The Analysis and PDE seminar will take place on Monday, November 27, from 4:10 to 5pm, in 740 Evans.

Title: Fine properties of Dirichlet energy minimizing multi-valued functions

Abstract: I will discuss the fine structure of the branch set of multivalued Dirichlet energy minimizing functions as developed by Almgren. It is well-known that the dimension of the interior singular set of a Dirichlet energy minimizing function on an $n$-dimensional domain is at most $n-2$. We show that the singular set is countably $(n-2)$-rectifiable and also prove the uniqueness of homogeneous tangent functions at almost every singular point. Our approach involves adapting a “blow up” method due to Leon Simon, which was originally applied to multiplicity one classes of minimal submanifolds. We apply Simon’s method in the higher multiplicity setting of multivalued energy minimizers using techniques from prior work of Neshan Wickramasekera together with new estimates. This is joint work with Neshan Wickramasekera.

The Analysis and PDE seminar will take place Monday, October 30, in 740 Evans from 4:10 to 5pm.

Title: Inverse scattering and the Davey-Stewartson II equation

Abstract: The aim of this talk is to describe a complete implementation of the inverse scattering approach to the study of the defocusing Davey-Stewartson equation.
This will involve dispersive quations, dbar pde’s, microlocal analysis and other fun stuff. This is joint work with Adrian Nachman and Idan Regev.

The Analysis and PDE seminar will take place Monday, October 16, in 740 Evans from 4:10 to 5pm.

Title: Fractal uncertainty for transfer operators

Abstract: I will present a new explanation of the connection between
the fractal uncertainty principle
of Bourgain–Dyatlov, a statement in harmonic analysis, and the
existence of zero free strips for Selberg zeta functions, which is a
statement in geometric scattering/dynamical systems. The connection is
proved using (relatively) elementary methods via the Ruelle transfer
operator which is a well known object in thermodynamical formalism of
chaotic dynamics. (Joint work with S Dyatlov.)

The Analysis and PDE seminar will take place Monday, October 9nd, in 740 Evans from 4:10 to 5pm.

Title: Gravity water waves and emerging bottom

Abstract: To understand the behavior of waves at a fluid surface in configurations where the surface and the bottom meet (islands, beaches…), one encounters a difficulty: the presence in the bulk of the fluid of an edge, at the triple line. To solve the Cauchy problem, we need to study elliptic regularity in such domains, understand the linearized operator around an arbitrary solution, and construct an appropriate procedure to quasi-linearize the equations. Using those tools, I will present some a priori estimates, a first step to a local existence result.

The Analysis and PDE seminar will take place Monday, October 2nd, in 740 Evans from 4:10 to 5pm.

Title: Convergence of phase-field models and thresholding schemes for multi-phase mean curvature flow

Abstract: The thresholding scheme is a time discretization for mean curvature flow. Recently, Esedoglu and Otto showed that thresholding can be interpreted as minimizing movements for an energy that Gamma-converges to the total interfacial area. In this talk I’ll present new convergence results, in particular in the multi-phase case with arbitrary surface tensions. The main result establishes convergence to a weak formulation of (multi-phase) mean curvature flow in the BV-framework of sets of finite perimeter. Furthermore, I will present a similar result for the vector-valued Allen-Cahn equation.

This talk encompasses joint works with Felix Otto, Thilo Simon, and Drew Swartz.

The Bay Area Microlocal Analysis Seminar will take place on Monday, September 25, in room 740, Evans Hall, with two talks, given by Kiril Datchev at 2:40 pm and Charles Hadfield at 4:10 pm.

Speaker: Kiril Datchev (2:40 pm)

Title: Semiclassical resolvent estimates away from trapping

Abstract: Semiclassical resolvent estimates relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. Roughly speaking, more trapping of particles corresponds to a larger resolvent near the trapping. If the trapping is mild, then propagation estimates imply that the larger norm occurs only there. However, in this talk I will show how the effects of heavy trapping can tunnel over long distances, implying that the resolvent can be very large far away as well. This is joint work with Long Jin.

Speaker: Charles Hadfield (4:10 pm)

Title: Resonances on asymptotically hyperbolic manifolds; the ambient metric approach

Abstract: On an asymptotically hyperbolic manifold, the Laplacian has essential spectrum. Since work of Mazzeo and Melrose, this essential spectrum has been studied via the theory of resonances; poles of the meromorphic continuation of the resolvent of the Laplacian (with modified spectral parameter). A recent technique of Vasy provides an alternative construction of this meromorphic continuation which dovetails the ambient metric approach to conformal geometry initiated by Fefferman and Graham. I will discuss the ambient geometry present in this construction, use it to define quantum resonances for the Laplacian acting on natural tensor bundles (forms, symmetric tensors), and mention an application showing a correspondence between Ruelle resonances and quantum resonances on convex cocompact hyperbolic manifolds.

The Analysis and PDE Seminar will take place on Monday, September 18, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Resolvent estimates and wave asymptotics for manifolds with cylindrical ends

Abstract: Wave oscillation and decay rates on a manifold M are are well known to be related to the geometry of M and to dynamical properties of its geodesic flow. When M is a closed system, such as a bounded Euclidean domain or a compact manifold, there is no decay and the connection is made via the eigenvalues of the Laplacian. When M is an open system, such as the complement of a bounded Euclidean domain, or a suitable more general manifold with large infinite ends, then the spectrum of the Laplacian is continuous and we look instead at resonances, which give rates of both oscillation and decay.

An interesting intermediate situation is a manifold with infinite cylindrical ends, which we call a mixed system. In this case the continuous spectrum has increasing multiplicity as energy grows, and in general it can have embedded resonances and eigenvalues accumulating at infinity, making wave asymptotics more mysterious. However, we prove that if geodesic trapping is sufficiently mild, then such an accumulation is ruled out, and moreover lossless high-energy resolvent bounds hold. We deduce from this the existence of resonance free regions and compute asymptotic expansions for solutions of the wave equation in terms of eigenvalues, resonances, and spectral thresholds. This is joint work with Tanya Christiansen.

The Analysis and PDE seminar will take place Monday, Sept 11, in Evans 740 from 4:10 to 5:00pm.

Title: Two-bubble dynamics for the equivariant wave maps equation.

Abstract: I will consider the energy-critical wave maps equation with values in the
sphere in the equivariant case, that is for symmetric initial data. It is
known that if the initial data has small energy, then the corresponding
solution scatters. Moreover, the initial data of any scattering solution
has topological degree 0. I try to answer the following question: what are
the non-scattering solutions of topological degree 0 and the least
possible energy? Such “threshold” solutions would have to decompose
asymptotically into a superposition of two ground states at different
scales, with no radiation.
In the first part I will show how to construct threshold solutions. In the
second part I will describe the dynamical behavior of any threshold
solution.
The second part is a joint work with Andrew Lawrie (MIT).

The Analysis and PDE seminar will take place Monday, May 8, in 740 Evans Hall, from 16:10-17:00.

Title: Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data

Abstract: Black holes are perhaps the most celebrated predictions of general relativity. Miraculously, stationary black hole spacetimes arise as explicit (i.e., exact expression can be written down!) solutions to the (vacuum) Einstein equation. Looking at these explicit solutions leads to an intriguing observation: While the black hole exterior looks qualitatively similar, the structure of the interior, in particular the nature of the `singularity’ inside the black hole, changes drastically depending on whether the black hole is spinning (Kerr) or not (Schwarzschild).

The topic of this talk is the strong cosmic censorship conjecture of R. Penrose, which is a proposed picture for what happens generically. In particular, I will present a recent work, joint with Jonathan Luk, where we establish a version of strong cosmic censorship for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry with two-ended asymptotically flat data, a toy model
that has been studied by physicists and mathematicians to understand issues connected to strong cosmic censorship in a simplified setting.

The Analysis and PDE Seminar will take place on Monday, April 17, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Control of water waves

Abstract: Water waves are disturbances of the free surface of a liquid. They are, in general, produced by the immersion of a solid body or by impulsive pressures applied on the free surface. The question we discuss in this talk is the following: which waves can be generated by blowing on a localized portion of the free surface. Our main result asserts that one can generate any small amplitude, periodic in x, two-dimensional, gravity-capillary water waves. This is a result from control theory. More precisely, we prove the local exact controllability of the incompressible Euler equation with free surface. This is a joint work with Pietro Baldi and Daniel Han-Kwan.