# Mihaela Ifrim (April 27th)

Speaker: Mihaela Ifrim (UC Berkeley)

Title: Long time solutions for two dimensional water waves

Abstract: This is joint work with Daniel Tataru, and in parts with John Hunter. My talk is concerned with the infinite depth water wave equation in two space dimensions, with either gravity or surface tension. Both cases will be discussed in parallel. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data.  For the gravity water waves there are several results available; they have been recently obtained by Wu, Alazard-Burq-Zuily and Ionescu-Pusateri using different coordinates and methods. In the capillary water waves case, we were the first to establish a global result.  Our goal is improve the understanding of these problems by providing a single setting for both cases, and  presenting simpler proofs. The talk will be as self contained as the time permits.

# Michal Wrochna (April 20th)

Speaker: Michal Wrochna (Stanford University)

Title: Scattering theory approach to the Feynman problem for the wave equation

Abstract: A classical result of Duistermaat and Hörmander provides four parametrices for the wave equation, distinguished by their wave front set. In applications in Quantum Field Theory one is interested in constructing the corresponding exact inverses, satisfying in addition a positivity condition. I will present a method (derived in a joint work with C. Gérard and dating back to W. Junker), where this is achieved by diagonalizing the wave equation in terms of elliptic pseudodifferential operators and solving the Cauchy problem with possible smooth remainders. I will then indicate possible ways of replacing Cauchy data by scattering data and comment on how this relates to global constructions of Feynman propagators.

# Tanya Christiansen (April 13th)

Speaker: Tanya Christiansen (University of Missouri)

Title: Resonances in even-dimensional Euclidean scattering

Abstract: Resonances may serve as a replacement for discrete spectral data for a class of operators with continuous spectrum. In odd-dimensional Euclidean scattering, the resonances lie on the complex plane, while in even dimensions they lie on the logarithmic cover of the complex plane. In even-dimensional Euclidean scattering there are some surprises for those who are more familiar with the odd-dimensional case. For example, qualitative bounds on the number of “pure imaginary” resonances are very different depending on the parity. Moreover, for Dirichlet or Neumann obstacle scattering or for scattering by a fixed-sign potential one can show there are many resonances in even dimensions. In fact, for these cases the $m$th resonance counting function ($m\in Z, m\neq 0$) has maximal order of growth.

Some of this talk is based on joint work with Peter Hislop.

AbstractWe consider active scalar equations $\partial_t \theta + \nabla \cdot (u\, \theta) = 0$, where $u = T[\theta]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator. We prove that when $T$ is not an odd multiplier, there are nontrivial, compactly supported solutions weak solutions, with Holder regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in $D’$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when $T$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected. This is a joint work with Phillip Isett (MIT).