MIT PDE/Analysis Seminar

Spring 2024

Tuesdays 4 PM in Room 2-136

February 13 Simons Lecture
February 20
(MIT Monday)
Montie Avery (Boston University)

Universal spreading into unstable states

Abstract: The emergence of complex spatial structures in physical systems often occurs after a simpler background state becomes unstable. Localized fluctuations then grow and spread into the unstable state, forming an invasion front which propagates with a fixed speed and selects a new stable state in its wake. The mathematical study of these invasion processes has historically been limited to systems with restrictive monotonicity properties (in PDE terms, a comparison principle). Such systems, however, inherently cannot describe the formation of complex spatiotemporal patterns, which is of particular interest both in nature and in manufacturing applications. On the other hand, formal calculations in the physics literature have long outlined a universal approach for predicting invasion speeds and associated selected states, valid for systems which do not obey comparison principles and instead exhibit complex spatiotemporal dynamics. This prediction scheme is often referred to as the marginal stability conjecture. In this talk, I will discuss the first proof of the marginal stability conjecture and explore applications to structure formation in physical systems.

February 27 Antoine Gloria (Sorbonne Université and Universitè Libre de Bruxelles)

Large-scale dispersive estimates for acoustic operators: homogenization meets localization

Abstract: At low frequencies the acoustic operator with random coefficients essentially behaves like a Laplacian (the so-called homogenized operator). We might thus expect the associated wave operator to display some dispersion. By blending standard dispersive estimates for homogenized operators and quantitative homogenization of the wave equation, we derive some "weak" (say, large-scale) dispersive estimates for waves in disordered media.

Applied to the spreading of low-energy eigenstates, they allow us to relate quantitatively homogenization to Anderson localization for acoustic operators in disordered media. This gives a short and direct proof that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic media, and it further provides new lower bounds on the localization length of possible eigenstates in case of quasiperiodic or random media.

March 5 Xuwen Zhu (Northeastern University)

Spectral gaps for large genus hyperbolic surfaces

Abstract: The study of "small" eigenvalues of the Laplacian on hyperbolic surfaces has a long history and has recently seen many developments. In this talk I will focus on the recent work (joint with Yunhui Wu and Haohao Zhang) on the higher spectral gaps, where we study the differences of consecutive eigenvalues up to $\lambda_{2g-2}$ for genus $g$ hyperbolic surfaces. We show that the supremum of such spectral gaps over the moduli space has infimum limit at least 1/4 as genus goes to infinity. The analysis relies on previous joint works with Richard Melrose on degenerating hyperbolic surfaces.

March 12 Yu Deng (USC)
March 19 Justin Holmer (Brown University)
April 2 Vedran Sohinger (University of Warwick)
April 9 Zhiyuan (Katherine) Zhang (Northeastern University)
April 16
Room 2-105
3-4pm Ahmed Bou-Rabee (Courant Institute)

4-5pm Aleksandr Logunov (MIT)
April 23 Simons Lecture
April 30 Dubi Kelmer (Boston College)
May 7 Larry Guth (MIT)
May 14 Francois Pagano (University of Geneva)