The talks are in room 2-190 (MIT, Building 2). Each talk is 60 minutes long.
The conference dinner is in the Mathematics Common Room, 2-290. It is limited to the participants who have signed up for it via the second registration form. No lunch is provided but there are many lunch places within a 10 minute walk from Building 2.
András Vasy (Stanford University) Microlocal analysis near null infinity on asymptotically flat spacetimes
There are a number of reasons due to which it is advantageous to have a phase space based, or microlocal, approach available for analyzing wave propagation. In this talk I will explain a microlocal framework for wave propagation on asymptotically flat spacetimes of arbitrary dimension which in particular includes operator corresponding to Lorentzian metrics arising from solutions of Einstein's equations in the 4 spacetime dimensional setting. On the compactification of Minkowski space that underlies this, which is a manifold with corners (with the usual null infinity, scri, being a boundary hypersurface), the operators lie in a combination of Melrose's totally characteristic (also called b), and Mazzeo's edge pseudodifferential operator algebras. I will give an introduction via a simpler setting (which includes Minkowski space and a different class of perturbations), and then explain the reasons for, and complications with, moving to the present setting. Along the way, I will also briefly describe the related Klein–Gordon work of Ethan Sussman. This is joint work with Peter Hintz.
Rafe Mazzeo (Stanford University) Bringing coals to Newcastle: a report on $\mathbb Z_2$ harmonic spinors
This talk will be partly historical, with thoughts on Melrose’s ideas which led to the development of geometric microlocal analysis, together with a report on some recent and not so recent results by various people concerning the existence and nature of $\mathbb Z_2$ harmonic spinors and 1-forms, as initiated by Taubes. These two parts of the talk are closely related.
Sun-Yung Alice Chang (Princeton University) On a conformal Einstein fill in problem
Given a manifold $(M^n,h)$, when is it the boundary of a conformally compact Einstein manifold $(X^{n+1},g)$, in the sense that there exists some defining function $r$ on $X$ so that $r^2 g$ is compact on the closure of $X$ and $r^2 g$ restricted to $M$ is the given metric $h$? The model example is the $n$-sphere as the conformal infinity of the hyperbolic $(n+1)$-ball.
In the special case when $n = 3$, one can formulate the problem as a Dirichlet to Neumann type inverse problem. In the talk, I will report on some progress made with Yuxin Ge on the issues of the `compactness', and as an application, the `existence' and `uniqueness' of the fill in problem for a class of metrics of positive scalar curvature defined on the 3-sphere.
Dan Freed (Harvard University) Index theory on pin manifolds
Peter Sarnak (Princeton University) The arithmetic structure of the spectrum of a metric graph
Endowing a finite combinatorial graph with lengths on its edges defines singular 1-dimensional Riemannian manifolds known as metric graphs. The spectra of their Laplacians have been widely studied. We show that these spectra have a structured linear part described in terms of arithmetic progressions and a nonlinear `random' part which is highly linearly and even algebraically independent over the rationals. These spectra give rise to exotic crystalline measures (`Generalised Poisson Summation Formulae') and resolve various open problems concerning the latter. Joint work with Pavel Kurasov.
Gunther Uhlmann (University of Washington)
TBA
Gigliola Staffilani (MIT) A curious phenomenon in wave turbulence theory
In this talk we will use the periodic cubic nonlinear Schrödinger equation to present some estimates of the long time dynamics of the energy spectrum, a fundamental object in the study of wave turbulence theory. Going back to Bourgain, one possible way to conduct the analysis is to look at the growth of high Sobolev norms. It turns out that this growth is sensitive to the nature of the space periodicity of the system. I will present a combination of old and very recent results in this direction.
Colin Guillarmou (Université Paris-Saclay and CNRS)
TBA
Maciej Zworski (UC Berkeley) Optimal enhanced dissipation for geodesic flows
We consider geodesic flows on negatively curved compact manifolds or more generally contact Anosov flows. The object is to show that if $ X $ is the generator of the flow and $ \Delta$, a (negative) Laplacian, then solutions to the convection diffusion equation, $ \partial_t u = X u + \nu \Delta u $ satisfy $ \| u(t) - \underline u \|_{L^2} \leq C \nu^{-K} e^{-\beta t } \| u(0) \|_{L^2} $ where $ \underline u $ is the (conserved) average of $ u ( 0 ) $ with respect to the contact volume form and $ K $ is a fixed constant. This provides many examples of very precise optimal enhanced dissipation in the sense of recent works of Bedrossian–Blumenthal–Punshon–Smith and Elgindi–Liss–Mattingly. The proof is based on results by Dyatlov and the speaker on stochastic stability of Pollicott–Ruelle resonances. The radial estimates introduced by Melrose in the context of scattering theory on asymptotically Euclidean manifolds are the crucial component of the proof of that result. The talk is based on joint work with Zhongkai Tao.
Local information
The conference will take place in Building 2 at Massachusetts Institute of Technology. It is a 5–10 minute walk from the MBTA Red Line subway station `Kendall/MIT'.
Click here for more information on getting to Building 2.
There are many hotels in the Kendall Square and Central Square neighborhoods of Cambridge, MA,
which are within walking distance of the conference building. There are more affordable options
in neighborhoods further down the Red Line, such as Porter Square and Davis Square.
Caution: we do not work with any third party companies to provide accommodations for the conference! The speakers have already been contacted regarding accommodations, the participants are generally expected to book their own accommodations. If you receive an email from someone other than us about `facilitating accommodation arrangements', it is a scam.
Note: for reasons related to various MIT policies regarding minors, you have to be at least 18 years old to attend the conference and at least 21 years old to attend the conference dinner. The conference organizers reserve the right to ask participants for identification to confirm their age.
