# Phillips 383

## February 2017

### Thomas Duyckaerts (University Paris 13), PDE Mini-School

Title:  Dynamics of the energy critical wave equations Abstract:  These lectures concern the energy-critical focusing nonlinear wave equation. It is conjectured that any solution of this equation that is bounded in the energy space is asymptotically the sum of a finite number of decoupled solitons and a radiation term. My goal is to prove this conjecture for radial solutions in space dimension 3 and to give partial results in the general case. This is based on joint works with Hao…

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## April 2017

### Andrew Stuart (Cal Tech), PDE/Analysis Seminar

Title:  (S) PDE Limits Arising in Graph Based Learning Abstract:  TBA

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### Lev Rozansky (UNC-CH), Physically Inspired Mathematics Seminar

Title: Flag varieties, Hilbert schemes on C^2 and link homology II   Abstract: This is a joint work with A. Oblomkov. I will explain (1) how our construction of the categorical (affine and ordinary) braid group action is related to permutations of eigenvalues of an upper-triangular matrix; (2) how this categorical action appears in the framework of geometric Langlands duality; (3) how it is related to quantum field theory and, very likely, to brane braiding within the string theory.

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## December 2017

### Michael Kemeny, Special Colloquium

Title: The equations defining curves and moduli spaces Abstract: A projective variety is a subset of projective space defined by polynomial equations. One of the oldest problems in algebraic geometry is to give a qualitative description of the equations defining a variety, together with the relations amongst them. When the variety is an algebraic curve (or Riemann surface), several conjectures made since the 80s give a fairly good picture of what we should expect. I will describe a new variational approach to these conjectures,…

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## December 2018

### Jared Wunsch, Northwestern University – Wunsch/PDE Mini-school

View Full Schedule here Title: Trapping, diffraction, and decay of waves Abstract: The long-time behavior of solutions to wave and Schrödinger equations is connected to geometry and dynamics via the correspondence principle, which states that at high-frequency, solutions propagate along classical particle orbits in phase space.  Making sense of the “high frequency” part of this statement often involves estimates for the resolvent operator family.  We will discuss some well-established results on how resolvent estimates and associated questions about distribution of scattering resonances are affected…

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### Oran Gannot, Northwestern University – Wunsch/PDE Mini-school

View Full Schedule here Title: Scattering resonances generated by diffractive trapping Abstract: In potential scattering where classical particles with a certain energy escape to infinity, it is expected that quantum states with neighboring energies decay rapidly in time. We will discuss some known results where the decay rate depends on the regularity of the potential. In cases where irregularities of the potential occur along an interface, the decay rate can be understood in terms of the strength of diffractive effects. We will illustrate this explicitly in one dimensional examples.

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### Perry Kleinhenz, Northwestern University – Wunsch/PDE Mini-school

View Full Schedule here Title: Stabilization rates for the damped wave equation with Hölder regular damping Abstract: We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. In particular we consider the case of a torus where the damping is $0$ on a strip and vanishes like a polynomial $x^{\beta}$. We prove that the semigroup is stable at rate at least as fast as $1/t^{(beta+2)/(\beta+4)}$ and sketch…

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### Jeffrey Galkowski, Northeastern University – Wunsch/PDE Mini-school

View Full Schedule here Title: Optimal Resolvent Estimates in Non-trapping Geometries Abstract: In this talk we discuss how propagation estimates can be used to prove resolvent estimates in non-trapping geometries. We will introduce the notion of defect measures and prove propagation estimates in this setting. We will then use these estimates to prove a non-trapping resolvent estimate and finally refine the estimates to give optimal bounds on the resolvent. Time permitting we discuss the necessary modification when the manifold has a boundary.

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