Seminars are usually held on Wednesdays, 13:30-14:30, in Burnside Hall Room 920

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[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.

Jessica Lin

Pengfei Guan

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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

Abstract: I 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

Abstract: In 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

Abstract: In 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

Abstract: In 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 Christodoulouâ€™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 holeâ€™â€™ emerge from a point? Can we find the boundary (apparent horizon) of a ``black holeâ€™â€™ region? The fourth result extends Christodoulouâ€™s famous example on formation of naked singularity for Einstein-scalar field system under spherical symmetry. With numerical and analytic tools, we generalize Christodoulouâ€™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.

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.