Thursday, November 12, 2015 - 11:00 , Location: Skiles 006 , Prof. Dr. Matthew Kahle , Ohio State University , firstname.lastname@example.org , Organizer: Molei Tao
The study of random topological spaces: manifolds, simplicial complexes, knots, and groups, has received a lot of attention in recent years. This talk will focus on random simplicial complexes, and especially on a certain kind of topological phase transition, where the probability that that a certain homology group is trivial passes from 0 to 1 within a narrow window. The archetypal result in this area is the Erdős–Rényi theorem, which characterizes the threshold edge probability where the random graph becomes connected. One recent breakthrough has been in the application of Garland’s method, which allows one to prove homology-vanishing theorems by showing that certain Laplacians have large spectral gaps. This reduces problems in random topology to understanding eigenvalues of certain random matrices, and the method has been surprisingly successful. This is joint work with Christopher Hoffman and Elliot Paquette.
Thursday, November 5, 2015 - 11:05 , Location: Skiles 006 , Caroline Uhler , MIT , cuhler@MIT.EDU , Organizer: Josephine Yu
Although the genetic information in each cell within an organism is identical, gene expression varies widely between different cell types. The quest to understand this phenomenon has led to many interesting mathematics problems. First, I will present a new method for learning gene regulatory networks. It overcomes the limitations of existing algorithms for learning directed graphs and is based on algebraic, geometric and combinatorial arguments. Second, I will analyze the hypothesis that the differential gene expression is related to the spatial organization of chromosomes. I will describe a bi-level optimization formulation to find minimal overlap configurations of ellipsoids and model chromosome arrangements. Analyzing the resulting ellipsoid configurations has important implications for the reprogramming of cells during development. Any knowledge of biology which is needed for the talk will be introduced during the lecture.
Thursday, October 15, 2015 - 11:00 , Location: Skiles 006 , Prof. Dr. Elisabeth Werner , Case Western Reserve University , email@example.com , Organizer: Molei Tao
How well can a convex body be approximated by a polytope? This is a fundamental question in convex geometry, also in view of applications in many other areas of mathematics and related fields. It often involves side conditions like a prescribed number of vertices, or, more generally, k-dimensional faces and a requirement that the body contains the polytope or vice versa. Accuracy of approximation is often measured in the symmetric difference metric, but other metrics can and have been considered. We will present several results about these issues, mostly related to approximation by “random polytopes”.
Thursday, September 17, 2015 - 11:00 , Location: Skiles 006 , Dr. Jinxin Xue , University of Chicago , firstname.lastname@example.org , Organizer: Molei Tao
Though the modern analytic celestial mechanics has been existing for more than 300 years since Newton, there are still many basic questions unanswered, for instance, there is still no rigorous mathematical proof explaining why our solar system has been stable for such a long time (five billion years) hence no guarantee that it would remain stable for the next five billion years. Instead, it is known that there are various instability behaviors in the Newtonian N-body problem. In this talk, we mention three types instability behaviors in Newtonian N-body problem. The first type we will talk about is simply chaotic motions, which include for instance the oscillatory motions, in which case, one body travels back and forth between neighborhoods of zero and infinity. The second type is “organized” chaotic motions, also known as Arnold diffusion or weak turbulence. Finally, we will talk about our work on the existence of the most wild unstable behavior, non collision singularities, also called finite time blow up solution. The talk is mostly expository. Zero background on celestial mechanism or dynamical systems is needed to follow the lecture.
Thursday, September 10, 2015 - 11:00 , Location: Skiles 006 , Prof. Dr. Hongkai Zhao , University of California, Irvine , email@example.com , Organizer: Molei Tao
Approximate separable representation of the Green’s functions for differential operators is a fundamental question in the analysis of differential equations and development of efficient numerical algorithms. It can reveal intrinsic complexity, e.g., Kolmogorov n-width or degrees of freedom of the corresponding differential equation. Computationally, being able to approximate a Green’s function as a sum with few separable terms is equivalent to the existence of low rank approximation of the discretized system which can be explored for matrix compression and fast solution techniques such as in fast multiple method and direct matrix inverse solver. In this talk, we will mainly focus on Helmholtz equation in the high frequency limit for which we developed a new approach to study the approximate separability of Green’s function based on an geometric characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness and implications for computation setups that are commonly used in practice. We will also make comparisons with other types of differential operators such as coercive elliptic differential operator with rough coefficients in divergence form and hyperbolic differential operator. This is a joint work with Bjorn Engquist.
Thursday, August 20, 2015 - 10:00 , Location: Skiles 005 , Prof. Dr. Ernst Stephan , Leibniz University Hannover , firstname.lastname@example.org , Organizer: Molei Tao
We consider the time-domain boundary element method for exteriorRobin type boundary value problems for the wave equation. We applya space-time Galerkin method, present a priori and a posteriori errorestimates, and derive an h-adaptive algorithm in space and time withmesh renement driven by error indicators of residual and hierarchicaltype.Numerical experiments are also given which underline our theoreticalresults. Special emphasis is given to numerical simulations of the soundradiation of car tyres.
Thursday, April 23, 2015 - 11:01 , Location: Skiles 005 , Yan Guo , Brown University , Organizer: Zhiwu Lin
As the cornerstone of two-fluid models in plasma theory, Euler-Maxwell (Euler-Poisson) system describes the dynamics of compressible ion and electron fluids interacting with their own self-consistent electromagnetic field. It is also the origin of many famous dispersive PDE such as KdV, NLS, Zakharov, ...etc. The electromagnetic interaction produces plasma frequencies which enhance the dispersive effect, so that smooth initial data with small amplitude will persist forever for the Euler-Maxwell system, suppressing any possible shock formation. This is in stark contrast to the classical Euler system for a compressible neutral fluid, for which shock waves will develop even for small smooth initial data. A survey along this direction for various two-fluid models will be given during this talk.
Thursday, April 16, 2015 - 11:00 , Location: Skiles 005 , Prof. Vili Totik , Szeged University (Hungary) and University of South Florida , email@example.com , Organizer: Doron Lubinsky
Bernstein's inequality connecting the norms of a (trigonometric) polynomial with the norm of its derivative is 100 years old. The talk will discuss some recent developments concerning Bernstein's inequality: inequalities with doubling weights, inequalities on general compact subsets of the real line or on a system of Jordan curves. The beautiful Szego-Schaake–van der Corput generalization will also be mentioned along with some of its recent variants.
Thursday, April 9, 2015 - 11:00 , Location: Skiles 005 , Haynes Miller , MIT , Organizer: Kirsten Wickelgren
Much effort in the past several decades has gone into lifting various algebraic structures into a topological context. I will describe one such lifting: that of the arithmetic theory of elliptic curves. The result is a rich and highly structured family of cohomology theories collectively known as elliptic cohomology. By forming "global sections" one is led to a topological enrichment of the ring of modular forms. Geometric interpretations of these theories are enticing but still conjectural at best.
Thursday, February 19, 2015 - 11:00 , Location: Skiles 005 , Professor Izabella Laba , University of British Columbia , Organizer: Martin Short
Singular and oscillatory integral estimates, such as maximal theorems and restriction estimates for measures on hypersurfaces, have long been a central topic in harmonic analysis. We discuss the recent work by the speaker and her collaborators on the analogues of such results for singular measures supported on fractal sets. The common thread is the use of ideas from additive combinatorics. In particular, the additive-combinatorial notion of "pseudorandomness" for fractals turns out to be an appropriate substitute for the curvature of manifolds.