Date	Speaker	Title	Abstract
9/12	Fabio Putsateri (Princeton)	The problem of global regularity for water waves.	The water waves equations are a system of evolution equations modeling the motion of waves, like those in the surface of the ocean. After introducing the system we will discuss some of the works done in recent years on the question of long-time regularity. We will then present a result, joint with Deng, Ionescu and Pausader, about global existence of smooth solutions for the 3d gravity-capillary water waves system in infinite depth. The main difficulties in this problem are the slow decay of linear solutions and the presence of large sets of resonant interactions.
9/19	Jonas Luerhmann (JHU)	Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation	The Maxwell-Klein-Gordon Equation is a system of evolution equations that describes the interaction of an electromagnetic field with a charged particle field. In this talk we discuss a proof of global regularity, scattering and a priori bounds for solutions to the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for essentially arbitrary smooth data of finite energy. The proof is based upon a novel "twisted" Bahouri-Gérard type profile decomposition and a concentration compactness/rigidity argument by Kenig-Merle, following the method developed by Krieger-Schlag in the context of energy critical wave maps. This is joint work with Joachim Krieger.
9/26	Guohuan Qiu (McGill)	Classical Neumann problems for Hessian equations and geometric applications	The classic Neumann problem for laplace equation has many geometric applications. For example, Reilly used its solution to give a new proof of Minkowski inequality. Recently, Xinan Ma and Guohuan Qiu, have proved the existence of the Neumann problems for Hessian equations in uniformly convex domain in Rn. Motivated from Reilly and Ma-Qiu's work, Chao Xia and I also find geometric applications about classical Neumann problems for Hessian equations. We will talk about how to prove the existence of classical Neumann problems under the uniformly convex domain. Then we use the solution of the classical Neumann problem to give a new proof of a family of Alexandrov-Fenchel inequalities arising from convex geometry.
10/3	Ruobing Zhang (Stony Brook)	Regularity and Degeneration of Einstein Metrics	In this talk, we will review some classical and new results in the regularity and degeneration theory of Einstein manifolds. That is, we will discuss the effective curvature estimates on a Einstein manifold and the limiting behavior for a sequence of Einstein manifolds. A reasonable starting point is the regularity theory of non-collapsing Einstein manifolds, which has been well established in recent years. However, the regularity and degeneration behavior of collapsing Einstein manifolds are much harder to handle. Without any extra assumption, there are few analytic tools in the general collapsed context. In this direction, we will introduce some new regularity theorems (joint work with Aaron Naber) and open questions.
10/10	Russell Schwab (MSU)	A min-max formula for Lipschitz operators that satisfy the global comparison principle	We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960’s, by Courr\`{e}ge, that if I is a linear mapping with the GCP, then I must be represented as a linear drift-jump-diffusion operator that may have both local and integro-differential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a min-min over linear and local drift-diffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study Dirichlet-to-Neumann mappings for fully nonlinear equations as integro-differential operators on the boundary. This is joint work with Nestor Guillen
10/17	Jonathan Zhu (Harvard U)	Entropy and self-shrinkers of the mean curvature flow	The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.
10/24	Philip Isett (MIT)	A Proof of Onsager’s Conjecture	In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured that weak solutions to the incompressible Euler equations may violate the law of conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof to be discussed of the full conjecture combines a new ingredient in the convex integration scheme due to Daneri-Székelyhidi with a new "gluing approximation" technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.
10/31	Yakov Shlapentokh-Rothman (Princeton U)	Scattering theory and blue-shift instabilities on Kerr exterior and interior spacetimes.	In the first part of the talk, which concerns joint work with Dafermos and Rodnianski, we will present results corresponding to the "existence and uniqueness of scattering states" and "asymptotic completeness" for the wave equation on Kerr exterior spacetimes. These results are stated with respect to function spaces which allow for the solutions to exhibit certain singular behaviors along the event horizon. In the second part of the talk, which concerns joint work with Dafermos, we will show that it is in fact necessary to allow such singular behavior, and furthermore, exploiting essentially the same mechanism, we will also show that solutions to the wave equation generically have a singular behavior along the Cauchy-Horizon in Kerr black hole interiors.
11/7	Nestor Guillen (Amherst)	Pointwise estimates for Generated Jacobian Equations	Generated Jacobian Equations (GJE) are a class of nonlinear, degenerate elliptic equations introduced by Trudinger, motivated by the analysis of reflecting surfaces in geometric optics. I will present a new regularity result for weak solutions of these equations, building in particular on work of Caffarelli for the real Monge-Ampère equation and work of Figalli, Kim, and McCann in optimal transport. An example of an equation covered by the theory is the point source near-field reflector problem, which already shows markedly different regularity phenomena when compared to optimal transport, as shown by the analysis and examples by Karakhanyan and Wang. A central tool in our approach is the derivation of certain pointwise inequalities connected to the Blaschke-Santaló inequality. This talk is based on joint works with Jun Kitagawa.
11/14	Nick Edelen (MIT)	The free-boundary Brakke flow	A surface has geometric free-boundary in a barrier hypersurface if its boundary meets the barrier orthogonally, like a bubble on a bathtub. We extend Brakke's weak notion of mean curvature flow to have a free-boundary condition, and using toy examples we show why this extension is necessary. Contrary to the classical flow, for which the barrier is ``invisible,'' our notion allows for the flow to ``pop'' or ``break up'' upon tangential contact with the barrier.
11/28	TBA
12/5	Sylvester Eriksson-Bique (NYU)	Quantitative connectivity, (weighted) Poincare inequalities and differentiability in doubling metric measure spaces.	Analysis on metric spaces concerns the study of first order analysis in non-smooth spaces, and their relations to the underlying geometry. This talk will focus on three related notions: notions of connectivity, Poincare inequalities and differentiability. I will outline how recent developments by myself and others have closely linked these seemingly unrelated concepts to each other. I will present two main theorems from my paper and the ideas involved in proving them. The first concerns the characterization of Poincare inequalities on doubling metric measure spaces in terms of a simple quantitative connectivity condition. The second theorem partially resolves an open question of Cheeger and Kleiner on the local geometry of Lipschitz differentiability spaces, and proves a new type of rectifiability result for a subclass of them. Finally we relate the new concepts to Muckenhoupt weights and new types of self-improvement results in the spirit of Keith-Zhong are presented."

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