Analysis and PDE seminar which will take place in 740 Evans Hall on Nov 16th

Speaker: Marina Iliopoulou (University of Birmingham)

Title: Algebraic aspects of harmonic analysis

Abstract: When we want to understand a geometric picture, finding the zero set of a polynomial hiding in it can be very helpful: it can reveal structure and allow computations. Polynomial partitioning, developed by Guth and Katz, is a technique to find such a nice algebraic hypersurface. Polynomial partitioning has revolutionised discrete incidence geometry in the recent years, thanks to the fact that interaction of lines with algebraic hypersurfaces is well-understood. Recently, however, Guth discovered agreeable interaction between tubes and algebraic hypersurfaces, and thus used polynomial partitioning to improve on the 3-dim restriction problem. In this talk, we will present polynomial partitioning via a discrete analogue of the Kakeya problem, and discuss its potential to be extensively used in harmonic analysis.

Monday, Nov 9th 2015

Evans Hall, Room 740

Speaker: Baoping Liu, (Peking University)

Place & Time : Evans Hall, room 740, Nov 2nd 2015, 4:10-5:00 pm.

Speaker: Richard B. Melrose (MIT)

Title: Differential operators undergoing adiabatic transitions

Abstract: I will describe a geometric type of degeneration of differential operators, which includes semiclassical and adiabatic limits. The most basic result  for elliptic operators of this type is the inheritance of invertibility from the limiting operators. I will discuss this and applications of it, in particular in differential topology.

Organizers: Mihaela and Peter

Speaker: Mohammad Reza Pakzad

Title: Rigidity of weak solutions to Monge-Ampere equations

Abstract: In this talk, we will explore rigidity of the weak solutions to the Monge-Amp\`ere equation, by replacing the Hessian determinant by other weaker variants, without any a priori convexity assumptions. Some past and recent results and their proofs concerning rigid behaviour (e.g. convexity or developabilty) of Sobolev solutions in two and higher dimensions will be discussed. We will also study the rigidity of solutions with H\”older continuous derivatives. We will contrast these results with some some non-rigidity statements recently proved by the speaker and M. Lewicka using convex integration.
 

Speaker: Vedran Sohinger (ETH Zurich)

Title: The Gross-Pitaevskii hierarchy on periodic domains

Abstract: The Gross-Pitavskii hierarchy is a system of infinitely many linear PDEs which occurs in the derivation of the nonlinear Schrodinger equation from the dynamics of many-body quantum systems. We will study this problem in the periodic setting. Even though the hierarchy is linear, it is non closed, in the sense that the equation for the k-th density matrix in the system depends on the (k+1)-st density matrix. This structure poses its challenges in the study of the problem, in particular in the understanding of uniqueness of solutions. Moreover, by randomizing in the collision operator, it is possible to use probabilistic techniques in order to study related hierarchies at low regularities. I will present some recent results obtained on these problems, partly in joint work with Philip Gressman, Sebastian Herr, and Gigliola Staffilani.

Speaker: Marta Lewicka

Title: “Convex integration for the Monge-Ampere equation in two dimensions”.

Abstract:

We discuss the dichotomy of rigidity vs. flexibility for the $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Ampere equation in two dimensions:

\begin{equation}

{\mathcal{D}et} \nabla^2 v := -\frac 12 \mbox{curl curl } (\nabla v \otimes \nabla v) = f \qquad \mbox{in } \Omega\subset\mathbb{R}^2.

\end{equation}

Firstly, we show that below the regularity threshold $\alpha<1/7$, the very weak $\mathcal{C}^{1,\alpha}(\bar\Omega)$ solutions to  the equation above, (\ref{MA}), are dense in the set of all continuous functions.

This flexibility statement is a consequence of the convex integration $h$-principle, whereas we directly adapt the iteration method of Nash and Kuiper in order to construct the oscillatory solutions.

Secondly, we prove that the same class of very weak solutions fails the above flexibility in the regularity regime $\alpha>2/3$.

Our interest in the regularity of Sobolev solutions to the Monge-Ampere equation is motivated by the variational description of shape formation, which I will also explain in the talk.

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.

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.

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.

Speaker: Vlad Vicol (Princeton University)

Title: Holder continuous solutions of active scalar equations