Seminars and Colloquia by Series

Math Problems in Gene Regulation

Series
School of Mathematics Colloquium
Time
Thursday, November 5, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Caroline UhlerMIT
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.

Best and random approximation of convex bodies by polytopes

Series
School of Mathematics Colloquium
Time
Thursday, October 15, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Elisabeth WernerCase Western Reserve University
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”.

How unstable is our solar system?

Series
School of Mathematics Colloquium
Time
Thursday, September 17, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Jinxin XueUniversity of Chicago
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.

Approximate separability of Green’s function and intrinsic complexity of differential operators

Series
School of Mathematics Colloquium
Time
Thursday, September 10, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Hongkai ZhaoUniversity of California, Irvine
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.

Time-Domain Boundary Element Methods for Acoustic Problems - Sound Radiation from Tyres

Series
School of Mathematics Colloquium
Time
Thursday, August 20, 2015 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Dr. Ernst StephanLeibniz University Hannover

Please Note: Special time.

We consider the time-domain boundary element method for exterior Robin type boundary value problems for the wave equation. We apply a space-time Galerkin method, present a priori and a posteriori error estimates, and derive an h-adaptive algorithm in space and time with mesh refinement driven by error indicators of residual and hierarchical type. Numerical experiments are also given which underline our theoretical results. Special emphasis is given to numerical simulations of the sound radiation of car tyres.

Absence of shocks in Euler-Maxwell system for two-fluid models in plasma

Series
School of Mathematics Colloquium
Time
Thursday, April 23, 2015 - 11:01 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yan GuoBrown University
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.

Polynomial Inequalities

Series
School of Mathematics Colloquium
Time
Thursday, April 16, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Vili TotikSzeged University (Hungary) and University of South Florida
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.

Elliptic moduli in algebraic topology

Series
School of Mathematics Colloquium
Time
Thursday, April 9, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Haynes MillerMIT
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.

Harmonic analysis and the geometry of fractals

Series
School of Mathematics Colloquium
Time
Thursday, February 19, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Izabella LabaUniversity of British Columbia
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.

Projections of probability distributions: A measure-theoretic Dvoretzky theorem.

Series
School of Mathematics Colloquium
Time
Thursday, February 12, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Elizabeth MeckesCase Western Reserve University
Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed:the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A natural question is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss a result showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!). The talk will not assume much background beyond basic probability and analysis; in particular, no prior knowledge of Dvoretzky's theorem is needed.

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