35-15:35, Burnside Hall 1104
Edward Chernysh (McGill University)  
Title: A Struwe-Type Decomposition for Caffarelli-Kohn-Nirenberg Equations
Abstract: In this talk, we establish a Struwe-type decomposition result for a class of critical $p$-Laplace equations of the Cafarelli-Kohn-Nirenberg type, in smoothly bounded domains for $n \ge 3$. In doing so, we highlight crucial differences between the weighted setting and the pioneering work of Michael Struwe in the unweighted model $p=2$ case.
Tuesday, December 5, 14:35-15:35, Burnside Hall 1104
Bartosz Syroka (McGill University)  
Title: A harmonic flow of geometric structures
Abstract: We will consider minimizers of a Dirichlet-type energy of tensor fields on a Riemannian manifold, left invariant by some Lie group H inside SO(n). To this end, we will explain how to deform these H-structures, define their torsion and some of its basic properties. The Euler-Lagrange equations for the Dirichlet-type energy functional can then be phrased in terms of a deformation of the structure by its torsion tensor. We will show some of the analytic results for the corresponding gradient flow of the energy, if time allows. Throughout the talk, we will reference the unitary group U(m) as a guiding example.


Wednesday, February 1, 15:15-16:15, Burnside Hall 1234 and Zoom
Hugues Auvray (Université Paris-Sud) 
Title: Bergman kernels on punctured Riemann surfaces
Abstract: In joint works with X. Ma (Paris 7) and G. Marinescu (Cologne) we obtain refined asymptotics for Bergman kernels computed from singular data. We work on the complement of a finite number of points, seen as punctures, on a compact Riemann surface, that we endow with a metric extending Poincaré's cusp metric around the puncture points. We moreover fix a holomorphic line bundle polarizing such a metric. I'll explain how an advanced description of the model (on the punctured unit disc) and weighted analusis techniques in a singular context allow to describe the Bergman kernels associated to such Riemann surfaces, up to the singularities.
Wednesday, February 8, 15:00-16:00, Burnside Hall 1234 and Zoom
Marcin Sroka
(McGill University)
Title: Gradient estimates for complex PDEs
Abstract: We will discuss the recent paper of Guo, Phong and Tong on the gradient estimate for the complex Monge-Ampere equation. In general we will outline troubles with obtaining this bound for more general PDEs on complex manifolds.
Wednesday, February 15, 15:00-16:00, Burnside Hall 1234 and Zoom
Fang Hong (McGill University) 
Title: Sharp Minkowski inequalities in Hadamard manifolds and their applications
Abstract: Extension of Minkowski inequality to hyperbolic space H^3 and Finding the sharp inequality have been a long standing problem. We will discuss the recent paper by M. Ghomi and J. Spruck, in which they generalized Minkowski inequality to Cartan-Hadamard manifolds via harmonic mean curvature flow. We will further discuss sharper inequality we get based on their results.
Wednesday, February 22, 15:00-16:00, Burnside Hall 1234 and Zoom
Pengfei Guan
(McGill University) 
Title: Remarks on the gradient estimate for real and complex Hessian type equations
Abstract: This is a follow-up of Marcin Sroka's talk. We will discuss various methods to obtain the global and interior the gradient estimates for the real Hessian type equations. We then switch the attention to global estimate of complex Hessian equations. There are still several open problems in this case, we will discuss why is so difficulty in complex. We will provide a proof of gradient estimate for geometric solutions of a class of complex Hessian equations on Hermitian manifolds.
Wednesday, March 8, 15:00-16:00, Burnside Hall 1234 and Zoom
Jacob Reznikov
(McGill University) 
Title: Entropy and singularities in mean curvature flow.
Abstract: We will discuss the basics of mean curvature flow and construct some examples of its singularities. We will then discuss entropy and Huisken’s monotonicity formula and some of its basic properties. We will further discuss a recent paper by Chodosh, Choi, Mantoulidis and Schulze on low-entropy initial data and further classification we get based on their results.
Wednesday, March 22, 15:00-16:00, Burnside Hall 1234 and Zoom
Min Chen
(McGill University) 
Title: Foliations by stable spheres with constant Gauss curvature in an asymptotically flat Riemannian manifold.
Abstract: We will discuss using a heat flow method to deform a coordinate sphere into a constant Gauss curvature surface. With the positivity of the mass, we then prove that the constructed surfaces form a stable constant Gauss curvature foliation. It defines a natural coordinate system near infinity. Finally, we will briefly introduce a method to obtain the uniqueness of the constant mean curvature foliation for ends with positive mass.
Wednesday, March 29 15:00-16:00, Burnside Hall 1234 and Zoom
Samuel Zeitler
(McGill University) 
Title: Sharp Sobolev inequalities of arbitrary order.
Abstract: Best constants for Sobolev inequalities on closed Riemannian manifolds have been the target of investigation for decades. We will introduce the first order case and discuss a natural extension of this problem to higher order embeddings. In particular we present the value of the best first constant and give examples of classes of manifolds where this constant is achieved.
Wednesday, April 5, 15:00-16:00, Burnside Hall 1234 and Zoom
Carlo Scarpa
Title: Kahler forms and B-fields
Abstract: Motivated by some constructions in Mirror Symmetry, we will consider the problem of finding a canonical representative of a complexified Kahler class on a compact complex manifold. In 2020, Schlitzer and Stoppa proposed a geometric PDE, whose solutions conjecturally give the required canonical representative of the class. I will explain a variational framework in which to consider their equation, focussing on an associated system of geodesic equations for Kähler potentials. In particular, I will explain how to prove uniqueness of solutions of the PDE in the toric setting.
Wednesday, April 19, 15:00-16:00, Burnside Hall 1234 and Zoom
Jérôme Vétois
(McGill University) 
Title: On the entire solutions to the critical p-Laplace equation
Abstract: We will discuss the problem of classifying the entire, positive solutions to the critical p-Laplace equation in the Euclidean space. In the case where p=2 (i.e. for the classical Laplace operator), this problem was solved by Caffarelli, Gidas and Spruck in 1989 (see also the work of Obata in 1971 for finite-energy solutions). I will review some recent work extending this result to the case of the p-Laplace operator for large values of p.
Thursday June 22, 15:00-16:00, Burnside Hall 920
YanYan Li (Rutgers University)

Title: Some recent results on conformally invariant equations
Abstract: We will present some recent work on conformally invariant nonlinear elliptic equations. This includes results on Liouville type theorems, derivative estimates, isolated singularities, existence and nonexistence of solutions.

FALL 2022

Wednesday, October 5, 15:00-16:00, Burnside Hall 1234 and Zoom
Bartosz Syroka (McGill University) 
Title: Balanced metrics on 6-manifolds of cohomogeneity one
Abstract: We will introduce the notions of a cohomogeneity one group action on a 6-manifold, and of balanced, non-Kahler SU(3) structures invariant with respect to the action. We will then state and describe an existence result contingent on the decomposition of the Lie algebra of the group. Such structures form a part of the Strominger system playing a role in string theory, inspiring recent work on the interplay of Lie group symmetries and the associated set of PDEs.
Wednesday, October 19, 15:00-16:00, Burnside Hall 1234 and Zoom
Borthwick (Université de Bourgogne-Franche Comté) 
Title: Projective differential geometry and asymptotic analysis in General Relativity
Abstract: Every Lorentzian manifold $(M,g)$ has a natural projective structure induced by its Levi-Civita connection. In some cases, $M$ can be embedded into a manifold with boundary $\overline{M}$, in which the projective structure extends to the boundary: $(M,g)$ is then said to be projectively compact. In this talk, we will discuss applications of the projective structure to the asymptotic analysis of partial differential equations, in particular a generalised Proca equation, on projectively compact Lorentzian manifolds.
Wednesday, October 26, 15:00-16:00, Burnside Hall 1234 and Zoom
Huangchen Zhou (McGill University) 
Title: From isometric embedding to a sum of squares theorem
Abstract: Motivated by an isometric embedding problem in the graph setting, we'll discuss a sum of squares theorem for Holder function. Given a non-negative C^{2,2\alpha} function f over R, can we find a function g in C^{1,\alpha} such that f=g^2? We've found a necessary and sufficient condition for this problem, which is related to the non-zero strict local minimum points of the function f.
Wednesday, November 2, 15:00-16:00, Burnside Hall 1234 and Zoom
Joshua Flynn (McGill University) 
Title: Sharp Hardy-Sobolev-Maz'ya inequalities for noncompact rank one symmetric spaces
Abstract: The Hardy-Sobolev-Maz'ya inequality combines the Hardy and Sobolev inequalities into a single inequality on the halfspace. Using conformal equivalence, this inequality is equivalent to the Poincare-Sobolev inequality on the real hyperbolic space. Using the Helgason-Fourier analysis, higher order versions of these inequalities were established by G. Lu and Q. Yang. We introduce these results and indicate how they may be extended to the other noncompact rank one symmetric spaces. Discussed works are that of J. Li, G. Lu, Q. Yang and myself.

Wednesday, November 9, 15:00-16:00, Burnside Hall 1234 and Zoom
Zhizhang Wang (Fudan University) 
Title: The prescribed curvature problem in Minkowski space
Abstract: In this talk, we will first discuss the existence of hypersurfaces with constant hessian curvature in Minkowski space. Using the similar method, we can obtain some existence theorems for the prescribed hessian curvature equations. We may further consider the hessian curvature flow in Minkowski space.
Wednesday, November 16, 15:00-16:00, Burnside Hall 1234 and Zoom
Marcin Sroka (McGill University) 
Title: On certain, geometrically motivated, Monge-Ampere type equation
Abstract: I will discuss the equation Alesker and Verbitsky introduced on hyperKahler with torsion manifolds as a device for proving “quaternionic version” of the Calabi conjecture. The equation is called quaternionic Monge-Ampere equation and has many common features with its real and complex counterparts. Its solvability has applications to obtaining Calabi-Yau type theorems for different classes of hermitian and hyperhermitian metrics.
Wednesday, November 23, 15:00-16:00, Burnside Hall 1234 and Zoom
Ramya Dutta (TIFR Bangalore) 
Title: Apriori decay estimates for Hardy-Sobolev-Maz'ya equations and application to a Brezis-Nirenberg problem
Abstract: In this talk we will discuss some qualitative properties and sharp decay estimates of solutions to the Euler-Lagrange equation corresponding to Hardy-Sobolev-Maz'ya inequality with cylindrical weight. Using these sharp asymptotics we will establish a Brezis-Nirenberg type existence result for class of $C^1$ sublinear perturbations of the p-Hardy-Sobolev equation with cylindrical weight in a bounded domain in dimensions $n > p^2$ and an appropriate notion of positivity for these perturbations.
Wednesday, November 30, 15:00-16:00, Burnside Hall 1234 and Zoom
Bruno Premoselli (Université Libre de Bruxelles) 
Title: (Non)-Compactness for sign-changing solutions of the Yamabe equation at the lowest energy level
Abstract: The goal of this talk is to describe the behavior of least energy sign-changing solutions of the Yamabe equation. Sign-changing solutions of the celebrated Yamabe equation naturally appear as extremals for the minimization problem of eigenvalues of the conformal Laplacian in a fixed conformal class. We will review their link with the original geometric problem and will describe in this talk their behavior at the lowest energy level. Our main focus will in particular be a compactness result in small dimensions or in the locally conformally flat case. The results in this talk have been obtained in collaboration with J. Vétois.
Wednesday, December 7, 15:00-16:00, Burnside Hall 1234 and Zoom
Clara Aldana (Universidad del Norte, Barranquilla)  
Title: Isospectral and quasi-isospectral Schrodinger operators 
Abstract: On this talk I will first briefly talk about the isospectral problem in geometry and about isospectrality of Strum-Liouville operators on a finite interval in the simplified form of a Schrodinger operator. I will mention very interesting known results about isospectral potentials. I will introduce quasi-isospectrality as a generalization of isospectrality. I will mention the history of the problem and how to construct quasi-isospectral potentials. I will present what we know so far about them. The work presented here is still on-going joint work with Camilo Perez.


Wednesday, February 9, 9:30-10:30, Zoom Meeting
Pengfei Guan (McGill University) 
Title: Diameter estimates for solutions of $L^p$-Minkowski problem with weak data
$L^p$-Minkowski problem corresponds to the following Monge-Ampere equation on $S^n$: $\det(u_{ij}(x)+u(x)\delta_{ij})=u^p(x)f(x)$. We establish estimate for solution $u$ with the prescribed data $f$ only in certain integrable space. In the talk, we will review literature and discuss some open problems.
Wednesday, February 16, 9:30-10:30, Zoom Meeting
Gantumur Tsogtgerel (McGill University) 
Title: Perron’s solution and the Weyl projection method
In this talks we will focus on the relationship between the Perron-Wiener method, and the variational method based on Sobolev spaces for constructing solutions of the classical Dirichlet problem. After talking about some facets of the history of the Dirichlet problem, we will discuss an elementary method to deal with the aforementioned issue
Wednesday, February 23, 9:30-10:30, Zoom Meeting
Jérôme Vétois (McGill University) 
Title: Stability and instability results for sign-changing solutions to second-order critical elliptic equations
In this talk, we will consider a question of stability (i.e. compactness of solutions to perturbed equations) for sign-changing solutions to second-order critical elliptic equations on a closed Riemannian manifold. I will present a stability result obtained in the case of dimensions greater than or equal to 7. I will then discuss the optimality of this result by constructing counterexamples in every dimension. This is a joint work with Bruno Premoselli (Université Libre de Bruxelles, Belgium).
Wednesday, March 9, 9:30-10:30, Zoom Meeting
Jiuzhou Huang (McGill University) 
Title: A warped product metric, Hilbert Einstein functional and Weyl’s problem
In this talk, we use a warped product metric introduced by Izmestiev and the Einstein Hilbert functional to study Weyl’s embedding problem. The warped metric can be used to give a new proof of the closeness of the problem, and the Hilbert-Einstein functional is related to the variational property and stability of the embedding.
Wednesday, March 16, 9:30-10:30, Zoom Meeting
Sisi Shen (Columbia University) 
Title: A Chern-Calabi flow on Hermitian Manifolds
We discuss the existence problem of constant Chern scalar curvature metrics on a compact complex manifold and introduce a Hermitian analogue of the Calabi flow on compact complex manifolds with vanishing first Bott-Chern class.
Wednesday, March 23, 9:30-10:30, Zoom Meeting
Bartosz Siroka (McGill University) 
Title: A cohomogeneity one approach to Kahler-Einstein metrics
We will discuss the work of Andrew Dancer and McKenzie Wang on the classification of solutions to the Kahler-Einstein equations using cohomogeneity one group actions, reducing the equations to a system of nonlinear ODEs. We will first cover the cohomogeneity one setup, then the explicit local solutions to the Einstein equations. Lastly, we will look at extending the local solutions to the global case.
Wednesday, March 30, 9:30-10:30, Zoom Meeting
Min Chen (McGill University) 
Title: Alexandrov-Fenchel type inequalities in the sphere
The Alexandrov-Fenchel inequalities for quermassintegrals in the Euclidean spaces are classical topics in differential geometry and convex geometry. The corresponding problem in non-convex hypersurfaces in space forms has gained much interest recently but remains largely unsettled. The application of curvature flows to prove the geometric inequalities is nowadays classical. In this talk, I will introduce a recent work about the Alexandrov-Fenchel inequalities in the sphere by employing suitable curvature flows.
Wednesday, April 6, 9:30-10:30, Zoom Meeting
Valentino Tosatti (McGill University) 
Title: The Chern-Ricci flow
The Chern-Ricci flow is a flow of Hermitian metrics by their Chern-Ricci form, which generalizes the Kähler-Ricci flow to the setting of non-Kähler metrics on complex manifolds, introduced by Weinkove and myself around 10 years ago. I will give an overview of known results for this flow, including a detailed discussion of the case of compact complex surfaces, and describe some open problems.
Wednesday, April 20, 9:30-10:30, Zoom Meeting
Huangchen Zhou (McGill University) 
Title: Carleman type estimates and uniqueness of Cauchy problem
Carleman estimate is a weighted estimate in proving the uniqueness of Cauchy problem. I will review some results and discuss an explicit example. In this example, we will see 1. How to get the estimate, 2. How to prove the uniqueness with this estimate.
Wednesday, April 27, 9:30-10:30, Zoom Meeting
Vladmir Sicca (McGill University)
Title: A prescribed scalar and boundary mean curvature problem on asymptotically Euclidean manifolds with boundary
(Joint work with Gantumur Tsogtgerel) We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.

Wednesday, May 4, 9:30-10:30, Zoom Meeting
Nick McCleerey (University of Michigan)
Title: Lelong Numbers of m-Subharmonic Functions Along Submanifolds
We study the possible singularities of an m-subharmonic function φ along a complex submanifold V of a compact Kähler manifold, finding a maximal rate of growth for φ which depends only on m and k, the codimension of V. When k < m, we show that φ has at worst log poles along V, and that the strength of these poles is moveover constant along V. This can be thought of as an analogue of Siu's theorem. This is joint work with Jianchun Chu.

Wednesday, May 11, 9:30-10:30, Zoom Meeting
Carlos Valero (McGill University) 
Title: Stability in the Inverse Steklov Problem for Warped Products
We review some results from a recent paper of Daudé, Kamran and Nicoleau in which it is shown that approximate knowledge of the Steklov spectrum, that is the spectrum of the Dirichlet-to-Neumann map, determines a warped product metric in a neighbourhood of the boundary; some stability estimates on the metric are also proved. We briefly touch on a possible extension to the spinor Laplacian.

FALL 2021

Wednesday, October 20, 12:30-13:30, Zoom Meeting
Bartosz Syroka (McGill University)
Title: An overview of the Hull-Strominger system
We will present the basics of conformally balanced Calabi-Yau manifolds, with a view towards the Hull-Strominger system of PDEs coming from heterotic string theory. We will discuss some of the known solutions, including the first non-Kähler cases such as Goldstein-Prokushkin fibrations. Finally, we will look at the approach to the system via the Anomaly flow.
Wednesday, October 27, 12:30-13:30, Zoom Meeting
Bartosz Syroka (McGill University)
Title: The Hull-Strominger system on Riemann surface fibrations
We present the construction of generalized Calabi-Gray manifolds, which are Riemann surface fibrations with hyperkähler fibres. The Anomaly flow reduces to a single scalar equation for a smooth function on the base surface. We will consider its properties of long time existence and convergence under the assumptions of large initial data.
WednesdayNovember 3, 12:30-13:30, Zoom Meeting
Pengfei Guan (McGill University)
Title: Gauss curvature type flows: an introduction
The talk is an introduction to some recent results on Gauss curvature type flows in $\mathbb R^{n+1}$. The main focus is the flow by powers of Gauss curvature $X_t=-K^{\alpha}\nu, \alpha>0$ in ambient space $\mathbb R^{n+1}$, where $\nu$ the outer normal and $K$ the Gauss curvature of the evolving convex hypersurfaces. After a brief review of curve shorting flow, we discuss the work of Guan-Ni, Andrews-Guan-Ni on entropies associated to the flows. The crucial entropy point estimate enable us to control the lower bound of the support function of normalized flow and deduce the convergence of the flow to solitons. Finally, we present the main arguments of the proof of beautiful uniqueness theorem of solitons for $\alpha>\frac{1}{n+2}$ by Brendle-Choi-Daskalopoulos.
Wednesday, November 10, 12:30-13:30, Zoom Meeting
Min Chen (McGill University)
Title: Flow by powers of the Gauss curvature in space forms
In this talk, we consider flow by powers of the Gauss curvature in space forms $\mathbb{N}^{n+1}(\kappa)$. Our approach to this flow (in space forms) is to deduce it to a flow in the Euclidean space by proper projections. The key in our proof is an almost monotonicity formula for associated entropies considered in Guan-Ni and Andrews-Guan-Ni. We could obtain a new monotone quantity $\mathcal{E}_{\alpha}(\hat{\Omega}_t)+C(n,\alpha,\tilde{X}_0)e^{-\frac{2(n+1)}{2n+1}t}$ along the normalized flow by modifying the monotone quantity used in Euclidean space. This allows us to extend the known results in Euclidean space to space forms completely.
WednesdayNovember 17, 12:30-13:30, Zoom Meeting
Jiuzhou Huang (McGill University)
Title: Flow by powers of the Gauss curvature in space forms (continued)
This is a continuation of Chen’s talk last week about our paper for flows by powers of the Gauss curvature in space forms. I this talk, I will go into more details of the proof. The focus will be on the difference between the space forms and the Euclidean space.
WednesdayNovember 24, 12:30-13:30, Zoom Meeting
Carlos Valero (McGill University)
Title: A Calderon problem for U(1)-connections coupled to spinors
We introduce the Dirichlet-to-Neumann (DN) map for the Dirac Laplacian coupled to a U(1)-connection A on a spin manifold with boundary, and show that it is a pseudodifferential operator of order 1 whose symbol determines the metric and connection to infinite order at the boundary. We go on to show that A can be recovered up to gauge equivalence from the DN map in the real analytic category if it satisfies a Yang-Mills-Dirac equation in the interior.
WednesdayDecember 8, 12:30-13:30, Zoom Meeting
Valentino Tosatti (McGill University)
Title: Immortal solutions of the Kähler-Ricci flow
I will discuss what is known, not known, and conjectured about solutions of the Kähler-Ricci flow on compact Kähler manifolds which exist for all positive time.


Wednesday, February 24, 13:30-14:30, Zoom Meeting
Bartosz Syroka (McGill University)
Title: Quasi-local mass in spacetimes
We will present some basic ideas of general relativity, and the problem of calculating mass and energy quantities associated to regions of spacetime. To this end, we will explain the equations of curvature for submanifolds of Lorentzian manifolds, and conformal Killing-Yano tensors which allow us access to Minkowski curvature formulas in the spacetimes which admit them.
Wednesday, March 10, 13:30-14:30, Zoom Meeting
Székelyhidi (University of Notre Dame)
Title: Uniqueness of certain cylindrical tangent cones
Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.
Wednesday, March 17, 13:30-14:30, Zoom Meeting
Sébastien Picard (University of British Columbia)
Title: Metrics Through Non-Kahler Transitions
It was proposed by Clemens, Friedman and Reid to connect Calabi-Yau threefolds of different topologies by an operation known as a conifold transition. However, this process may produce a non-Kahler complex manifold with trivial canonical bundle. We will consider conifold transitions from the point of view of differential geometry and discuss passing special metrics through a non-Kahler transition. This is joint work with T.C. Collins and S.-T. Yau.
Wednesday, March 24, 13:30-14:30, Zoom Meeting
Jianchun Chu (Northwestern University)
Title: The k-Ricci curvature in Kahler geometry
In 2018, Lei Ni introduced the definition of k-Ricci curvature, which can be regarded as a natural generalization of holomorphic sectional curvature and Ricci curvature. There are some connections between these curvatures and the properties of the underlying manifold. In this talk, I will show that a compact Kahler manifold with quasi-negative k-Ricci curvature is projective. This is a joint work with Man-Chun Lee and Luen-Fai Tam.
Wednesday, March 31, 13:30-14:30, Zoom Meeting
Marc-Andrew Lavigne (McGill University)
Title: Concentration Compactness Principle for Variable Exponent Spaces
In 1985, Lions published his paper on the concentration compactness principle, which became very useful when proving existence of solutions for PDE with critical growth (with respect to Sobolev embeddings). In the fields of electro-rheological fluids and image processing, the need for variable exponents in PDEs gave rise to many new questions. Several results were obtained for nonlinear elliptic equations when the growth is subcritical. As for the critical case, a concentration compactness principle for variable exponents proves again very useful. In this talk, I will prove this principle by combining the proofs of Fu (2009) and of Bonder and Silva (2010) into a shorter one.
Wednesday, April 7, 13:30-14:30, Zoom Meeting
Valentino Tosatti (McGill University)
Title: Higher order estimates for collapsing complex Monge-Ampère equations
I will consider a family of complex Monge-Ampère equations on a compact Calabi-Yau manifold which has a fibration structure, with fiber size that is shrinking to zero, whose solutions of these equations give Ricci-flat Kähler metrics on the total space. Each equation is uniquely solvable by classical work of Yau, but understanding their asymptotic behavior as the fiber size shrinks is a very challenging problem. I will discuss very recent work with Hans-Joachim Hein where we prove a priori Ck estimates for all k, away from the singular fibers of the fibration. This is a consequence of an aymptotic expansion for the solution, which relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
Wednesday, April 14, 13:30-14:30, Zoom Meeting
Jérôme Vétois (McGill University)
Title: Sign-changing blow-up for the Moser-Trudinger equation
In this talk, we will discuss a question of stability for the energy levels of the Moser-Trudinger functional on a smooth, bounded domain of R^2. This functional involves an exponential non-linearity, which is critical with respect to Sobolev embeddings. We will present an existence result of sign-changing blowing-up solutions, which stands in sharp contrast to the quantization result for positive solutions, recently obtained by Olivier Druet and Pierre-Damien Thizy. This is a joint work with Luca Martinazzi (University of Padua) and Pierre-Damien Thizy (University of Lyon).
Wednesday, May 5, 13:30-14:30, Zoom Meeting, joint with the Analysis seminar
Frédéric Naud (Sorbonne Université)
Title: The spectral gap of random hyperbolic surfaces
We will start with a survey on (some very recent) results on the low spectrum of "random" compact hyperbolic surfaces, for various models including discrete and continuous Teichmuller spaces. We will then give some ideas of the proofs by emphasizing the analogy with radom graphs. We will also dicuss non compact situations where similar results on resonances can be obtained. Joint works with Michael Magee and Doron Puder.
Wednesday, May 19, 13:30-14:30, Zoom Meeting, joint with the Analysis seminar
Laurent Moonens (Université Paris-Saclay)
Title: Solving the divergence equation with measure data in non-regular domains
In this talk, we shall present a recent joint work with E. Russ (Grenoble) concerning the equation $\mathrm{div}\,v=\mu$ in a (rather general) open domain $\Omega$, where $\mu$ is a (signed) Radon measure in $\Omega$ satisfying $\mu(\Omega)=0$. We show in particular that, under mild assumptions on the geometry of $\Omega$ (and some assumptions on $\mu$), one can provide a constructive way to build solutions $v$ in a weighted $L^\infty$ space enjoying weak Neumann-type boundary conditions.

FALL 2020

Wednesday, October 7, 13:30-14:30, Zoom Meeting
Moritz Reintjes (University of Konstanz)
Title: Uhlenbeck compactness and optimal regularity in Lorentzian geometry
We resolve two problems of Mathematical Physics. First, we prove that any L^\infty connection \Gamma on the tangent bundle of an arbitrary differentiable manifold with L^\infty Riemann curvature can be smoothed by coordinate transformation to optimal regularity \Gamma\in W^{1,p} (one derivative smoother than the curvature), any p<\infty. This implies in particular that Lorentzian metrics of shock wave solutions of the Einstein-Euler equations are non-singular-geodesic curves, locally inertial coordinates and the resulting Newtonian limit all exist in a classical sense. This result is based on a system of nonlinear elliptic partial differential equations, the Regularity Transformation equations, and an existence theory for them at the level of L^\infty connections. Secondly, we prove that this existence theory suffices to extend Uhlenbeck compactness from the case of connections on vector bundles over Riemannian manifolds, to the case of connections on the tangent bundle of arbitrary manifolds, including Lorentzian manifolds of General Relativity.
Wednesday, October 21, 13:30-14:30,  Zoom Meeting
Fengrui Yang (McGill University)
Title: Prescribed curvature measure problem in hyperbolic space
The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this talk, we are going to talk about our recent result about prescribed curvature measure problem in hyperbolic space. We obtained the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.
Wednesday, October 28, 13:30-14:30,  Zoom Meeting
Jiawei Liu (McGill University)
Title: The Kähler-Ricci flows with cusp singularity
In this talk, I will talk about the existence, uniqueness and convergence of the Kähler-Ricci flow with cusp singularity on a compact Kähler manifold M which carries a smooth hypersurface D such that the twisted canonical bundle K_M+D is ample. We deduce this flow by limiting a sequence of conical Kähler-Ricci flows as the cone angles tend to zero.
Wednesday, November 11, 13:30-14:30,  Zoom Meeting
Valentino Tosatti (McGill University)
Title: Regularity of envelopes on Kähler manifolds
I will give an introduction to the topic of envelopes of quasi-plurisubharmonic functions (also known as extremal functions) on compact Kähler manifolds. I will discuss the optimal C1,1 regularity of such envelopes, by approximating the envelope by a family of complex Monge-Ampère equations and using an a priori C1,1 estimate developed by Chu, Weinkove and myself. I will also mention some related open questions.
Wednesday, November 18, 13:30-14:30, Zoom Meeting
Bruno Premoselli
(ULB, Brussels)
Title: Towers of bubbles for Yamabe-type equations in dimensions larger than 7
In this talk we consider perturbations of Yamabe-type equations on closed Riemannian manifolds. In dimensions larger than 7 and on locally conformally flat manifolds we construct blowing-up solutions that behave like towers of bubbles concentrating at a critical point of the mass function. Our result does not assume any symmetry on the underlying manifold. We perform our construction by combining finite-dimensional reduction methods with a linear blow-up analysis. Our approach works both in the positive and sign-changing case. As an application we prove the existence, on a generic bounded open set of R^n, of blowing-up solutions of the Brézis-Nirenberg equation that behave like towers of bubbles of alternating signs.
Wednesday, November 25, 13:30-14:30,  Zoom Meeting
Vladmir Sicca (McGill University)
Title: A prescribed scalar and boundary mean curvature problem on compact manifolds with boundary
In this talk I will present our recent result in the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature, on a compact manifold with boundary. We established a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures, which we call the relative Yamabe invariant of the set.  
Wednesday, December 2, 13:30-14:30,  Zoom Meeting
Jiuzhou Huang (McGill University)
Title: Approximation of convex surfaces by Ricci flow
Ricci flow has been an important tool in geometric analysis since Hamilton’s seminal paper in 1982. In this talk, we are going to discuss an approximation of general convex surfaces by Ricci flow and mention some of its possible applications.


Friday, January 17, 13:30-14:30, Burnside Hall 1104, joint with the Analysis seminar
Henrik Matthiesen (University of Chicago)
Title: Handle attachment and the normalised first eigenvalue
I will discuss asymptotic lower bounds of the first eigenvalue for two constructions of attaching degenerating handles to a given closed Riemannian surface. One of these constructions is relatively simple but often fails to strictly increase the first eigenvalue normalized by area. Motivated by this negative result, we then give a much more involved construction that always strictly increases the first eigenvalue normalized by area. As a consequence we obtain the existence of a metric that maximizes the first eigenvalue among all unit area metrics on a given closed surface. This is based on joint work with Anna Siffert.
Wednesday, January 22, 13:30-14:30, Burnside Hall 920
Dmitry Jakobson (McGill University)
Title: Zero and negative eigenvalues of conformally covariant operators, and nodal sets in conformal geometry
We first review some old results about conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, as well as applications to curvature prescription problems. Next, we discuss related results on manifolds with boundary. We relate Dirichlet and Neumann eigenvalues for conformally covariant boundary value problems. If time permits, we shall discuss related results for boundary operators of arbitrary order, as well as for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire.
Wednesday, January 29, 13:30-14:30, Burnside Hall 920
Jérôme Vétois (McGill University)
Title: Blowing-up solutions for critical elliptic equations in low dimensions: the impact of the mass and the scalar curvature
In this talk, we will consider the question of existence of positive blowing-up solutions to a class of elliptic equations with critical Sobolev growth on a closed Riemannian manifold. A result of Olivier Druet provides necessary conditions for the existence of such solutions. We will present new results showing the optimality of Druet's conditions. We will see that the scalar curvature of the manifold plays a crucial role in this question. Furthermore, we will give special attention to the case of dimensions 4 and 5, where a mass term arises and plays an important role in the analysis. This is a joint work with Frédéric Robert (Université de Lorraine).
Wednesday, February 5, 13:30-14:30, Burnside Hall 920
Pengfei Guan (McGill University)
Title: Locally constrained flows, isoperimetric type inequalities, and open problems
We discuss a new type of hypersurface flows with constraints. This type of flows enjoy certain monotonicity properties which make them a natural PDE tool to prove various isoperimetric geometric inequalities. Yet, there are several challenging problems for the longtime existence and regularity of these flows. We will discuss their background and open problems arising from these new flows. The talk is aiming for graduate students and young researchers.
Wednesday, February 19, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McMaster University)
Title: Monge-Ampere equation with bounded periodic data
We consider the Monge-Ampere equation det(D^2u) = f in R^n, where f is a positive bounded periodic function. We prove that u must be the sum of a quadratic polynomial and a periodic function. For f =1, this is the classic result by Jorgens, Calabi and Pogorelov. For f \in C^\alpha, this was proved by Caffarelli and Li. This is a joint work with Y.Y. Li.
Wednesday, February 26, 13:30-14:30, Burnside Hall 920
Niky Kamran (McGill University)
Title: Solving the Einstein equations holographically
We will discuss the problem of solving the Einstein equations with boundary data at infinity close to the conformal infinity of anti-de-Sitter space and present some recent well-posedness results obtained in collaboration with Alberto Enciso (ICMAT). We will also list some open problems. The talk will have a significant introductory component and should be of interest to graduate students and post-docs.
Wednesday, March 11, 13:30-14:30, Burnside Hall 920
Bartosz Syroka (McGill University)
Title: Minkowski formulas and quasi-local mass
The classical Minkowski formulas for curvature were extended to spacetimes in a paper of Wang, Wang, and Zhang, making use of the presence of conformal Killing-Yano tensors. We will look at the use of Minkowski formulas in spacetime to analyze a quasi-local mass definition given by Wang and Yau. We will then consider some applications to rigidity theorems if time allows.
Wednesday, April 29, 13:30-14:30,  Zoom Meeting, joint with the Analysis seminar
Julian Scheuer (University of Freiburg)
Concavity of solutions to elliptic equations on the sphere
An important question in PDE is when a solution to an elliptic equation is concave. This has been of interest with respect to the spectrum of linear equations as well as in nonlinear problems. An old technique going back to works of Korevaar, Kennington and Kawohl is to study a certain two-point function on a Euclidean domain to prove a so-called concavity maximum principle with the help of a first and second derivative test. To our knowledge, so far this technique has never been transferred to other ambient spaces, as the nonlinearity of a general ambient space introduces geometric terms into the classical calculation, which in general do not carry a sign. In this talk we have a look at this situation on the unit sphere. We prove a concavity maximum principle for a broad class of degenerate elliptic equations via a careful analysis of the spherical Jacobi fields and their derivatives. In turn we obtain concavity of solutions to this class of equations. This is joint work with Mat Langford, University of Tennessee Knoxville.
Wednesday, May 6, 13:30-14:30,  Zoom Meeting
Jiawei Liu (McGill University)
Stability of the Conical Kähler-Ricci flows on Fano manifolds
I will talk about the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle 2πβ along the divisor, then for any β' sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle 2πβ' along the divisor. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures.
Wednesday, May 13, 13:30-14:30,  Zoom Meeting
Sébastien Picard (Harvard University) 
Non-Kahler Calabi-Yau manifolds and nonlinear PDE
We will discuss a certain class of manifolds introduced by string theorists C. Hull and A. Strominger. These spaces are non-Kahler Calabi-Yau threefolds. We propose to study this geometry by using the Anomaly flow, which is a nonlinear flow of non-Kahler metrics. This talk will contain joint work with T. Fei, D.H. Phong, and X.-W. Zhang.
Wednesday, May 20, 13:30-14:30, Zoom Meeting
Frédéric Robert (Université de Lorraine) 
Impact of localization of the Hardy potential on the stability of Pohozaev obstructions
The Pohozaev obstruction yields a sufficient condition (say (C)) for the absence of positive solutions to critical nonlinear elliptic equations on domains of the flat space. On the round sphere, this condition is essentially the Kazdan-Warner obstruction. Condition (C) is not stable under reasonable perturbations of the potential. However, for the classical Brezis-Nirenberg problem, the absence of solutions is preserved in small dimension only, see Druet-Laurain. In this talk, I will address the same issue when a Hardy-type potential is added. When the Hardy potential is centered in the interior of the domain, there is also a small dimensions phenomenon. Surprisingly, when the Hardy potential is centered on the boundary of the domain, the obstruction is satble, whatever the dimension.
Wednesday, May 27, 13:30-14:30, Zoom Meeting
Edward Chernysh (McGill University) 
A Compactness Theorem for Weighted Critical p-Laplace Equations
We investigate the compactness of Palais-Smale sequences for a class of critical p-Laplace equations with weights. More precisely, we establish a Struwe-type decomposition result for Palais-Smale sequences extending a recent result of Mercuri-Willem to weighted equations. In sharp contrast to the model case of the critical p-Laplace equation, all bubbling must occur at the origin.
Wednesday, June 3, 13:30-14:30,  Zoom Meeting, joint with the Analysis seminar
Sagun Chanillo (Rutgers University)
Bourgain-Brezis inequalities, applications and Borderline Sobolev inequalities on Riemannian Symmetric spaces of non-compact type.
Bourgain and Brezis discovered a remarkable inequality which is borderline for the Sobolev inequality in Eulcidean spaces. In this talk we obtain these inequalities on nilpotent Lie groups and on Riemannian symmetric spaces of non-compact type. We obtain applications to Navier Stokes eqn in 2D and to Strichartz inequalities for wave and Schrodinger equations and to the Maxwell equations for Electromagnetism. These results were obtained jointly with Jean Van Schaftingen and Po-lam Yung.

FALL 2019

Wednesday, August 7, 13:30-14:30, Burnside Hall 1104
Karoly Boroczky (Central European University, Budapest)
Title: The logarithmic Minkowski problem and some relatives
Some exciting recent developments are surveyed concerning the celebrated logarithmic Brunn-Minkowski conjecture and the related conjectured uniqueness  of the solution of the even logarithmic Minkowski problem.
Friday, August 9, 13:30-14:30, Burnside Hall 1104 , joint with the Analysis seminar
Ali Aleyasin (Stony Brook University)
Title: Singular and degenerate Monge-Ampère equations
It is well known that several non-linear elliptic partial differential equations have applications in various fields of geometry and analysis, including but not limited to the Calabi problem, the Weyl, and Minkowski problems, and optimal transport. An important class of such non-linear equations are the real and complex Monge-Ampère equations. Although the case of strictly elliptic equations with smooth source term has been rather well-understood, the behaviour of solutions in the vicinity of possible singularities or degeneracies of the source term is far from being understood. This corresponds to the vanishing or blowing up of the prescribed curvature in the Weyl problem. In this talk, I will present an application of a differential geometric approach to the study of certain singularities and degeneracies of elliptic complex Monge-Ampère equations. This approach will allow new estimates for solutions to be derived. I shall also outline how the idea works in the case of several other important geometric partial differential equations.
Wednesday, August 21, 13:30-14:30, Burnside Hall 1104
Guofang Wang (University of Freiburg)
Title: Alexandrov-Fenchel type inequalities and applications
In the first part of this talk we review our previous work on Alexandrov-Fenchel inequalities and weighted Alexandrov-Fenchel inequalities in the hyperbolic space and applications on a higher order hyperbolic, which leads to a Penrose type inequality. In the second part, we prove the remaining open case of weighted Alexandrov-Fenchel inequalities and introduce an extrinsic mass. For this mass, we also obtain a Penrose type inequality.
Wednesday, August 21, 14:40-15:40, Burnside Hall 1104
Julian Scheuer (University of Freiburg)
Title: New estimates for the Willmore flow and applications
: The Minkowski inequality for a mean-convex closed surface in three-dimensional Euclidean space provides a lower bound on the total mean curvature in terms of surface area, which is optimal on round spheres. Until today, it is only known to be valid under additional assumptions, for example starshapedness or outward minimality. It remains open, if these additional assumptions can be dropped. In this talk we prove a so-called "almost-Minkowski-inequality" within the class of closed surfaces with L^2-small traceless second fundamental form. No further curvature assumptions are made. This result is based on new asymptotic estimates for the Willmore flow with small energy, which were achieved in a recent joint work with Ernst Kuwert. Further direct applications will be discussed as well.
Wednesday, October 23, 13:30-14:30, Burnside Hall 920
Thierry Daudé (Université de Cergy-Pontoise)
Title: Some non-uniqueness results in the Calderon inverse problem with local or disjoint data
: In dimension 3 or higher, the anisotropic Calderon inverse problem amounts to recovering a Riemannian metric on a compact connected manifold with boundary from the knowledge of the Dirichlet to Neumann operator (modulo diffeomorphisms that fix the boundary). In this talk, I will prove that there is non uniqueness in the Calderon problem when : 1) the Dirichlet and Neumann data are measured on the same proper subset of the boundary provided the metric is only Holder continuous. 2) the Dirichlet and Neumann data are measured on distinct subsets of the boundary (for smooth metrics). This is a joint work with N. Kamran (McGill) and F. Nicoleau (Nantes).
Wednesday, October 30, 13:30-14:30, Burnside Hall 920
Jiuzhou Huang (McGill University)
Title: A normal mapping method for the isoperimetric inequality
: In this talk, I will introduce a proof of the isoperimetric inequality by using the normal mapping. I'll illustrate this method via Brendle's paper "The isoperimetric inequality for a minimal submanifold in Euclidean space".
Wednesday, November 6, 13:30-14:30, Burnside Hall 920
Spiro Karigiannis (University of Waterloo)
Title: Towards higher dimensional Gromov compactness in $G_2$ and $Spin(7)$ manifolds
: Let $(M, \omega)$ be a compact symplectic manifold. If we choose a compatible almost complex structure $J$ (which in general is not integrable) then we can study the space of $J$-holomorphic maps $u : \Sigma \to (M, J)$ from a compact Riemann surface into $M$. By appropriately “compactifying” the space of such maps, one can obtain powerful global symplectic invariants of $M$. At the heart of such a compactification procedure is understanding the ways in which sequences of such maps can degenerate, or develop singularities. Crucial ingredients are conformal invariance and an energy identity, which lead to to a plethora of analytic consequences, including: (i) a mean value inequality, (ii) interior regularity, (iii) a removable singularity theorem, (iv) an energy gap, and (v) compactness modulo bubbling. Riemannian manifolds with closed $G_2$ or $Spin(7)$ structures share many similar properties to such almost Kahler manifolds. In particular, they admit analogues of $J$-holomorphic curves, called associative and Cayley submanifolds, respectively, which are calibrated and hence homologically volume-minimizing. A programme initiated by Donaldson-Thomas and Donaldson-Segal aims to construct similar such “counting invariants” in these cases. In 2011, a somewhat overlooked preprint of Aaron Smith demonstrated that such submanifolds can be exhibited as images of a class of maps $u : \Sigma \to M$ satisfying a conformally invariant first order nonlinear PDE analogous to the Cauchy-Riemann equation, which admits an energy identity involving the integral of higher powers of the pointwise norm $|du|$. I will discuss joint work with Da Rong Cheng (Chicago) and Jesse Madnick (McMaster) in which we establish the analogous analytic results of (i)-(v) in this setting. arXiv:1909.03512
Wednesday, November 13, 13:30-14:30, Burnside Hall 920
François Fillastre (Université de Cergy-Pontoise, UMI-CRM)
Title: Equivariant Minkowski problem in Minkowski space
: We present a class of convex bodies, which are invariant under the action of affine deformations of cocompact lattices of SO(n,1). They appear naturally in general relativity. Classical geometric problems can be brung into that setting, and their intrinsic formulations are on compact hyperbolic manifolds rather than on the round sphere. In dimension (2+1), those convex sets are related to the tangent space of Teichmueller space.
Wednesday, November 20, 13:30-14:30, Burnside Hall 920
Fengrui Yang (McGill University)
Title: W2,1 regularity for solutions of the Monge Ampere equation
: In this talk, I will introduce the thoughts and ideas of how to prove W2,1 regularity for Monge Ampere equation. This comes from a paper written by Guido De Philippis and Alessio Figalli.
Wednesday, November 27, 13:30-14:30, Burnside Hall 920
Vladmir Sicca Goncalves (McGill University)
Title: The Yamabe Invariant of a Measurable Set
: In this talk I'll present the construction of a Yamabe invariant for an arbitrary measurable set in an asymptotically Euclidean manifold. If time allows, I'll also hint how that can help solve the prescribed scalar curvature problem in this setting. The talk is based on the paper "Yamabe Classification and Prescribed Scalar Curvature in the Asymptotically Euclidean Setting" by Dilts and Maxwell (2015).


Thursday, January 24, 14:00-15:00, joint seminar at Concordia University LB 921-4
Almut Burchard (University of Toronto)
Title: A geometric stability result for Riesz-potentials
Riesz' rearrangement inequality implies that integral functionals (such as the Coulomb energy of a charge distribution) that are defined by a pair interaction potential (such as the Newton potential) which decreases with distance are maximized (under appropriate constraints) only by densities that are radially decreasing about some point. I will describe recent and ongoing work with Greg Chambers on the stability of this inequality for the special case of the Riesz-potentials in n dimensions (given by the kernels |x-y|^-(n-s), for densities that are uniform on a set of given volume. For 1<s<n we bound the square of the symmetric difference of a set from a ball by the difference in energy of the corresponding uniform distribution from that of the ball.
Wednesday, January 30, 13:30-14:30, Burnside Hall 1104
Wubin Zhou (Tongji University, China)
Title: The Existence of Constant Scalar Curvature PMY Type Kähler Metrics
Let M be a compact Kähler manifold and N consist some points of M. In this talk, we will discuss the existence of constant scalar curvature PMY type Kähler metrics on non-compact Kähler manifold M-N.
Wednesday, February 6, 13:30-14:30, Burnside Hall 1104
Pengfei Guan (McGill University)
Title: Some open problems in geometric analysis
We will discuss some open problems in geometric analysis, related to isoperimetric type inequalities, isometric embedding problems, rigidity problems, evolution of hypersurfaces, and regularity of nonlinear elliptic and parabolic equations. Main emphasis is the role of nonlinear PDEs in geometric settings.

Wednesday, February 20, 13:30-14:30, Burnside Hall 1104
Edward Chernysh (McGill University)
Title: A global compactness result for p-Laplace equations with critical nonlinearities
In this talk, we will discuss a representation theorem of Mercuri-Willem (2010) for Palais-Smale sequences involving critical p-Laplace equations in smoothly bounded domains with negative parts vanishing at infinity.
Wednesday, February 27, 13:30-14:30, Burnside Hall 1104
Changyu Ren (Jilin University, China)
Title: An inequality for C^2 estimates to k-Hessian equations (Case k=3,n=5)
In this talk, I will introduce an inequality about the C^2 estimates to k-Hessian equations. We can show the inequality is true in case k=3, n=5. For general cases, further discussions are needed.
Wednesday, March 13, 13:30-14:30, Burnside Hall 1104
Jiuzhou Huang (McGill University)
Title: A subsolution method in Monge-Ampere equation
In this talk, I will give a brief introduction to a subsolution method in  Guan-Spruck's paper "Boundary-value Problems on S^n for surfaces of constant Gauss Curvature.
Thursday, March 14, 15:00-16:00, Burnside Hall 1214
Valentino Tosatti (Northwestern University)
Title: Higher order estimates for collapsing Ricci-flat metrics
I will discuss the problem of obtaining uniform C^k estimates for solutions of a degenerating family of complex Monge-Ampere equations where the ellipticity is degenerating along the fibers of a fibration. Geometrically, the solutions of this family of PDE give Ricci-flat Kahler metrics on a Calabi-Yau manifold with a holomorphic fibration onto a lower-dimensional space. I will describe new estimates that prove a uniform C^{2,alpha} bound in general, and C^k bounds for all k when the smooth fibers are all pairwise isomorphic. This is joint work with H.-J. Hein.
Thursday, March 2115:00-16:00, Burnside Hall 1214
Sergio Almaraz (Fluminense Federal University, Brazil)
Title: The mass of asymptotically hyperbolic manifolds with non-compact boundary
We define a mass-type geometric invariant for Riemannian manifolds asymptotic to the hyperbolic half-space and prove a positive mass theorem for spin manifolds. This is a joint work with Levi Lima (UFC-Brazil).
Wednesday, March 27, 13:30-14:30, Burnside Hall 1104
Saikat Mazumdar (McGill University)
Title: Q-curvature, Paneitz operator and a maximum principle
In this talk, I will discuss the higher-order version of the Yamabe problem: "Given a compact Riemannian manifold (M,g), does there exist a metric conformal to g with constant Q-curvature"? The behaviour of Q-curvature under conformal changes of the metric is governed by certain conformally covariant powers of the Laplacian. The problem of prescribing the Q-curvature in a conformal class then amounts to solving a nonlinear elliptic PDE involving the powers of Laplacian called the GJMS operator. In general the explicit form of this GJMS operator is not explicitly known. This together with a lack of maximum principle for polyharmonic operators makes the problem challenging. In this talk, I will mainly focus on the biharmonic case and survey some recent developments.
Wednesday, April 10, 13:30-14:30, Burnside Hall 1104
Liangming Shen (McGill University)
Title: A compactness result along a general continuity path in the study of Kahler-Einstein problem on Fano manifolds
Recent years, Tian and CDS proved the folklore Yau-Tian-Donaldson conjecture based on the study in the continuity path of conical Kahler-Einstein metrics. After that G. Szekelyhidi showed that similar work could be established along Aubin's continuity path. In this talk I will consider a more general continuity path mixed with conic singularities and a torsion term. I will focus on the compactness along the continuity path and show the geometric structure of the limit space. If time permits I will briefly discuss how these results lead to a new proof of Yau-Tian-Donaldson conjecture based on this general continuity path. This is joint with Feng and Ge.
Wednesday, April 17, 13:30-14:30, Burnside Hall 1104
Fengrui Yang  (McGill University)
Title: Isoperimetric inequalities for quermassintegrals of k-star shaped domains
This content comes from a lecture note of Prof. Guan. Here I will briefly introduce the idea of how to solve Isoperimetric inequalities for quermassintegrals of k-star shaped domains.
Wednesday, May 1st, 13:30-14:30, Burnside Hall 1104
Ronan Conlon  (Florida International University)
Title: Classification Results for Expanding and Shrinking gradient Kahler-Ricci solitons
A complete Kahler metric g on a Kahler manifold M is a "gradient Kahler-Ricci soliton" if there exists a smooth real-valued function f:M-->R with \nabla f holomorphic such that Ric(g)-Hess(f)+\lambda g=0 for \lambda a real number. I will present some classification results for such manifolds. This is joint work with Alix Deruelle (Université Paris-Sud) and Song Sun (UC Berkeley).
Friday, May 3, 13:30-14:30, Burnside Hall 1104 , joint with the Analysis seminar
Luca Martinazzi (University of Padua, Italy)
Title: News on the Moser-Trudinger inequality: from sharp estimates to the Leray-Schauder degree
The existence of critical points for the Moser-Trudinger inequality for large energies has been open for a long time. We will first show how a collaboration with G. Mancini allows to recast the Moser-Trudinger inequality and the existence of its extremals (originally due to L. Carleson and A. Chang) under a new light, based on sharp energy estimate. Building upon a recent subtle work of O. Druet and P-D. Thizy, in a work in progress with O. Druet, A. Malchiodi and P-D. Thizy, we use these estimates to compute the Leray-Schauder degree of the Moser-Trudinger equation (via a suitable use of the Poincaré-Hopf theorem), hence proving that for any bounded non-simply connected domain the Moser-Trudinger inequality admits critical points of arbitrarily high energy. In a work in progress with F. De Marchis, O. Druet, A. Malchiodi and P-D. Thizy, we expect to use a variational argument to treat the case of a closed surface.
Wednesday, May 15, 13:30-14:30, Burnside Hall 1104
Bruno Premoselli (ULB, Brussels)
Title: Existence of infinitely many solutions for the Einstein-Lichnerowicz system
We will consider in this talk the Einstein-Lichnerowicz system of equations. It originates in General Relativity as a way to determine initial-data sets for the evolution problem. This system takes the form of a strongly coupled, supercritical, nonlinear system of elliptic PDEs. We will investigate its blow-up properties and show that, under some assumptions on the physics data, it possesses a non-compact family of solutions. This family of solutions will be constructed by combining toplogical methods with a finite-dimensional reduction approach; due to the non-variational structure of the system, the latter has to be carried on in strong spaces and relies of a priori blow-up estimates that we shall describe.

FALL 2018

Wednesday, August 8, 13:30-14:30, Burnside Hall 920
Qun Li (Wright State University)
Title: The Constant Rank Theorems in Complex Geometry
Abstract: In this talk, we will present some constant rank results on the Hessian of solutions to some fully nonlinear equations in complex variables. We will then discuss some geometric applications including the connections to some Hermitian curvature flows.
Wednesday, August 15, 13:30-14:30, Burnside Hall 920
Yiyan Xu (Nanjing University)
Title: Classification of shrinking Ricci solitons
Abstract: We will discuss some classification results for shrinking Ricci solitons. In particular, we will present new proofs of some known results from the equation point of view.
Wednesday, August 22, 13:30-14:30, Burnside Hall 920
Yu Yuan (University of Washington)
Title: Hessian estimates for convex solutions to quadratic Hessian equations
Abstract: We present Hessian estimates for semi-convex solutions to quadratic Hessian equations by compactness arguments. This is based on a new strip argument, a known constant rank theorem, and also a joint work with Chang on Liouville type results for quadratic Hessian equation in general dimensions. This is joint work with McGonagle and Song.
Thursday, September 20, 14:30-15:30, Burnside Hall 1104
Liangming Shen (McGill University)
Title: The Kahler-Ricci flow with log canonical singularities
Abstract: In this talk we will introduce how to construct solutions to the Kahler-Ricci flow with log canonical singularities. First, as Song-Tian did for klt singularites, we will transform the flow equation to a Monge-Ampere type equation with singularites. Then we will establish a potential estimate based on approximations with respect to several parameters. Finally if we have time we will briefly discuss the high order estimate and relations to the minimal model program.  This work is joint with A. Chau, H. Ge and K. Li.

Thursday, September 27, 14:30-15:30, Burnside Hall 1104
Casey Kelleher (Princeton University)
Title: Symplectic curvature flow revisited
Abstract: We continue studying a parabolic flow of almost Kähler structures introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions of canonical quantities, in particular those related to the Chern connection. Using this, we give an extended characterization of fixed points of the flow originally performed by Streets and Tian.
Thursday, October 4, 14:30-15:30, Burnside Hall 1104
Jérôme Vétois (McGill University)
Title: Compactness of sign-changing solutions to critical elliptic equations with bounded negative part
Abstract: In this talk, we will look at the question of compactness of sign-changing solutions to a class of critical elliptic Schrödinger equations on a closed Riemannian manifold. We will present a sharp compactness result for the set of sign-changing solutions with bounded negative part. We obtained this result in dimensions greater than or equal to 7 when the potential function is below the geometric threshold of the conformal Laplacian. The whole set of sign-changing solutions is non-compact in general. We will also discuss constructions of counterexamples in the case of the sphere in dimensions less than or equal to 6 and for potentials above the geometric threshold in higher dimensions. This is a joint work with Bruno Premoselli (ULB, Bruxelles).
Thursday, October 11, 14:30-15:30, Burnside Hall 1104
Fengrui Yang (McGill University)
Title:  (Zhejiang University of Technology)
Title:  Curvature estimates for minimal hypersurfaces
Abstract: In this talk, I will present the main ideas and techniques of paper by Schoen-Simon-Yau "Curvature estimates for minimal hypersurfaces". I will mainly focus on how they obtain a number of new estimates for the curvature of stable minimal hypersurface M which is immersed in a Riemannian manifold N.
Thursday, October 18, 14:30-15:30, Burnside Hall 1104
Chuanqiang Chen (Zhejiang University of Technology)
Title: Smooth solutions to the $L_p$-Dual Minkowski problem
Abstract: In this talk, we consider the $L_p$-dual Minkowski problem. By studying the a priori estimates and curvature flows, we establish the existence theorem of the smooth solutions. This is a joint work with Yong Huang, and Yiming Zhao.
Thursday, October 25, 14:30-15:30, Burnside Hall 1104
Saikat Mazumdar (McGill University)
Title: Compactness results for elliptic equations with critical growth and Hardy weight
Abstract: In this talk we will consider a class of elliptic PDEs with Hardy weight and Sobolev critical growth, which are in general non-compact due to scale invariance. We want to arrive at suitable conditions which would ensure the compactness and this in turn will help establish the existence of solutions to these equations. We will start by describing the blow-up behaviour of a sequence of approximating solutions approaching our PDE and obtain optimal control on such a sequence. Next we will look at the interaction of the various terms in the Pohozaev identity and calculate the blow-up rates. The compactness theorems will follow from this. We will see that the location of the singularity, be it in the interior of the domain or on its boundary, affects the analytical properties of the equation and makes the two situations quite different. When the singularity is in the interior, then a lower order perturbation suffices for high dimensions, while the curvature of the boundary plays a crucial role if the singularity is on the boundary for high dimensions. This is a joint work with Nassif Ghoussoub (UBC) and Frédéric Robert (Université de Lorraine).

Thursday, November 1, 14:40-15:40, Burnside Hall 1104
Wubin Zhou (Tongji University)
Title: The Futaki Invariant for Poincaré-Mok-Yau type Kähler metrics
Abstract: In this talk, we recall the definition of PMY(Poincaré-Mok-Yau) type Kähler metric and then define its Futaki invariant which is an obstruction to the existence of constant scalar curvature Kähler metrics. Also we will give some explicit PMY type metrics on some noncompact Kähler manifolds with non-vanishing Futaki invariant. This work is joint with Jixiang Fu.
Thursday, November 8, 14:40-15:40, Burnside Hall 1104
Shaodong Wang (McGill University)
Title: A compactness theorem for boundary Yamabe problem in the scalar-flat case
Abstract: In this talk, I will present some recent results on the compactness of the solutions to the Yamabe problem on manifolds with boundary. The compactness of Yamabe problem was introduced by Schoen in 1988. There have been a lot of works on this topic later on. This is a joint work with Sergio Almaraz and Olivaine Queiroz.
Thursday, November 15, 14:40-15:40, Burnside Hall 1104
Jiuzhou Huang (McGill University)
Title: Two different skills in double normal derivative estimates for Monge-Ampere equations
Abstract: In this talk, I will illustrate two different skills in double normal derivative estimates for Monge-Ampere equation. The materials are Caffarelli-Nirenberg-Spruck's classical paper: “The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampere Equations” and Trudinger's lecture “Lecture on Nonlinear Elliptic Equations of Second Order”. I will focus on illustrating the basic ideas behind their estimates.
Thursday, November 22, 14:40-15:40, Burnside Hall 1104
Vladmir Sicca (McGill University)
Title: The Lichnerowicz Equation for Einstein's General Relativity
Abstract: A solution to Lichnerowicz equation gives a Riemannian manifold that can be an initial condition for Einstein's equation in General Relativity. In this talk I will introduce how Lichnerowicz equation fits in the context of General Relativity and present an existence result for the equation proved by Hebey, Pacard and Pollack. The core of the result relies on the mountain pass lemma, which will also be presented in the talk.
Thursday, December 6, 14:40-15:40, Burnside Hall 1104
Peter Hintz (MIT)
Title: Stability of Minkowski space and polyhomogeneity of the metric
Abstract: I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a compactification of R^4 to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity. I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. Joint work with András Vasy.


Monday, January 8, 16:00-17:00, Burnside Hall 719A
Tristan Collins
(Harvard University)
Title: Sasaki-Einstein metrics and K-stability
Abstract: I will discuss the connection between Sasaki-Einstein metrics and algebraic geometry in the guise of K-stability. In particular, I will give a differential geometric perspective on K-stability which arises from the Sasakian view point, and use K-stability to find infinitely many non-isometric Sasaki-Einstein metrics on the 5-sphere. This is joint work with G. Szekelyhidi.
Monday, January 15, 16:00-17:00, Burnside Hall 719
Brent Pym
(University of Edinburgh)
Title: Geometry and quantization of Poisson Fano manifolds.
Abstract:  Complex Poisson manifolds and the noncommutative algebras that "quantize" them appear in many parts of mathematics, but their structure and classification remain quite mysterious, especially in the positively curved case of Fano manifolds. I will survey recent breakthroughs on several foundational conjectures in this area, which were formulated by Artin, Bondal, Kontsevich and others in the 80s and 90s. For instance, we will see that the curvature of a Poisson manifold has a strong effect on the singularities of its associated foliation, and that the remarkable transcendental numbers known as multiple zeta values arise naturally as universal constants in the corresponding quantum algebras.
Thursday, January 18, 16:00-17:00, Burnside Hall 1205
Philip Engel
(Harvard University)
Title: Cusp Singularities
Abstract:  In 1884, Klein initiated the study of rational double points (RDPs), a special class of surface singularities which are in bijection with the simply-laced Dynkin diagrams. Over the course of the 20th century, du Val, Artin, Tyurina, Brieskorn, and others intensively studied their properties, in particular determining their adjacencies---the other singularities to which an RDP deforms. The answer: One RDP deforms to another if and only if the Dynkin diagram of the latter embeds into the Dynkin diagram of the former. The next stage of complexity is the class of elliptic surface singularities. Their deformation theory, initially studied by Laufer in 1973, was largely determined by the mid 1980's by work of Pinkham, Wahl, Looijenga, Friedman and others. The exception was a conjecture of Looijenga's regarding smoothability of cusp singularities---surface singularities whose resolution is a cycle of rational curves. I will describe a proof of Looijenga's conjecture which connects the problem to symplectic geometry via mirror symmetry, and summarize some recent work with Friedman determining adjacencies of a cusp singularity.
Monday, January 22, 16:00-17:00, Burnside Hall 1205
Alex Waldron
(Stony Brook University)
Title: Yang-Mills flow in dimension four
Abstract: Among the classical geometric evolution equations, YM flow is the least nonlinear and best behaved. Nevertheless, curvature concentration is a subtle problem when the base manifold has dimension four. I'll discuss my proof that finite-time singularities do not occur, and briefly describe the infinite-time picture.
Thursday, January 25, 16:00-17:00, Burnside Hall 1205
Kiumars Kaveh
(University of Pittsburgh)
Title: Convex bodies in algebraic geometry and symplectic geometry
Abstract:  We start by discussing some basic facts about asymptotic behavior of semigroups of lattice points (which is combinatorial in nature). We will see how this allows one to assign convex bodies to projective algebraic varieties encoding important "intersection theoretic" data. Applying inequalities from convex geometry to these bodies (e.g. Brunn-Minkowski) one immediately obtains Hodge inequalities from algebraic geometry. This is in the heart of theory of Newton-Okounkov bodies. It generalizes the extremely fruitful correspondence between toric varieties and convex polytopes, to arbitrary varieties. We then discuss connection with symplectic (and Kahler) geometry and in particular regarding these bodies as images of moment maps for Hamiltonian torus actions. For "spherical varieties" (or "multiplicity-free spaces") these constructions become very concrete and they bring together algebraic geometry, symplectic geometry and representation theory. For the most part the talk is accessible to anybody with just a basic knowledge of algebra and geometry.
Wednesday, January 31, 13:30-14:30, Burnside Hall 920
Bruno Premoselli
(ULB, Brussels)
Title: Examples of Compact Einstein four-manifolds with negative curvature
Abstract:  We construct new examples of closed, negatively curvedEinstein four-manifolds. More precisely, we construct Einstein metrics of negative sectional curvature on ramified covers of compact hyperbolic four-manifolds with symmetries, initially considered by Gromov and Thurston. These metrics are obtained through a deformation procedure. Our candidate approximate Einstein metric is an interpolation between a black-hole Riemannian Einstein metric near the branch locus and the pulled-back hyperbolic metric. We then deform it into a genuine solution of Einstein's equations, and the deformation relies on an involved bootstrap procedure. Our construction yields the first example of compact Einstein manifolds with negative sectional curvature which are not locally homogeneous. This is a joint work with J. Fine (ULB, Brussels).
Wednesday, February 7, 13:30-14:30, Burnside Hall 920
Gantumur Tsotgerel
Title: Some scaling estimates in Besov and Triebel-Lizorkin spaces
  This is continuation of my previous talk, where we presented elliptic estimates for operators with rough coefficients. The whole theory depended on certain scaling properties of functions. Here we will discuss ways to establish those properties.
Wednesday, February 14, 13:30-14:30, Burnside Hall 920
Luca Capogna
(Worcester Polytechnic Institute)
Title: A Liouville type theorem in sub-Riemannian geometry, and applications to several complex variables
Abstract:  The Riemann mapping theorem tells us that any simply connected planar domain is conformally equivalent to the disk. This provides a classification of simply connected domains via conformal maps. This classification fails in higher dimensional complex spaces, as already Poincare' had proved that bi-discs are not bi-holomorphic to the ball. Since then, mathematicians have been looking for criteria that would allow to tell whether two domains are bi-holomorphic equivalent. In the early 70's, after a celebrated result by Moser and Chern, the question was reduced to showing that any bi-holomorphism between smooth, strictly pseudo-convex domains extends smoothly to the boundary. This was established by Fefferman, in a 1974 landmark paper. Since then, Fefferman's result has been extended and simplified in a number of ways. About 10 years, ago Michael Cowling conjectured that one could prove the smoothness of the extension by using minimal regularity hypothesis, through an argument resting on ideas from the study of quasiconformal maps. In its simplest form, the proposed proof is articulated in two steps: (1) prove that any bi-holomorphism between smooth, strictly pseudoconvex domains extends to a homeomorphisms between the boundaries that is 1-quasiconformal with respect to the sub-Riemannian metric associated to the Levi form; (2) prove a Liouville type theorem, i.e. any $1-$quasiconformal homeomorphism between such boundaries is a smooth diffeomorphism. In this talk I will discuss recent work with Le Donne, where we prove the first step of this program, as well as joint work with Citti, Le Donne and Ottazzi, where we settle the second step, thus concluding the proof of Cowling's conjecture. The proofs draw from several fields of mathematics, including nonlinear partial differential equations, and analysis in metric spaces.
Friday, February 16, 13:30-14:30, Burnside Hall 920
Loredana Lanzani
Title: Harmonic Analysis techniques in Several Complex Variables
Abstract: This talk concerns the application of relatively classical tools from real harmonic analysis (namely, the T(1)-theorem for spaces of homogenous type) to the novel context of several complex variables. Specifically, I will present recent joint work with E. M. Stein (Princeton U.) on the extension to higher dimension of Calderon's and Coifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset C$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel: $H(w, z)=\frac{1}{2\pi i}\frac{dw}{w-z}$ is that it is holomorphic (that is, analytic) as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogue of H(w, z). This is because of geometric obstructions (the Levi problem), which in dimension 1 are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Leray in the context of a $C^\infty$-smooth, convex domain D: while these conditions on D can be relaxed a bit, if the domain is less than C^2-smooth (never mind Lipschitz!) Leray's construction becomes conceptually problematic. In this talk I will present (a), the construction of the Cauchy-Leray kernel and (b), the L^p(bD)-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called ``T(1)-theorem technique'' from real harmonic analysis. Time permitting, I will describe applications of this work to complex function theory - specifically, to the Szego and Bergman projections (that is, the orthogonal projections of L^2 onto, respectively, the Hardy and Bergman spaces of holomorphic functions). References:
[C] Calderon A. P, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. 74 no. 4, (1977) 1324-1327.
[CMM] Coifman R., McIntosh A. and Meyer Y., L'integrale de Cauchy definit un operateur borne sur L^2 pour les courbes Lipschitziennes, Ann. of Math. 116 (1982) no. 2, 361-387.
[L] Lanzani, L. Harmonic Analysis Techniques in Several Complex Variables, Bruno Pini Mathematical Analysis Seminar 2014, 83-110, Univ. Bologna Alma Mater Studiorum, Bologna.
[LS-1] Lanzani L. and Stein E. M., The Szego projection for domains in C^n with minimal smoothness, Duke Math. J. 166 no. 1 (2017), 125-176.
[LS-2] Lanzani L. and Stein E. M., The Cauchy Integral in C^n for domains with minimal smoothness, Adv. Math. 264 (2014) 776-830.
[LS-3] Lanzani L. and Stein E. M., The Cauchy-Leray Integral: counter-examples to the L^p-theory, Indiana Math. J., to appear.
Wednesday, February 21, 13:30-14:30, Burnside Hall 920
Benoît Pausader
Title: Stability of Minkowski space for the Einstein equation with a massive scalar field
Abstract: This is joint work with A. Ionescu. We consider the stability of the Minkowski space for the Einstein model equations with a matter model given by a massive scalar field. This problem was already studied under more stringent conditions by LeFloch-Ma and Q. Wang. After apropriate parametrization, this is a quasilinear problem involving a wave equation and a Klein-Gordon equation for which one proves a small data-global existence result. Part of the complication comes from the fact that, as per the constraint equation, the ``initial data'' has a rather poor behavior at infinity, and that we do not specify a priori the main term in the fall-off decay (e.g. the data is not necessarily Schwartschild outside a bounded ball).
Wednesday, February 28, 13:30-14:30, Burnside Hall 920
Jessica Lin
Title: Regularity Estimates for the Stochastic Homogenization of Elliptic Nondivergence Form Equations
Abstract: I will present some regularity estimates related to the stochastic homogenization for nondivergence form equations. In a joint work with Scott Armstrong, we show that in the stochastic homogenization for linear uniformly elliptic equations in random media, solutions actually exhibit improved regularity properties in light of the homogenization process. In particular, we show that with extremely high probability, solutions of the random equation have almost the same regularity as solutions of the deterministic homogenized equation. The argument is similar to the proof of the classical Schauder estimates, however it utilizes the random structure of the problem to obtain improvement. 
Wednesday, March 21, 13:30-14:30, Burnside Hall 920
Pengfei Guan
Title: An inverse curvature type hypersurface in space form
Abstract: We introduce a new type of hypersurface flow for bounded starshaped domains in space form. An interesting property of this type of flow is monotonicity for corresponding quermassintegrals. The focus is the long time existence and convergence of the flow. We will discuss some recent progress and open problems arising from the regularity estimates
Wednesday, March 28, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill)
Title: Interior estimates for Special Lagrangian equations and Scalar curvature equations
Abstract: I will reveal some connections between special Lagrangian equations and Scalar curvature equation. Then I will discuss how to get interior curvature estimates for Scalar curvature equation in dimension three. 
Wednesday, April 4, 13:30-14:30, Burnside Hall 920
Fengrui Yang (McGill)
Title: The Dirichlet problem for Hessian equation
Abstract: In this talk, I will present the main ideas and techniques of the classical paper, "The Dirichlet Problem for nonlinear Second-Order elliptic Equations, Functions of the eigenvalues of the Hessian". I will mainly focus on the estimate of double normal second derivatives. 
Wednesday, April 25, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill)
Title: Regularity Estimates for the Penalized Parabolic Boundary Obstacle Problem
Abstract: We will discuss regularity estimates for the solution to the time dependent penalized boundary obstacle problem. We will obtain using geometric arguments Lipschitz estimates in time and Holder Regularity in space independent of the permeability constant of interest in the context of semipermeable membrane theory.
Wednesday, May 2, 13:30-14:30, Burnside Hall 920
Jérôme Vétois (McGill)
Title: Examples of singular solutions to an elliptic equation with a sign-changing non-linearity
Abstract: We will examine the possible behaviors of singular solutions to an elliptic equation with a sign-changing non-linearity in a punctured ball. I will present new existence results of radial solutions with prescribed behavior. This is a joint work with Florica Cîrstea and Frédéric Robert.
Thursday, May 31, 10:00-11:00, Burnside Hall 920
Siyuan Lu (Rutgers) 
Title: Exterior Dirichlet problem for Monge-Ampere equation
Abstract: We consider exterior Dirichlet problem for Monge-Ampere equation with prescribed asymptotic behavior. Based on earlier work by Caffarelli and Li, we complete the characterization of existence and nonexistence of solutions in terms of their asymptotic behaviors. This is a joint work with Y.Y. Li.
Wednesday, June 13, 13:30-14:30, Burnside Hall 920
Laurent Moonens (Paris-Sud)
Title: Differentiation along rectangles
Abstract: Lebesgue’s differentiation theorem states that, when $f$ is a locally integrable function in Euclidean space, its average on the ball $B(x,r)$ centered at $x$ with radius $r$, converges to $f(x)$ for almost every $x$, when $r$ approaches zero. Many questions arise when the family of balls $\{B(x,r)\}$ is replaced by a \emph{differentiation basis} $\mathcal{B}=\bigcup_x \mathcal{B}_x$ (where, for each $x$, $\mathcal{B}_x$ is, roughly speaking, a collection of sets shrinking to the point $x$). In this case, one looks for conditions on $\mathcal{B}$ such that the average of $f$ on sets belonging to $\mathcal{B}_x$ are known to converge to $f(x)$ for a.e. $x$, when those sets shrink to the point $x$. Many interesting phenomena happen when sets in $\mathcal{B}$ have a \emph{rectangular} shape (Lebesgue’s theorem may or may not hold in this case, depending on the geometrical properties of sets in $\mathcal{B}$). In this talk, we shall discuss some of the history around this problem, as well as recent results obtained with E. D’Aniello and J. Rosenblatt in the planar case, when the rectangles in $\mathcal{B}$ are only allowed to lie along a fixed sequence of directions.

FALL 2017

Wednesday, August 2, 13:30-14:30, Burnside Hall 920
Joshua Ching (Sydney)
Title: Singular solutions to nonlinear elliptic equations with gradient dependency
Abstract: Let $N \geq 2$ be the dimension. Let $\Omega \subseteq \mathbb{R}^N$ be a domain containing the origin. We consider non-negative $C^1(\Omega \setminus \{ 0\})$ solutions to the following elliptic equation: ${\rm div} (|x|^{\sigma} |\nabla u|^{p-2}\nabla u)=|x|^{-\tau} u^q |\nabla u|^m$ in $\Omega \setminus \{ 0 \}$, where we impose appropriate conditions on the parameters $m,p,q,\sigma,\tau,N$. We study these solutions from several perspectives including existence, uniqueness, radial symmetry, regularity and asymptotic behaviour. In the model case where $p=2$ and $\sigma=\tau=0$, we impose the conditions $q>0$, $m+q>1$ and $0<m<2$. Here, we provide a sharp classification result of the asymptotic behaviour of these solutions near the origin and infinity. We also provide corresponding existence results in which we emphasise the more difficult case of $m \in (0,1)$ where new phenomena arise. A key step in these proofs is to obtain gradient estimates. Using a technique of Bernstein's and some other ideas, we find a new gradient estimate that is independent of the domain and is applicable in a more general setting than the model case. Via these gradient estimates, we will show a Liouville-type result that extends a theorem of Farina and Serrin (2011). Time permitting, we will also look at further applications of this gradient estimate. In this talk, we present results from Ching and Cîrstea (2015, Analysis & PDE), results from my PhD thesis as well as ongoing research.
Wednesday, August 2, 14:45-15:45, Burnside Hall 920
Laurent Moonens (Paris-Sud)
Title: Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields
Abstract: In this talk, we shall discuss a characterization, obtained with T.H. Picon, of all the distributions $F \in \calD’(\Omega)$ for which one can locally solve by a \emph{continuous} vector field $v$ the divergence-type equation  $$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F$$ where  $\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on $\Omega \subset \R^{N}$. In case where $(L_1,\dots, L_n)$ is the usual gradient field on $\R^N$, we recover a classical result for the divergence equation, obtained previously by T. De Pauw and W.F. Pfeffer.
Wednesday, August 9, 13:30-14:30, Burnside Hall 920
Florica Cîrstea (Sydney)
Title: Nonlinear elliptic equations with isolated singularities
Abstract: In this talk, I will review recent developments on isolated singularities for various classes of nonlinear elliptic equations, which could include Hardy-Sobolev type potentials. In particular, we shall look at fully classifying the behaviour of all positive solutions in different contexts that underline the interaction of the elliptic operator and the nonlinear part of the equation. We also provide sharp results on the existence of solutions with singularities, besides optimal conditions for the removability of all singularities. I will discuss results obtained with various collaborators including T.-Y. Chang (University of Sydney) and F. Robert (University of Lorraine).
Wednesday, August 16, 13:30-14:30, Burnside Hall 920
Chao Xia (Xiamen University)
Title: Uniqueness of stable capillary hypersurfaces in a ball
Capillary hypersurfaces in a  ball $B$ is minimal or CMC hypersurfaces whose boundary intersects $\partial B$ at a constant angle. They are critical points of  some energy functional under volume preserving variation. The study of stability of capillary hypersurfaces in $B$ was initiated by Ros-Vergasta and Ros-Souam in 90's. An open problem is whether any immersed stable capillary hypersurfaces in a  ball in space forms are totally umbilical. In this talk, we will give a complete affirmative answer. We remark that the related uniqueness result for closed hypersurfaces is due to Barbosa-Do Carmo-Eschenburg. The talk is based on a joint work with Guofang Wang.
Wednesday, August 23, 13:30-14:30, Burnside Hall 920
Xinan Ma (University of Science and Technology of China)
Title: The Neumann problem of special Lagrangian equations with supercritical phase
Abstract: In this talk, we establish the global $C^2$ estimates of the Neumann problem of special Lagrangian equations with supercritical phase and the existence theorem by the method of continuity, we also mention the complex version. This is the joint work with Chen chuanqiang and Wei wei.
Wednesday, September 20, 13:30-14:30, Burnside Hall 920
Pengfei Guan (McGill)
Title: Gauss curvature flows and Minkowski type problems
Abstract: We discuss a class of isotropic flows by power of Gauss curvature of convex hypersurfaces. For each flow, there is an entropy associated to it, and it is monotone decreasing. For this entropy, there is an unique entropy point. The flow preserves the enclosed volume. The main question is to control the entropy point. This was done for standard flows in joint works with Lei Ni, and Ben Andrews and Lei Ni. For isotropic flows, under appropriate assumptions, one prove that the entropy point will keep as origin. From there, one may deduce regularity and convergence. The self-similar solutions are the solutions to corresponding Minkowski type problem. Similar results were also obtained by Bryan-Ivaki-Scheuer via inverse type flows.
Wednesday, September 27, 13:30-14:30, Burnside Hall 920
Gantumur Tsogtgerel (McGill)
Title: Elliptic estimates for operators with rough coefficients
We will discuss a possible approach to establish elliptic estimates for operators with barely continuous coefficients in a Sobolev-Besov and Triebel-Lizorkin scale. The result would obviously be not new but the proposed approach is relatively elementary and therefore of interest. Please be warned that this is an ongoing project, and the talk is going to be more of a discussion than a polished lecture.
Wednesday, October 4, 13:30-14:30, Burnside Hall 920
Jerome Vetois (McGill)
Title: Blowing-up solutions for critical elliptic equations on a closed manifold
 In this talk, we will look at the question of existence of blowing-up solutions for smooth perturbations of positive scalar curvature-type equations on a closed manifold. From a result of Druet, we know that in dimensions different from 3 and 6, a necessary condition for the existence of blowing-up solutions is that the limit equation agrees with the Yamabe equation at least at one blow-up point. I will present new existence results in situations where the limit equation is different from the Yamabe equation away from the blow-up point. I will also discuss the special role played by the dimension 6. This is a joint work with Frederic Robert.
Wednesday, October 11, 13:30-14:30, Burnside Hall 920
Shaya Shakerian (University of British Columbia)
Title: Borderline Variational problems for fractional Hardy-Schrödinger operators
 In this talk, we investigate the existence of ground state solutions associated to the fractional Hardy-Schrödinger operator on Euclidean space and its bounded domains. In the process, we extend several results known about the classical Laplacian to the non-local operators described by its fractional powers. Our analysis show that the most important parameter in the problems we consider is the intensity of the corresponding Hardy potential. The maximal threshold for such an intensity is the best constant in the fractional Hardy inequality, which is computable in terms of the dimension and the fractional exponent of the Laplacian. However, the analysis of corresponding non-linear equations in borderline Sobolev-critical regimes give rise to another threshold for the allowable intensity. Solutions exist for all positive linear perturbations of the equation, if the intensity is below this new threshold. However, once the intensity is beyond it, we had to introduce a notion of Hardy-Schrödinger Mass associated to the domain under study and the linear perturbation. We then show that ground state solutions exist when such a mass is positive. We then study the effect of non-linear perturbations, where we show that the existence of ground state solutions for large intensities, is determined by a subtle combination of the mass (i.e., the geometry of the domain) and the size of the nonlinearity of the perturbations. 
Wednesday, October 18, 13:30-14:30, Burnside Hall 920
Shaodong Wang (McGill)
Title: A compactness theorem for Yamabe problem on manifolds with boundary
 In this talk, I will present a compactness result on Yamabe problems on manifolds with boundary. This is from a paper by Zheng-chao Han and Yanyan Li. 
Wednesday, October 25, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill)
Title: One-dimensional convex integration
 Convex integration theory was introduced by M.Gromov in his thesis dissertation in 1969 which is a powerful tool for solving differential relations. An important application of the convex integration theory is that it can recover the Nash-Kuiper result on C^{1} isometric embeddings. I will briefly mention the history of h-principle and rigidly theorem for isometric embedding. Then some details about one-dimensional convex integration.
Friday, November 10, 13:30-14:30, Burnside Hall 920
Daniel Pollack (University of Washington)
Title: On the geometry and topology of initial data sets with horizons
One of the central and most fascinating objects which arise in general relativity are black holes. From a mathematical point of view this is closely related to questions of "singularities" and "horizons" which arise in the study of the Einstein equations. We will present a number of results which relate the presence (or absence) of horizons to the topology and geometry of the "exterior region" of an initial data set for Einstein equations. Time permitting we will also connect these results with previous work of Galloway and Schoen on the topology of the black holes themselves. This is joint work with Lars Andersson, Mattias Dahl, Michael Eichmair and Greg Galloway.
Wednesday, November 15, 13:30-14:30, Burnside Hall 920
Fengrui Yang
Title: The Dirichlet Problem for Monge-Ampere equation.
Abstract: In this talk, I will present the main ideas and techniques of the classical paper, ' The Dirichlet Problem for nonlinear Second-Order Elliptic Equations I, Monge-Ampere Equation'. Firstly, I will give a brief introduction of the history of estimating third-order derivatives of Monge-Ampere equation, and then focus on the proof of boundary C2 and C2,a estimates.
Wednesday, November 22, 13:30-14:30, Burnside Hall 920
Saikat Mazumdar
(University of British Columbia)
Title: Blow-up analysis for a critical elliptic equation with vanishing singularity
 In this talk, we will examine the asymptotic behavior of a sequence of ground state solutions of the Hardy-Sobolev equations as the singularity vanishes in the limit.  If this sequence is uniformly bounded in L-infinity,  we obtain a minimizing solution of the stationary Schrödinger equation with critical growth. In case the sequence blows up, we obtain C0 control on the blow up sequence, and we localize the point of singularity and derive precise blow up rates. 
Wednesday, November 29, 13:30-14:30, Burnside Hall 920
Edward Chernysh
Title: Weakly Monotone Decreasing Solutions to an Elliptic Schrödinger System
 In this talk we study positive super-solutions to an elliptic Schrödinger system in R^n for n\geq3. We give conditions guaranteeing the non-existence of positive solutions and introduce weakly monotone decreasing functions. We establish lower-bounds on the decay rates of positive solutions and obtain upper-bounds when these are weakly monotone decreasing. 
Wednesday, December 6, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill)
 The Two-Penalty Boundary Obstacle Problem
 Inspired by a problem of fluid flow through a semi-permeable membrane we study optimal regularity estimates for solutions as well as some structural properties of the free boundary for a two-penalty boundary obstacle problem. This is ongoing work with Thomas Backing and Donatella Danielli.


Wednesday, January 25, 13:30-14:30, Burnside Hall 920
Gantumur Tsogtgerel (McGill)
Title: A prescribed scalar-mean curvature problem
Abstract: In this talk, we will be concerned with a problem of prescribing scalar curvature and boundary mean curvature of a compact manifold with boundary. This is an ongoing work motivated by the study of the Einstein constraint equations on compact manifolds with boundary, and builds on the results of Rauzy and of Dilts-Maxwell.

Wednesday, February 1st, 13:30-14:30, Burnside Hall 920
Mohammad Najafi Ivaki (Concordia)
Title: Harnack estimates for curvature flows
AbstractI will discuss Harnack estimates for curvature flows in the Riemannian and Lorentzian manifolds of constant curvature and that "duality" allows us to obtain a certain type of inequalities, "pseudo"-Harnack inequalities.

Wednesday, February 8, 13:30-14:30, Burnside Hall 920
Jerome Vetois (McGill)
Title: Decay estimates and symmetry of solutions to elliptic systems in R^n
AbstractIn this talk, we will look at a class of coupled nonlinear Schrödinger equations in R^n. I will discuss a notion of finite energy solutions for these systems and I will present some recent qualitative results on these solutions.
Wednesday, February 22, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill)
Title: Rigidity of closed self-similar solution to the Gauss curvature flow
AbstractIn the seminar, I will present Choi and Daskalopoulos's recent [arXiv:1609.05487v1] rigidity result about Gauss curvature flow. They proved that a convex closed solution to the Gauss curvature flow in R^n becomes a round sphere after rescaling.
Wednesday, March 8, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McGill)
Title: Minimal hypersurface and boundary behavior of compact manifolds with nonnegative scalar curvature
AbstractIn the study of boundary behavior of compact Riemannian manifolds with nonnegative scalar curvature, a fundamental result of Shi-Tam states that, if a compact manifold has nonnegative scalar curvature and its boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary of the manifold is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In this talk, we give a supplement to Shi-Tam's result by considering manifolds whose boundary includes the outermost minimal hypersurface of the manifold. Precisely speaking, given a compact manifold \Omega with nonnegative scalar curvature, suppose its boundary consists of two parts, \Sigma_h and \Sigma_o, where \Sigma_h is the union of all closed minimal hypersurfaces in \Omega and \Sigma_o is isometric to a suitable 2-convex hypersurface \Sigma in a Schwarzschild manifold of positive mass m, we establish an inequality relating m, the area of \Sigma_h, and two weighted total mean curvatures of \Sigma_o and $ \Sigma, respectively. This is a joint work with Pengzi Miao from Miami.
Wednesday, March 16, 2:00pm-3:3:00pm, Burnside 1234
Yuanwei Qi (University of Central Florida)
Title: Traveling Wave of Gray-Scott model: Existence, Multiplicity and Stability.
Abstract: In this talk, I shall present some recent works I have done with my collaborators in rigorously proofing the existence of traveling wave solution to the Gray-Scott model, which is one of the most important models in Turing type of pattern formation after the experiments in early 1990s to validate his theory. We shall also discuss some interesting features of traveling wave solutions. This is a joint work with Xinfu Chen.
Wednesday, March 22, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill)
Title: Regularity estimates for Semi-permeable membrane Flow
Abstract: We study a boundary value problem modeling flow through the semi-permeable boundary $\Gamma$ with finite thickness $\lambda$ and an applied fluid pressure $\phi(x)$. We study optimal regularity estimates for the solution as well as asymptotic estimates as $\lambda \to 0$.
Wednesday, March 29, 13:30-14:30, Burnside Hall 920
Kyeongsu Choi (Columbia)
Title: Free boundary problems in the Gauss curvature flow
Abstract: We will discuss the optimal C^{1,1/(n-1)} regularity of the Gauss curvature flow with flat sides, and the C^{\infty} regularity of the flat sides. Moreover, we will study connections between the free boundary problems, the classification to the self-shrinkers, and the prescribed curvature measure equations.
Wednesday, April 5, 13:30-14:30, Burnside Hall 920
Shaodong Wang (McGill)
Title: Infinitely many solutions for cubic Schrödinger equation in dimension 4
Abstract: In this talk, I will present some recent results in the existence of blow-up solutions to a cubic Schrödinger equation on the standard sphere in dimension four. This is a joint work with Jerome Vetois.
Friday, April 7, 13:30-14:30, Burnside Hall 920
Xinliang An (University of Toronto)
Title: On Gravitational Collapse in General Relativity
Abstract: In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, I will present four results with regard to gravitational collapse for Einstein vacuum equation. The first is a simplified approach to Christodoulou’s monumental result which showed that trapped surfaces can form dynamically by the focusing of gravitational waves from past null infinity. We extend the methods of Klainerman-Rodnianski, who gave a simplified proof of this result in a finite region. The second result extends the theorem of Christodoulou by allowing for weaker initial data but still guaranteeing that a trapped surface forms in the causal domain. In particular, we show that a trapped surface can form dynamically from initial data which is merely large in a scale-invariant way. The second result is obtained jointly with Jonathan Luk. The third result answered the following questions: Can a ``black hole’’ emerge from a point? Can we find the boundary (apparent horizon) of a ``black hole’’ region? The fourth result extends Christodoulou’s famous example on formation of naked singularity for Einstein-scalar field system under spherical symmetry. With numerical and analytic tools, we generalize Christodoulou’s result and construct an example of naked singularity formation for Einstein vacuum equation in higher dimension. The fourth result is obtained jointly with Xuefeng Zhang.
Wednesday, April 19, 13:30-14:30, Burnside Hall 920
Ben Weinkove (Northwestern)
Title: The Monge-Ampere equation, almost complex manifolds and geodesics
Abstract: I will discuss an existence theorem for the Monge-Ampere equation in the setting of almost complex manifolds. I will describe how techniques for studying this equation can be used to prove a regularity result for geodesics in the space of Kahler metrics. This is joint work with Jianchun Chu and Valentino Tosatti.
Wednesday, April 26, 13:30-14:30, Burnside Hall 920
Chen-Yun Lin (University of Toronto)
Title: An embedding theorem: differential analysis behind massive data analysis
Abstract: High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

FALL 2016

Wednesday, September 21, 13:30-14:30, Burnside Hall 920
Pengfei Guan (McGill)
Title: A volume preserving flow and the isoperimetric problem in warped product spaces with general base
Abstract: A flow was introduced in a previous work to handle the isoperimetric problem in sapce forms. We propose to study a similar normalized hypersurface flow in the more general ambient setting of warped product spaces with general base. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains. This is a join work with Junfang Li and Mu-Tao Wang.
Wednesday, September 28, 13:30-14:30, Burnside Hall 920
Dylan Cant (McGill)
Title: A Curvature flow and application to an isoperimetric inequality
Abstract: Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under weak assumption on the warp product of surface). This curvature flow preserves the area enclosed by the curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumption on the warp potential.
Wednesday, October 5, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill)
Title: Geometric Methods in Obstacle-Type Free Boundary Problems I
Abstract: Obstacle-type free boundary problems naturally appear as mathematical models in science and engineering with some particular motivations arising from contact problems in elasticity, options pricing in financial mathematics, and phenomenological models in superconductor physics. The first talk will focus on geometric methods that have been used to study regularity estimates in Obstacle-Type Free Boundary Problems. The regularity theory for obstacle-type problems (and other type of free boundary problems as well) was much inspired by the regularity theory for minimal surfaces. We will discuss the basic existence, uniqueness and regularity questions in the classical obstacle problem. We will point out generalizations and current problems of interest in this field of research. In the second talk we will focus on an obstacle-type problem arising in stochastic impulse control theory that appeared first as a model for cash management and portfolio optimization under transaction costs. Here the underlying theory for the obstacle problem has to be suitably modified to consider obstacle problems with an implicit and nonlocal obstacle. Regularity estimates will be presented and natural directions for future research discussed.
Wednesday, October 12, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill)
Title: Geometric Methods in Obstacle-Type Free Boundary Problems II
Abstract: We will continue studying Geometric Methods in Obstacle-Type Free Boundary Problems. In the second talk we will focus on an obstacle-type problem arising in stochastic impulse control theory that appeared first as a model for cash management and portfolio optimization under transaction costs. Here the underlying theory for the obstacle problem has to be suitably modified to consider obstacle problems with an implicit and nonlocal obstacle. Regularity estimates for the solution and the free boundary will be presented.
Wednesday, October 19, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill)
Title: Hessian estimate for the Sigma-2 Equation in dimension Three (After Michah Warren and Yu Yuan)
Abstract: Heinz derived a Hessian bound for the two dimensional Monge-Ampere equation by using Uniformization Theorem. Sigma-2=1 in three dimension can be viewed as a equation of a special lagranian graph in C^3. Which is also a three dimensional minimal surface in R^6. Michah Warren and Yu Yuan used this observation and Michael-Simon's sobolev inequalities on generalized submanifolds of R^n to prove a priori interior Hessian estimates for Sigma_2 =1 in three dimension. We will go through their proof in this seminar.
Wednesday, November 2, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McGill)
Title: Isoperimetric inequality in warped product manifold.
Abstract: We consider isoperimetric inequality in warped product manifold. We discuss two results by Montiel and Bray-Morgan. The paper by Montiel shows that under natural assumption of the warped function, a star shaped constant mean curvature hypersurface must be a coordinate slice. The paper by Bray-Morgan shows that under stronger assumption of the warped function, isoperimetric domain must be a coordinate slice.
Thursday, November 10, 14:30-15:30, Burnside Hall 920
Tatiana Toro (University of Washington)
Title: Almost minimizers with free boundary
Abstract: In recent work with G. David, and ongoing work with G. David and M. Engelstein, we study almost minimizer for functionals which yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem which explicitly takes noise into account. In this talk we will discuss regularity results for these almost minimizers and as well as the structure of the corresponding free boundary. A key ingredient in the study of the 2-phase problem is the existence of almost monotone quantities.
Wednesday, November 16, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McGill)
Title: Isoperimetric inequality in warped product manifold II.
Abstract: We will continue to discuss the isoperimetric inequality in warped product manifold. We'll focus on Bray-Morgan's result using comparison to obtain the isoperimetric inequality without the assumption of starshapedness.
Wednesday, December 14, 13:30-14:30, Burnside Hall 920
Pengzi Miao (University of Miami)
Title: Boundary effect of scalar curvature
Abstract: Manifolds with nonnegative scalar curvature arise naturally as maximal slices of physical spacetimes in general relativity. When the manifold is noncompact, there are the Riemannian positive mass theorem and Penrose inequality which give global results on how scalar curvature affects the manifold geometry near infinity. When the manifold is compact, it models bounded domains in such spacetime slices and how the scalar curvature affects its boundary geometry is tied to the quasi-local mass problem. In this talk, I will survey known results on boundary behavior of compact manifolds with nonnegative scalar curvature, and if time permits, I will discuss related open questions.

Previous Talks