Each week, the Michael Zhao Memorial Student Colloquium holds 45-minute talks by Columbia mathematics faculty about their own research. The talks are intended for current PhD students in mathematics at Columbia. If you are an undergraduate student or external graduate student and would like to come, please email firstname.lastname@example.org or email@example.com.
This Spring the Seminar is organized online. It meets on Wednesdays at 4:30p.m. EDT virtually in Zoom, followed by informal gathering through a platform Gather Town.
Organizers: Georgy Gaitsgori and Patrick Lei.
|Date and time||Speaker||Title and abstract|
|Wednesday, March 3, 4:30p.m. EDT||Aleksander Doan||The Yang-Mills equations and higher-dimensional geometry.
The Yang-Mills equations are a generalization of Maxwell's equations of electromagnetism. While they originated from the theory of nuclear forces in physics, in the 1980s they were discovered to be intimately related to the geometry and topology of four-dimensional manifolds. In my talk, I will outline some of these advances and discuss a recent line of research whose goal is to develop applications of the Yang-Mills equations to the study of higher-dimensional Riemannian manifolds with special holonomy.
|Wednesday, March 10, 4:30p.m. EDT||Ioannis Karatzas||A trajectorial approach to the gradient flow. Properties of conservative diffusions.
We provide a detailed, probabilistic interpretation for the variational characterization of conservative diffusion as entropic gradient flow. Jordan, Kinderlehrer, and Otto showed in 1998 that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features; these are valid along almost every trajectory of the motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by "aggregating", i.e., taking expectations. As a bonus of the analysis, the HWI inequality of Otto and Villani relating relative entropy, Fisher information, and Wasserstein distance, falls in our lap; and with it the celebrated log-Sobolev, Talagrand and Poincare inequalities of functional analysis. (Joint work with Walter Scahchermayer and Bertram Tschiderer, University of Vienna.)
|Wednesday, March 24, 4:30p.m. EDT||Daniela De Silva||Free boundary problems.
I will discuss questions of existence and regularity for a prototype elliptic free boundary problem appearing in the applications. The purpose will be to highlight the type of questions, toos, techniques which arise in this branch on PDEs.
|Wednesday, March 31, 4:30p.m. EDT||Dorian Goldfeld||Orthogonality relations for GL(n).
In 1837 Dirichlet proved infinitely many primes in an arithmetic progression: a, a+q, a+2q, ..., where a,q are integers without a common factor. His proof relied on an orthogonality relation on GL(1). We show how Dirichlet's orthogonality can be generalized to higher rank groups.
|Wednesday, April 14, 4:30p.m. EDT||Igor Krichever||Algebraic geometry and Integrable systems.
The famous Novikov conjecture which asserts that the Jacobians of smooth algebraic curves are precisely those indecomposable principally polarized Abelian varieties whose theta-functions provide explicit solutions of the Kadomtsev-Petviashvili (KP) equation, fundamentally changed the relations between the classical algebraic geometry of Riemann surfaces and the theory of soliton equations. It turns out that the finite-gap, or algebro-geometric, theory of integration of non-linear equations developed in the mid-1970s can provide a powerful tool for approaching the fundamental problems of the geometry of Abelian varieties.
|Wednesday, April 21, 4:30p.m. EDT||Konstantin Aleshkin||Curve counting and mirror symmetry.
Typical curve counting questions are "how many lines pass through two points" (1) or "how many lines are there on a generic degree 5 hypersurface in P^4" (2875). Remarkably, there is a lot of structure behind this type of questions. In particular, one can often obtain such numbers from power series expansions of interesting analytic functions. In the talk I will give examples of this correspondence and explain how one can use it.