The seminar meets Wednesdays in WH-100E at 3:30-4:30 p.m. There are refreshments and snacks in WH-102 at 3:15.
The seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.
* August 29th, Wednesday (3:30-4:30pm)
Topic: organizational meeting
* September 5th, Wednesday (3:30-5:00pm)(No talk due Monday schedule)
Topic: No talk
* September 12th, Wednesday (3:30-4:55pm)
Speaker : Gang Zhou (Binghamton University)
Topic: A description of generic singularities formed by mean curvature flow
Abstract: In this talk I will present the progresses my collaborators, including Michael Sigal and Dan Knopf, and I made in the past few years. We developed a new way of studying mean curvature flow, and I am trying to use it to understand the evolution of hypersurfaces under mean curvature flow.
* September 19th, Wednesday (3:30-4:55pm)(Holiday, Yom Kippur)
* September 26th , Wednesday (3:30-4:55pm)
Speaker: Adam Weisblatt (Binghamton University)
Topic: The heat equation on planar diagrams.
Abstract: The heat kernel on a surface helps to describe its geometry. However, solving the heat equation explicitly and extracting the geometric information can be difficulty. In this talk, I will offer a new approach to the heat equation using planar diagrams. The heat kernel constructed will not be the authentic heat kernel for the surface, but we will show how it captures geometry.
* October 3rd, Wednesday (3:30-4:55pm)
Speaker: Brian Allen (West Point)
Topic: Stability Questions and Convergence of Riemannian Manifolds
Abstract: We will start by surveying the stability of the scalar torus rigidity theorem, a result about the impact of geometry on topology, and the stability of the positive mass theorem, an important theorem in mathematical relativity. Since stability requires a notion of closeness this will lead us naturally to consider various notions of distance between and convergence of Riemannian manifolds. We will end by discussing theorems and important examples which aim at contrasting these notions of convergence which have been, and will continue to be, applied to stability problems.
* October 10th, Wednesday (3:30-4:55pm)
* October 17th, Wednesday (3:30-4:55pm)
Speaker: Shengwen Wang (Binghamton University)
Topic: Mean curvature flow with surgery and applications
Abstract: I will first review about mean curvature flow with surgery for 2-convex hypersurfaces. Then I will report on joint work with Mramor for mean curvature flow with surgery for low entropy mean-convex hypersurfaces and an application to the classification of self-shrinkers. I will also discuss what elements we still lack to do surgery for generic mean curvature flow.
* October 24th, Wednesday (3:30-4:55pm)
Speaker: Lu Zhang (Binghamton University)
Topic: Some useful methods for Fourier multipliers
Abstract: I will give a introduction of some methods that have been recently used to study the Lp bounds for the multi-parameter Fourier multipliers, which include one method that was applied in my recent work.
* October 31st, Wednesday (3:30-4:55pm)
Speaker: Xiangjin Xu (Binghamton University )
Topic: New heat kernel estimates on manifolds with negative Ricci curvature
Abstract: In this talk, we first introduce some new sharp Li–Yau type gradient estimates, both in local and global version, for the positive solution $u(x,t)$ of the heat equations $$\partial_t u-\Delta u=0$$ on a complete manifold with $Ric(M)\geq -k$. As applications, some new parabolic Harnack inequalities, both in local and global version, are derived. Based on the new parabolic Harnack inequalities, some new sharp Gaussian type lower bound and upper bound of the heat kernel on a complete manifold with $Ric(M)\geq -k$ are proved, which are new even for manifold $M$ with nonnegative Ricci curvature, $Ric(M)\geq 0$. An upper bound of $\mu_1 (M) \geq 0$, the greatest lower bound of the $L^2$-spectrum of the Laplacian on a complete noncompact manifold $M$, is achieved. At the end, we discuss some open questions related to the sharp Li–Yau type estimates.
* November 7th, Wednesday (3:30-4:55pm)
* November 14th, Wednesday (3:30-4:55pm)
Speaker: Phil Sosoe, Cornell University
Topic: Applications of CLTs and homogenization for Dyson Brownian Motion to Random Matrix Theory
Abstract: I will explain how two recent technical developments in Random Matrix Theory allow for a precise description of the fluctuations of single eigenvalues in the spectrum of large symmetric matrices. No prior knowledge of random matrix theory will be assumed.
(Based on joint work with B Landon and HT Yau).
* November 21st, Wednesday (3:30-4:55pm)(Thanksgiving break)
* November 28th, Wednesday (3:30-4:55pm)
Speaker: Martin Fraas, Virginia Tech
Topic: Perturbation Theory of Quantum Trajectories
Abstract: Quantum trajectories are certain Markov processes on a complex projective space. They describe the evolution of a quantum system subject to a repeated indirect measurement. For a given set of matrices $A$ and a unit vector $x$, a probability of a sequence of matrices $V_1, V_2, \dots , V_n$, $Vj \in A$ is proportional to $||V_n \dots V_1x||^2$. The Markov process is given by $x_n \sim V_n \dots V_1 x||. In this talk, I will review the basic properties of this process, in particular, conditions that guarantee the uniqueness of the stationary measure. Then I will discuss how the measure and the process change if the underlying set of matrices A changes.
* December 5th, Wednesday (3:30-4:55pm)
Speaker: Kunal Sharma (Binghamton University)