Geometric Analysis Seminar

Organizers: Pengfei Guan and Jerome Vetois.
Seminars are usually held on Wednesdays, 13:30-14:30, in Burnside Hall Room 920


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
Wednesday, March 28, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill)
Wednesday, April 4, 13:30-14:30, Burnside Hall 920
Fengrui Yang (McGill)
Wednesday, April 11, 13:30-14:30, Burnside Hall 920
Vladimir Sicca (McGill)
Wednesday, April 18, 13:30-14:30, Burnside Hall 920
Shaodong Wang (McGill)
Wednesday, April 25, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill)
Wednesday, June 13, 13:30-14:30, Burnside Hall 920
Laurent Moonens (Paris-Sud)

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.

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