Past seminars

Tittle: On uniqueness of excited states and related questions

Abstract: This talk will present the long-standing problem of excited states uniqueness for the nonlinear Schroedinger equation. We will describe the history of the problem, it's relevance to long-term dynamics of nonlinear wave equations, related spectral problems, and progress on the uniqueness question via rigorous numerics. The recent breakthrough by Moxun Tang, who found an analytical proof, will be discussed. 

Title:

Fundamental solution to the heat equation with a dynamical boundary condition

Abstract:

We give an explicit representation of the fundamental solution to the heat equation on a half-space with the homogeneous dynamical boundary condition, and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space with the homogeneous dynamical boundary condition. This talk is based on a joint work with Kazuhiro Ishige (University of Tokyo) and Sho Katayama (University of Tokyo).

Title: A Counterexample to the Mizohata-Takeuchi Conjecture

Abstract: We derive a family of Lp estimates of the X-Ray transform of positive measures in Rd, which we use to construct a logR-loss counterexample to the Mizohata-Takeuchi conjecture for every C2 hypersurface in Rd that does not lie in a hyperplane. In particular, multilinear restriction estimates at the endpoint cannot be sharpened directly by the Mizohata-Takeuchi conjecture

Title: Regularity theory for fractional elliptic equations in nondivergence form 

Abstract: We present recent advances in the regularity theory for solutions to fractional nonlocal equations driven by fractional powers of nondivergence form elliptic operators in bounded domains, where sharp regularity on the coefficients is assumed. The results include Harnack inequality, H\”older regularity of solutions and Schauder estimates. These are proved using the characterization of solutions by a degenerate/singular extension problem. The sharp form of the extension problem has been recently obtained in joint work with Animesh Biswas (Missouri State University). This is joint work with Mary Vaughan (Texas State University). 

Title: Critical sets of harmonic functions and Almgren's frequency function

Abstract: Given a harmonic function on a nice domain in n-dimensional Euclidean space, its critical set (the set of points where its gradient vanishes) is known to have dimension at most n-2, with locally finite (n-2)-dimensional Hausdorff measure. For harmonic polynomials, the Hausdorff measure can be bounded in terms of the degree of the polynomial. For more general harmonic functions, Almgren's frequency function serves as an analog of the degree of a polynomial, measuring local growth properties of the harmonic function.

Lin conjectured that for a harmonic function with frequency at most N, the (n-2)-dimensional Hausdorff measure of the critical set is (locally) at most CN^2. The first (and only) quantitative bound in this direction was due to Naber and Valtorta, who showed a bound of the form exp(CN^2). In this talk, I'll discuss an improvement of this bound to the nearly polynomial threshold N^(C log N) via a multiscale analysis, some linear algebra, and geometric combinatorics. This is joint work with Josep Gallegos and Eugenia Malinnikova.

Title: A degenerate one-phase free boundary problem arising from the Alt-Phillips equation for negative exponents 

Abstract: We study viscosity solutions for a degenerate one-phase free boundary problem of the form $\Delta w = \frac{h(\nabla w)}{w}$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This type of degenerate one-phase free boundary problem arises when a canonical transformation is performed to a semilinear equation $\Delta u = f(u)$, and $f$ morally behaves like $u^{-(\gamma + 1)}$ for some $\gamma > 0$. In this case, known as the Alt-Phillips equation for negative exponents, $h(\nabla u) = c(|\nabla u|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ degenerates to $2$, the free boundary becomes a minimal surface.  

Title: Rigidity of critical points of hydrophobic capillary functionals

Abstract: We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between 90◦ and 120◦. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the “tangential” part of the capillary boundary is H^n-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between 90◦ and 180◦. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds. This is joint work with R. Neumayer and R. Resende.

Title: Ricci Solitons and Hamiltonian Dynamics

Abstract: In this talk, we will discuss a specific connection between a Kähler Gradient Ricci soliton (GRS) and Hamiltonian dynamics. The former is a singularity model of the Kahler version of Ricci flows and a generalization of Einstein metrics while the latter comes from classical mechanics. In particular, we show that a Kähler GRS with a non-trivial potential function in complex dimension two is an integrable Hamiltonian system. This novel perspective directly exploits the non-triviality of the potential function, a distinctive feature of non-trivial GRS. Thus, it leads to strong rigidity which might not hold for the trivial case of Einstein metrics. For instance, we discuss a conjecture stating that a Kahler GRS in complex dimension two has toric symmetry and a positive confirmation under a generic condition. 

Title: Bounds for spectral projectors on the three-dimensional torus

Abstract: Periodic functions can be expanded in Fourier series, a linear combination of exponentials indexed by the integers. Despite the explicitness of the involved expressions, it is a difficult problem to take information about the frequency support of a function and deduce information on the physical side. We study the L^p norms of spectral projection operators on the torus, with special attention given to the three dimensional case, and prove sharp estimates for various values of p and sizes of the window. Our methods include harmonic analysis via decoupling, the geometry of numbers, exponential sum estimates, and bilinear techniques. This work is joint with Pierre Germain and Simon Myerson.

Title: On singular points in the supercooled Stefan problem

Abstract: The supercooled Stefan problem models how water, cooled below freezing, forms into ice. Experiments have shown that this process can exhibit nucleation (pieces of ice forming spontaneously) and that fractal-like patterns can emerge in the ice. This stands in sharp contrast with the melting problem, in which the ice is expected to be smooth at most points and times. This was recently demonstrated rigorously by Figalli, Ros Oton and Serra. 

In joint work with Inwon Kim (UCLA) and Sebastian Munoz (UCLA), we analyze a subclass of solutions to the supercooled problem which are connected to a parabolic obstacle-type problem. For this subclass we show that some of the experimentally observed behavior can occur (such as nucleation) but that the singular set cannot be completely pathological (i.e. we give bounds on the size of the singular points). 

Title: Control of eigenfunctions on negatively curved manifolds

Abstract: Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. They have a long history of study going back to the Quantum Ergodicity theorem and the Quantum Unique Ergodicity conjecture. I will speak about the work with Jin and Nonnenmacher, proving that on a negatively curved surface, every semiclassical measure has full support. I will also discuss applications of this work to control for the Schrödinger equation and decay for the damped wave equation.

Our theorem was restricted to dimension 2 because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and myself), was only known for subsets of the real line. I will talk about more recent joint work with Athreya and Miller in the setting of complex hyperbolic quotients and the work by Kim and Miller in the setting of real hyperbolic quotients of any dimension. In these works there are potential obstructions to the full support property which can be classified by Ratner theory and geometrically described in terms of certain totally geodesic submanifolds. Time permitting, I will also mention a recent counterexample to Quantum Unique Ergodicity for higher-dimensional quantum cat maps, due to Kim and building on the previous counterexample of Faure-Nonnenmacher-De Bièvre.

Title: Weighted Fourier Extension Estimates

Abstract: In this talk, we will survey recent results on weighted Fourier extension estimates and its variants. Such estimates ask for L^2 bounds of the Fourier extension operator on $\alpha$-dimensional sets, and they have applications to several problems in PDEs and geometric measure theory, including size of divergence set of Schrodinger solutions, spherical average Fourier decay rates of fractal measures, and Falconer’s distance set problem.

Title: Positive scalar curvature and exotic structures on simply connected four manifolds.

Abstract: We address Gromov’s band width inequality and Rosenberg’s S1-stability conjecture for smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless, we show that both of these results hold upon considering simply connected smooth four manifolds up to homeomorphism. We also obtain a weaker related result in the general case. Based on joint work with B. Sen.

Title: Arnold-Thom conjecture for the arrival time of surfaces

Abstract: Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. In this talk, I will explain the proof of Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions of mean convex mean curvature flows of surfaces. This is joint work with Tang-Kai Lee.

Title: Some recent developments on the fully nonlinear Yamabe problems 

Abstract: In recent joint work with YanYan Li and Zongyuan Li, we broaden the scope of fully nonlinear Yamabe problems by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor. Our results accommodate conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities of the associated equations. In addition to above results, I will briefly describe our developments on the fully nonlinear Yamabe problems on manifolds with boundary, discussing both boundary mean curvature and boundary curvatures arising from the Chern–Gauss–Bonnet formula.

Title: Finite time explosion for SPDEs

Abstract: The classical existence and uniqueness theorems for SPDEs prove that, under appropriate assumptions, SPDEs with globally Lipschitz continuous forcing terms have unique global solutions. This talk outlines recent results about SPDEs exposed to superlinearly growing deterministic and stochastic forcing terms. I describe sufficient conditions that guarantee that, despite the superlinear growth, the SPDEs have unique global solutions.

Title: Minimal Surfaces in Negative Curvature

Abstract: It follows from work by Kahn-Markovic that every closed negatively curved 3-manifold contains essential minimal surfaces in great abundance.  Since then the goal of better understanding these minimal surfaces has been a focus of activity, both in analogy to the geodesic flow one dimension lower and the more positive-curvature-centric min-max theory of minimal surfaces.   This talk will survey recent developments in this area, which brings together techniques from dynamical systems, geometric analysis, and hyperbolic geometry.

Title: Restriction and local smoothing estimates using decoupling and two-ends inequalities.

Abstract: I will discuss restriction estimates and local smoothing estimates derived from decoupling theorems and two-ends inequalities, with a focus on the differences between these two problems. Specifically, I will introduce the wave packet density, which is the key to the induction on scales argument for the local smoothing problem. Also, I will show how an additional careful analysis of L^2 orthogonality leads to the sharp local smoothing estimates for wave equations on compact Riemannian manifolds.

Title: Triple junction solution for the Allen-Cahn system

Abstract: In this talk, I will discuss recent results on the vector-valued Allen-Cahn system with a triple-well potential. We establish the existence of an entire minimizing solution that asymptotically converges to a triple junction, corresponding to a planar minimal cone, at infinity.  Furthermore, we prove the uniqueness of the blow-down limit at infinity by deriving precise estimates for the location and size of the diffuse interface. We also show that the solution is almost invariant along the diffuse interface and, at infinity, closely resembles one dimensional heteroclinic connections between two energy wells. Our proof is primarily variational and does not assume any symmetry of the solution. These results are based on joint works with Nicholas Alikakos.

Title: Generic Regularity of Minimal Submanifolds with Isolated Singularities

Abstract: Singularities commonly arise in geometric variational problems such as minimal submanifolds, where they are locally modeled by minimal cones. Although a wide variety of singularity models have been constructed in the literature, it is conjectured that under generic setting, the they are substantially simpler.

In this talk, we will present a characterization of regular minimal cones that can appear as singularity models for minimal submanifolds in a generic Riemannian manifold, without any restrictions on the dimension or codimension. As an application, we will discuss a generic finiteness result for low-area minimal hypersurfaces in nearly round 4-spheres. This is based on joint work with Alessandro Carlotto and Yangyang Li.

Title: Moment growth and intermittency for SPDEs in the sublinear-growth regime

Abstract: In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming some concavity of the diffusion coefficient, we establish non-trivial moment upper bounds for the solution. These moment bounds shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. The method we employ is highly robust and can be extended to a wide range of stochastic partial differential equations, including the one-dimensional stochastic wave equation. This talk is based on a joint work with Panqiu Xia from the University of Cardiff.

Title: Strichartz estimates for the Schrödinger equation on the sphere.

 Abstract: We will discuss Strichartz estimates for solutions of the Schrödinger equation on the standard round sphere, which is related to the results of Burq, Gérard and Tzvetkov (2004). The proof is based on the arithmetic properties of the spectrum of the Laplacian on the sphere along with some local bilinear estimate in harmonic analysis. We shall also discuss some related results on the flat tori and on manifolds with negative curvature. This is based on ongoing joint work with Christopher Sogge.

Title: The geometry and topology of lower bounds on scalar curvature

Abstract: In this talk, I will discuss some recent results about controlling the geometry and topology of manifolds from a (positive) lower bound on their scalar curvature. 

Title: On isolated singularities and generic regularity of min-max CMC hypersurfaces

Abstract: Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away. This talk will be accessible to graduate students with a light background in differential geometry and PDE. 

Tittle: Partial regularity for Lipschitz solutions to the minimal surface system 

Abstract: The minimal surface system is the Euler-Lagrange system for the area functional of a high codimension graph and reduces to the minimal surface equation in the case that the codimension is one. Though the regularity theory for the minimal surface equation is well understood, much less is known in the systems case. This is largely due to the examples provided by Lawson and Osserman in 1977 and the lack of a maximum principle for the system. In this talk, we discuss recent progress on the regularity theory for Lipschitz stationary solutions and Lipschitz viscosity solutions to the minimal surface system with an emphasis on partial regularity results. 

Title: The Lawson-Osserman conjecture for the minimal surface system

Abstract: In their seminal work on the minimal surface system, Lawson and Osserman conjectured that Lipschitz graphs that are critical points of the area functional with respect to outer variations are also critical with respect to domain variations. We will discuss the proof of this conjecture for two-dimensional graphs of arbitrary codimension. This is joint work with J. Hirsch and R. Tione. 

Title: The Nevo-Thangavelu spherical maximal function on two step nilpotent Lie groups.

Consider R^d \times R^m with the group structure of a 2-step Carnot  Lie group and natural parabolic dilations.  The maximal operator   originally introduced by Nevo and Thangavelu  in the setting of the Heisenberg groups is generated by (noncommutative)  convolution  associated with  measures on  spheres or generalized spheres in R^d.  We discuss a number of approaches that have been taken to prove L^p boundedness and then talk about  recent work with Jaehyeon Ryu in which we drop  the nondegeneracy condition in the known results on Métivier groups. The new results have the sharp L^p boundedness range for all  two step Carnot  groups with d\geq 3. 

Title: Optimal observability times for wave and Schrodinger equations on really simple domains

Abstract: Observability for evolution equations asks: if I take a partial measurement in a system, can it “see” some physical quantity, such as energy?  In a series of papers with E. Stafford, Z. Lu, and S. Carpenter, we showed that energy for the wave or Schrodinger equation can be observed from any one side of a simplex in any dimension.  For this talk we will consider the wave and Schrodinger equations on intervals and triangles and ask how long does it take to observe the energy.  The computations use only elementary integrations and multivariable calculus.  This is work in progress with Claire Isham.

Title: Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

Abstract: In 1989, Brian White conjectured that every Riemannian 3-sphere contains at least five embedded minimal tori. The number five is optimal, corresponding to the Lyusternik-Schnirelmann category of the space of Clifford tori. I will present recent joint work with Adrian Chu, where we confirm this conjecture for 3-spheres of positive Ricci curvature. Our proof is based on min-max theory, with heuristics largely inspired by mean curvature flow.

Title: Restriction estimates for spectral projections

Abstract: Restriction estimates for spectral projections have been widely studied since the work of Burq, Gérard, and Tzvetkov as a method for investigating eigenfunction concentration. The problem of establishing the optimal $L^p$ bounds for the restriction of Laplace-Beltrami eigenfunctions remains open, particularly when the restriction submanifold is of codimension 1 or 2. This talk will explore how these optimal bounds are characterized by the geometry of the underlying manifold $M$ and its submanifold $H$, focusing on the case where $M$ is two-dimensional and $H$ is a smooth curve in $M$.

Title: Nonnegative Ricci curvature and linear volume growth revisited 

Abstract: Manifolds with nonnegative Ricci curvature and linear (minimal) volume growth were extensively studied in the beginning of this century. The recent advance of RCD theory and the study of isoperimetric problem require new understandings of the limit spaces at infinity in the above setup. In this talk, we will survey through classical results and discuss the isoperimetric problem in this setting. We will reveal the interactions between structure at infinity, properties of Busemann functions and the existence of isoperimetric sets. 

Title: Recent Progress on Fractional GJMS Operators

Abstract: The Fractional GJMS operators are one-parameter family of conformally invariant operators, defined by the renormalized scattering operators on the conformal infinity of Poincare-Einstein Manifolds. These operators provide a bridge to transfer the information of interior Einstein geometry to the boundary conformal geometry. In this talk, I will first introduce too types of comparison theorems for them: integral type and pointwise type. Then I will give some applications of these comparison theorems, including the estimates for interior Gromov-Bishop volume ratio and the positive mass theorem. 

Title: L^p bounds for spectral projectors on hyperbolic surfaces

Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces, focusing on the case where the spectral window has small width. I will show that the negative curvature assumption leads to improvements over the universal bounds of C. Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. The proof relies on new Strichartz and smoothing estimates for the Schrödinger semi-group on locally symmetric spaces, thus illustrating how dispersive PDE techniques can lead to new results in classical harmonic analysis. 

This is based on joint work with Jean-Philippe Anker and Pierre Germain.

Title: Oscillatory integrals on manifolds and related Kakeya and Nikodym problems. 

Abstract: This talk is about oscillatory integrals on manifolds and their connections to  Kakeya and Nikodym problems on manifolds. 

There are two types of manifolds that are particularly interesting: manifolds of constant sectional curvature and manifolds satisfying Sogge's chaotic curvature conditions. I will discuss these two types of manifolds and related results. 

Title: Maximal Subellipticity

Abstract: The theory of elliptic PDE stands apart from many other areas of PDE because sharp results are known for very general linear and fully nonlinear elliptic PDE.  Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDE leans heavily on the Fourier transform and Riemannian geometry.

Starting with work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced:  now known as maximal subellipticity or maximal hypoellipticity.  In the intervening years, many authors have adapted results from elliptic PDE to various special cases of maximally subelliptic PDE.

Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry.  The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis.

In this talk, we present the sharp regularity theory of general linear and fully nonlinear maximally subelliptic PDEs.

Title: On the isoperimetric profile of the hypercube 

Abstract: The isoperimetric problem in the hypercube is a strikingly simple topic which is not yet completely understood. We will review the known facts on the problem, focusing on how the gaussian isoperimetric inequality provides a lower bound for the isoperimetric profile of the hypercube.

We will exploit this observation to prove that, if the volume is close to 1/2, then the isoperimetric set within the cube is a half-space (intersected with the cube). Furthermore, we will show that the lower bound provided by the gaussian isoperimetric inequality is not sharp when the dimension of the hypercube goes to infinity (contrasting with the case of the sphere).

Title: Oscillatory integrals on manifolds and related Kakeya and Nikodym problems. 

Abstract: This talk is about oscillatory integrals on manifolds and their connections to  Kakeya and Nikodym problems on manifolds. 

There are two types of manifolds that are particularly interesting: manifolds of constant sectional curvature and manifolds satisfying Sogge's chaotic curvature conditions. I will discuss these two types of manifolds and related results. 

Title: Three theorems about asymptotic expansions of harmonic functions at the boundary

Abstract: Consider a harmonic function on a domain, vanishing along a part of its boundary, and a point on that part of the boundary which is asymptotically conical. I will explain that under a very mild notion of "asymptotically conical," the harmonic function has a unique (but maybe infinite) order of vanishing. Under a stronger notion, the harmonic function has a unique blow-up limit and a specific rate of convergence to it, giving an asymptotic expansion. Finally, I will show how to build examples of convex, C^1 domains in 3D where a harmonic function has non-unique blow-up limits. This is based on joint work with Zongyuan Li. 

Title: Boundary behavior of the Allen-Cahn equation and the construction of free boundary minimal hypersurfaces

Abstract: In the late 70s, the work of Modica, Mortola, De Giorgi, and many others established deep connections between the Allen-Cahn equation, a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions, and minimal hypersurfaces, i.e. critical points of the area functional. Based on these ideas, in recent years, the combined work of Guaraco, Hutchinson, Tonegawa, and Wickramasekera established the existence of (optimally regular) minimal hypersurfaces in compact manifolds without boundary. In this talk we will consider the Allen-Cahn equation on manifolds with boundary, and describe geometric and analytic aspects of the boundary behavior of the associated limit interfaces. The end goal of this line of investigation is the construction of free boundary minimal hypersurfaces in manifolds with boundary, i.e. submanifolds with vanishing mean curvature and meeting the boundary orthogonally. I will present progress towards this goal, based on joint work with Martin Li and Lorenzo Sarnataro.  

Title: Soap films, Plateau's laws, and the Allen-Cahn equation 

Abstract: Plateau's problem of minimizing area among surfaces with a common boundary is the basic model for soap films and leads to the theory of minimal surfaces. In this talk we will discuss a modification of Plateau's problem in which surfaces are replaced with regions of small but positive volume. We will also discuss the PDE approximation of this problem via an Allen-Cahn free boundary problem and its relation to Plateau's laws, which govern singularities in soap films.   

Title: First eigenvalue estimates on asymptotically hyperbolic manifolds and their submanifolds.  

Abstract: I will report on joint work with Samuel Pérez-Ayala. We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within $\mathbb{H}^{n+1}(-1)$, $\lambda_{1,p}(Y)=\left(\frac{k}{p}\right)^{p}$. We then obtain lower bounds on the first eigenvalue of these submanifolds in the case where minimality is replaced with a weaker mean curvature assumption and where the ambient space is a general Poincar\'e-Einstein space whose boundary is of non-negative Yamabe type. In the process, we introduce an invariant $\hat \beta^Y$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung. 

Title: Recovery of time-dependent coefficients in hyperbolic equations on Riemannian manifolds from partial data.

Abstract: In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent coefficients of the wave equation uniquely in a space-time cylinder. We shall discuss two problems: (1) Recovery of a potential appearing in the wave equation, when the Dirichlet and Neumann values are measured on opposite parts of the lateral boundary of the space-time cylinder. (2) Recovery of both a damping coefficient and a potential appearing in the wave equation, when the Dirichlet values are measured on the whole lateral boundary and the Neumann data is collected on roughly half of the boundary. This talk is based on joint works with Teemu Saksala (NC State University) and Lili Yan (University of Minnesota). 

Title: A family of Kahler flying wing steady Ricci solitons

Abstract: Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension n>2. This is joint work with Pak-Yeung Chan and Yi Lai.

Title: Stability of perturbed wave equations on Kerr black hole spacetimes

Abstract: I will discuss a recent work with Gustav Holzegel, in which we prove integrated decay bounds for solutions of the geometric wave equation with small linear perturbations on Kerr black hole spacetimes.

Our proof adapts the framework introduced by Dafermos, Rodnianski, and Shlapentokh-Rothman for the homogeneous wave equation on Kerr spacetimes. When adding the perturbative term one must also compensate for obstructions caused by the necessary degeneration of Morawetz-type estimates in these spacetimes, which is due to the presence of trapped null geodesics.

Mathematically, the key mechanism to our approach is the construction of a pseudodifferential commutator W, such that for the commuted equation one may obtain a nondegenerate Morawetz-type estimate.

Title:  Maximal Subellipticity

Abstract:  The theory of elliptic PDE stands apart from many other areas of PDE because sharp results are known for very general linear and fully nonlinear elliptic PDE.  Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDE leans heavily on the Fourier transform and Riemannian geometry.

Starting with work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced:  now known as maximal subellipticity or maximal hypoellipticity.  In the intervening years, many authors have adapted results from elliptic PDE to various special cases of maximally subelliptic PDE.

Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry.  The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis.

In this talk, we present the sharp regularity theory of general linear and fully nonlinear maximally subelliptic PDEs.

Title: A Generalization of the Geroch Conjecture with Arbitrary Ends

The Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles. In particular, I will talk about why the connected sum of a Schoen-Yau-Schick n-manifold with an arbitrary n-manifold does not admit a complete metric of positive scalar curvature for n <=7; this generalizes work of Chodosh and Li. I will also discuss about how to generalize Brendle-Hirsch-Johne’s non-existence result for metrics of positive m-intermediate curvature on N^n = M^{n-m} x T^m to to manifolds with arbitrary ends for n <= 7 and certain m. Here, m-intermediate curvature is a new notion of curvature interpolating between Ricci and scalar curvature.

Title: On the existence of isoperimetric sets on nonnegatively curved spaces 

Abstract: In this presentation, I will deal with the isoperimetric properties of spaces with nonnegative curvature, placing special emphasis on the existence of isoperimetric sets in the non-compact case. 

Isoperimetric sets of any volume do exist in compact Riemannian manifolds. Conversely, it is possible to construct simple examples of non-compact Riemannian manifolds without isoperimetric sets. However, when one imposes a lower bound on the Ricci curvature and a uniform lower bound on the volumes of unit balls, a "generalized existence" theorem can be proven. Specifically, for every volume, a "generalized isoperimetric set" exists in the disjoint union of the space and, possibly, a finite number of its limits at infinity. 

Following a review of recent literature concerning the existence of isoperimetric sets in spaces with nonnegative (sectional/Ricci/scalar) curvature, I will leverage the aforementioned generalized existence result to establish new existence theorems. In particular, I will demonstrate that isoperimetric sets with large volumes always exist in manifolds with nonnegative sectional curvature and nondegenerate asymptotic cones. Finally, I will construct a 3-dimensional manifold with positive sectional curvature and nondegenerate asymptotic cone, where isoperimetric sets do not exist for small volumes. 

The results of this presentation will be presented within the framework of smooth Riemannian manifolds. However, the techniques employed in the proofs are adaptable to non-smooth settings (such as Alexandrov or RCD spaces), and therefore, similar results hold in the non-smooth realm. 

This research is based on collaborative work with Bruè, Fogagnolo, Glaudo, Nardulli, and Pozzetta.

Title: On mean-field super-Brownian motions

Abstract: The mean-field stochastic partial differential equation

(SPDE) corresponding to a mean-field super-Brownian motion (sBm) is

obtained and studied. In this mean-field sBm, the branching-particle

lifetime is allowed to depend upon the probability distribution of the

sBm itself, producing an SPDE whose space-time white noise coefficient

has, in addition to the typical sBm square root, an extra factor that

is a function of the probability law of the density of the mean-field

sBm. This novel mean-field SPDE is thus motivated by population models

where things like overcrowding and isolation can affect growth. A two

step approximation method is employed to show the existence for this

SPDE under general conditions. Then, mild moment conditions are imposed

to get uniqueness.

This talk is based on a joint work with Yaozhong Hu, Michael A.

Kouritzin at the University of Alberta in Canada and Jiayu Zheng at

Shenzhen MSU-BIT University in China.

Title: Genus one singularities in mean curvature flow

Abstract: We show that for certain one-parameter families of initial conditions in ℝ³, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. Among applications, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut, and we proved that the fourth lowest entropy self-shrinker in ℝ³ can not be rotationally symmetric. Based on joint work with Adrian Chu.

Title: Carleson ε^2 conjecture in higher dimensions

Abstract: I will talk about a joint work with Xavier Tolsa and Michele Villa on a higher dimensional analogue of the Carleson ε^2 conjecture. In this work, we characterise tangent points of certain domains in Euclidean space via a novel "spherical" square function which measures whether the common boundary of the open sets is close to a plane in the sphere. Beyond its intrinsic geometric appeal, this result is motivated by connections to Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula.

Title: Frobenius-type results on submanifolds and currents

Abstract: The question of producing a foliation of the n-dimensional Euclidean space with k-dimensional submanifolds which are tangent to a prescribed k-dimensional simple vectorfield is part of the celebrated Frobenius theorem: a decomposition in smooth submanifolds tangent to a given vectorfield is feasible (and then the vectorfield itself is said to be integrable) if and only if the vectorfield is involutive. In this seminar I will summarize the results obtained in collaboration with G. Alberti, A. Merlo and E. Stepanov when the smooth submanifolds are replaced by weaker objects, such as (integral or normal) currents or even contact sets with "some" boundary regularity. I will also provide Lusin-type counterexamples to the Frobenius property for rectifiable currents. Finally, I will try to highlight the connection between involutivity/integrability à la Frobenius and Carnot-Carathéodory spaces and how to apply our techniques in this framework.

Title: Singularities in minimal submanifolds

Abstract: In the last few years there have been significant developments in techniques used to understand singularities within minimal submanifolds. I will discuss this circle of ideas and explain how they enable us to reconnect the study of geometric singularities with more classical PDE techniques, such as those used in unique continuation.

Title: Liouville theorems for conformally invariant fully nonlinear elliptic equations

Abstract:  A fundamental theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be constant. A remarkable Liouville type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of $-\Delta u =u^{ (n+2)/(n-2) }, n\ge 3$, are unique modulo Mobius transformations. Far-reaching extensions were established for general fully nonlinear conformally invariant equations through the works of Chang-Gursky-Yang, Li-Li, Li, and Viaclovsky. In this paper, we derive necessary and sufficient conditions for the validity of such Liouville-type theorems. This leads to necessary and sufficient conditions for local gradient estimates of solutions to hold, assuming a one-sided bound on the solutions, for a wide class of fully nonlinear elliptic equations involving Schouten tensors.

This is a joint work with Baozhi Chu and Zongyuan Li.

Title: Robin harmonic measure in rough domains

Abstract: I will describe the construction of a harmonic measure that reproduces a harmonic function from its Robin boundary data, which is a combination of the value of the function and its normal derivative. I shall discuss the surprising fact that this measure turns out to be (quantitatively) mutually absolutely continuous with respect to surface measure on a wide class of domains that includes the complement of certain fractals. Based on joint work with Guy David, Max Engelstein, Svitlana Mayboroda and Marco Michetti.

Title: Stable minimal hypersurfaces in 4-manifolds

Abstract: In 1982, Schoen and Yau showed that 3-manifolds with positive Ricci curvature do not admit complete stable minimal hypersurfaces. Using this fact, they showed that Euclidean space is the only noncompact 3-manifold admitting a complete metric with positive Ricci curvature. I will discuss some features of the analogous problem in 4-manifolds, based on joint work with Otis Chodosh and Chao Li. In particular, I will discuss ambient curvature conditions on 4-manifolds that prohibit complete stable minimal hypersurfaces, as well as some topological consequences of this result.

Title: Analysis on Isotropic-Nematic Phase Transition and Liquid Crystal Droplet

Abstract: In this talk, I will discuss the phase transition phenomena between the isotropic and nematic states within the framework of Ericksen theory of liquid crystals with variable degrees of orientations.  Treating it as the singular perturbation problems within the Gamma convergence theory, we will show that the sharp interface formed between isotropic and nematic states is an area minimizing surface. Under suitable assumptions either on the strong anchoring boundary values on the boundary of a bounded domain or the volume constraint of nematic regions in the entire space, we also show that the limiting nematic liquid configuration in the nematic region is a minimizer of the corresponding Oseen-Frank energy with either homeotropic or planar anchoring on the free sharp interface pending on the relative sizes of leading Frank elasticity coefficients. This is a joint work with Fanghua Lin.

Title: The ASD deformation complex and a conjecture of Singer.

Abstract: In this talk I will give a brief overview of the deformation complex for anti-self-dual 4-manifolds, and formulate a conjecture (attributed to Singer) about the vanishing of one of the cohomology groups. I will then report on some recent progress on resolving this complex. This is joint work with Rod Gover.

Title: Differential inclusions, entropies and the Aviles Giga functional 

Abstract: We will outline some elementary questions and theorems about differential inclusions. Then make a "discontinuous jump" and talk about the concept of entropies from scalar conservation laws and the adaption of this concept to the Aviles Giga functional. Then we show how these topics connect and how the connection has application to both differential inclusions and to the Aviles Giga

Title: Second order elliptic operators on triple junction surfaces

Abstract: In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can show that the classical PDE theory holds on triple junction surfaces. As a consequence, we can prove a type of Generalized Bernstein Theorem and talk about the Morse index on minimal triple junction surfaces.

Title: Fourier coefficients of restricted eigenfunctions

Abstract:  We will discuss the growth of Laplace eigenfunctions on a compact manifold when restricted to a submanifold. We analyze the behavior of the restricted eigenfunctions by studying their Fourier coefficients with respect to an arbitrary orthonormal basis for the submanifold. We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions, the eigenfunctions and the basis, relate.

Title: Classifying sufficiently connected manifolds with positive scalar curvature

Abstract: I will describe the proof of the following classification result for manifolds with positive scalar curvature. Let M be a closed manifold of dimension $n=4$ or $5$ that is "sufficiently connected", i.e. its second fundamental group is trivial (if $n=4$) or second and third fundamental groups are trivial (if $n=5$). Then a finite covering of $M$ is homotopy equivalent to a sphere or a connect sum of $S^{n-1} \times S^1$. The proof uses techniques from minimal surfaces, metric geometry, geometric group theory. This is a joint work with Otis Chodosh and Chao Li.

Title: Diameter estimates in Kahler geometry

Abstract: I will present a recent joint work with D.H. Phong, J. Song and J. Sturm on the diameter estimate and noncollapsing for Kahler metrics, which are uniform in a bounded set of the Kahler cone and require only certain integral bound of the volume form, and no assumptions on Ricci curvature are needed.  As applications, diameter bounds are obtained for long-time solutions of the Kähler-Ricci flow and finite-time solutions when the limiting class is big.

Title: Interior regularity for stationary two-dimensional multivalued maps

Abstract: $Q$-valued maps minimizing a suitably defined Dirichlet energy were introduce by Almgren in his proof of the optimal regularity of area minimizing currents in any dimension and codimension. In this talk I will discuss the extension of Almgren's result to stationary $Q$-valued maps in dimension $2$. This is joint work with Jonas Hirsch (Leipzig).

Title: Recent results on the sigma-2 equation

Abstract: The sigma-2 equation is the remaining member of the Monge-Ampere family of fully nonlinear elliptic equations to be understood from the point of view of interior regularity of viscosity solutions. We discuss some recent results in this direction with Yu Yuan.

Title: The asymptotic behavior of eigenfunctions on symmetric spaces

Abstract: Let X be a compact locally symmetric space, and Y a locally symmetric subspace.  Let f be an eigenfunction of the invariant differential operators on X with eigenvalue tending to infinity.  I will present bounds for the period and Fourier coefficients of f along Y, and the L^p norms of f restricted to Y, for a range of different X and Y.  These results are be based on a combination of techniques from harmonic analysis and the theory of automorphic forms.

Title: A fourth-order Escobar-Yamabe problem on a half-ball

Abstract: The celebrated Yamabe problem asks us to make a conformal change on a compact Riemannian manifold such that the scalar curvature becomes constant. The (Type-II) Escobar-Yamabe problem is to make a conformal change on a compact Riemannian manifold with boundary so that the scalar curvature vanishes and the boundary mean curvature is constant. Both of these problems have been generalized to higher order: the Q-Yamabe problem is a Yamabe-type problem for a fourth-order curvature invariant, while the (Q,T) Yamabe-Escobar problem adds a third-order boundary curvature. In the critical dimension n = 4, Q and T appear as integrands in the Gauss-Bonnet formula, giving the problem a particularly nice interpretation. We review this story, then introduce a form of the Gauss-Bonnet theorem on a four-manifold with corners for which the curvatures have nice conformal transformation properties. A natural question of Escobar-Yamabe type is: can one make a conformal change so that all of the Gauss-Bonnet integrands vanish except on the corner, and are there constant? We answer this in the affirmative for the half-ball in four-space. This is joint work with Jeffrey Case, Tzu-Mo Kuo, Yueh-Ju Lin, Cheikh Ndiaye, Andrew Waldron, and Paul Yang.

Title: Restriction of the Schrodinger operator.

Abstract: Studying eigenfunctions of the Laplace-Beltrami operator has been an interesting topic in harmonic analysis. There was a seminal work of Sogge about Lp bounds of the eigenfunction estimates on compact Riemannian manifolds. Its restriction version was also studied by Burq-Gerard-Tzvetkov, and Hu. Recently, Blair-Sire-Sogge and Blair-Huang-Sire-Sogge found the analogues of Sogge's Lp bounds for the Schrodinger operator with singular potentials. In my joint work with Blair, we found an analogue of restriction estimates for the Schrodinger operator with singular potentials when the dimension of the ambient manifold is 2. In this talk, we briefly review previous results and discuss why it is hard to find higher dimensional analogues of restriction estimates for the Schrodinger operator with singular potentials, which is still an ongoing project.

Title: A quantitative stability result on semidiscrete optimal transport

Abstract: In this talk I will discuss some stability results on the transport map in the optimal transport (Monge-Kantorovich) problem in the semidiscrete case. Specifically, we consider the setting where the target measure is supported on a fixed, finite set, and discuss stability of the so-called Laguerre cells under perturbations of the target masses. There are two results, one where the stability is in a measure sense, and the other under Hausdorff distance. This talk is based on joint work with Mohit Bansil (UCLA).

Title: Rotational Symmetry of Mean Curvature Flows coming out of a double cone.

Abstract: We show that any initially smooth mean curvature flow (MCF) coming out of a rotationally symmetric double cone must stay rotationally symmetric for all time. Examples of such flows include (smooth) self-expanders, which are potential models of continuations of MCFs through conical singularities. I will also discuss non-self-similar, possibly singular, MCFs coming out of such cones that fit into our theorem.

Dual-Frame Generalized Harmonic Gauge on Hyperboloidal Slices

Both for studies of cosmic censorship and for practical purposes in gravitational wave astronomy, it is desirable to include future null-infinity in the computational domain. Extending formulations of general relativity known to behave well in the strong-field regime out to infinity with compactification is, however, a subtle game. In my presentation I will explain how the competition between decay of fields near infinity and growth of coefficients (due to hyperboloidal compactification) plays out in dual-frame generalized harmonic gauge. I will discuss the numerical implementation of the resulting PDEs.

Title: A strong maximum principle for minimizers of the one-phase Bernoulli problem

Abstract: We prove a strong maximum principle for minimizers of the one-phase Alt-Caffarelli functional. We use this to construct a Hardt-Simon-type foliation associated to any 1-homogenous global minimizer.

The black hole stability problem — an introduction and results

Analysis and geometry in the black hole stability problem

Title: “A degenerate fully nonlinear free transmission problem with variable exponents”.

Abstract: "In this talk we will discuss optimal regularity for equations with a degeneracy rate which varies in a discontinuous fashion over the domain. Moreover, this discontinuity depends in an implicit way on the solution itself. We introduce the notion of pointwise sharp regularity which requires an alternative characterization of Holder spaces. These ideas are combined with a geometric tangential analysis argument to obtain the optimal regularity."

Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations

Abstract: We will start with a quick introduction to the weak cosmic censorship conjecture and the problem of constructing naked singularities for the Einstein vacuum equations. Then we will explain our discovery of a new type of self-similarity and explain how this allows us to construct the desired naked singularity solutions.

Title: "Relatively non-degenerate estimates on Kerr (de Sitter) spacetimes"

Abstract: "I will start discussing how to prove exponential decay for the solutions of the wave equation on a Schwarzschild de Sitter black hole spacetime by exploiting a novel "relatively non-degenerate" estimate. This estimate does not degenerate at trapping. The main ingredient in proving this estimate is to commute with a novel vector field that "sees" trapping. Then, we will discuss how to use this black box estimate to prove stability and exponential decay for the solutions of a quasilinear wave equation on Schwarzschild de Sitter. Finally, I will discuss the generalization of these methods to the Kerr de Sitter black hole spacetime, by commuting with a pseudodifferential vector field. There are more technicalities because of the elaborate nature of trapping."

Title: Translating mean curvature flow with simple end.

Abstract: Translators are known as candidates of Type II blow-up model for mean curvature flows.  Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical cases, most of which has either infinite entropy or higher multiplicity asymptotics near infinity.  In this talk, we shall present the construction of a new family of translators with prescribed end.  This is the joint work with Ao Sun.

Title: On asymptotic stability of solitons in classical scalar field theories on the line

Abstract: We discuss the asymptotic stability problem for solitons in classical scalar field theories on the line. Prime examples include kinks in the sine-Gordon and phi^4 models as well as the solitons of the 1D quadratic and cubic Klein-Gordon equations. We begin with a general introduction to scalar field theories on the line and their soliton solutions. In the second part of the talk we try to elucidate the role that threshold resonances of the linearized operators play for the long-time dynamics of perturbations of these classical solitons.

This talk is based on joint works with Y. Li (on the 1D quadratic Klein-Gordon equation) and with W. Schlag (on the sine-Gordon equation and on the 1D cubic Klein-Gordon equation).