Seminars and Colloquia by Series

Non-asymptotic approach in random matrix theory

Series
School of Mathematics Colloquium
Time
Friday, November 16, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Rudelson – University of Michigan –

Please Note: please note special time!

Random matrices arise naturally in various contexts ranging from theoretical physics to computer science. In a large part of these problems, it is important to know the behavior of the spectral characteristics of a random matrix of a large but fixed size. We will discuss a recent progress in this area illustrating it by problems coming from combinatorics and computer science: Condition number of “full” and sparse random matrices. Consider a system of linear equations Ax = b where the right hand side is known only approximately. In the process of solving this system, the error in vector b gets magnified by the condition number of the matrix A. A conjecture of von Neumann that with high probability, the condition number of an n × n random matrix with independent entries is O(n) has been proven several years ago. We will discuss this result as well as the possibility of its extension to sparse matrices. Random matrices in combinatorics. A perfect matching in a graph with an even number of vertices is a pairing of vertices connected by edges of the graph. Evaluating or even estimating the number of perfect matchings in a given graph deterministically may be computationally expensive. We will discuss an application of the random matrix theory to estimating the number of perfect matchings in a de- terministic graph. Random matrices and traffic jams. Adding another highway to an existing highway system may lead to worse traffic jams. This phenomenon known as Braess’ paradox is still lacking a rigorous mathematical explanation. It was recently explained for a toy model, and the explanation is based on the properties of the eigenvectors of random matrices.

Random walks with relocations and memory through random recursive trees

Series
Stochastics Seminar
Time
Thursday, November 15, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geronimo Uribe – UNAM –
(Based on joint work with Cécile Mailler)Consider a stochastic process that behaves as a d-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in the physics literature under the name "random walk with preferential relocations", where it is argued that the position of the walker after n steps, scaled by log(n), converges to a Gaussian random variable; because of the log spatial scaling, the process is said to undergo a "slow diffusion". We generalize this model by allowing the underlying random walk to be any Markov process and the random run-lengths (time between two relocations) to be i.i.d.-distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently. We prove rigorously the central limit theorem described above by associating to the process a growing family of vertex-weighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a typical vertex in the random tree. This typical height can range from doubly-logarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.

Finding small simple cycle separators for 2-connected planar graphs

Series
Graph Theory Working Seminar
Time
Wednesday, November 14, 2018 - 16:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Michael Wigal – Georgia Tech
Continuation of last week's talk. For a graph on n vertices, a vertex partition A,B,C is a f(n)-vertex separator if |C|≤f(n) and |A|,|B|≤2n/3 and (A,B)=∅. A theorem from Gary Miller states for an embedded 2-connected planar graph with maximum face size d there exists a simple cycle such that it is vertex separator of size at most 2√dn. This has applications in divide and conquer algorithms.

THE GROUND STATE OF A MAGNETOPOLARON BOUND TO A COULOMB POTENTIAL

Series
Math Physics Seminar
Time
Wednesday, November 14, 2018 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rohan Ghanta – SoM Georgia Tech
We shall consider a three-dimensional Quantum Field Theory model of an electron bound to a Coulomb impurity in a polar crystal and exposed to a homogeneous magnetic field of strength B > 0. Using an argument of Frank and Geisinger [Commun. Math. Phys. 338, 1-29 (2015)] we can see that as B → ∞ the ground- state energy is described by a one-dimensional minimization problem with a delta- function potential. Our contribution is to extend this description also to the ground- state wave function: we shall see that as B → ∞ its electron density in the direction of the magnetic field converges to the minimizer of the one-dimensional problem. Moreover, the minimizer can be evaluated explicitly.

Existence of contact structures in 3-manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, November 14, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyunki Min – Georgia Tech
Unlike symplectic structures in 4-manioflds, contact structures are abundant in 3-dimension. Martinet showed that there exists a contact structure on any closed oriented 3-manifold. After that Lutz showed that there exist a contact structure in each homotopy class of plane fields. In this talk, we will review the theorems of Lutz and Martinet.

Cotlar’s identity for Hilbert transforms---old and new stories.

Series
Analysis Seminar
Time
Wednesday, November 14, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tao Mei – Baylor University –
Cotlar’s identity provides an easy (maybe the easiest) argument for the Lp boundedness of Hilbert transforms. E. Ricard and I discovered a more flexible version of this identity, in the recent study of the boundedness of Hilbert transforms on the free groups. In this talk, I will try to introduce this version of Cotlar’s identity and the Lp Fourier multipliers on free groups.

Estimating High-dimensional Gaussian Tails

Series
High Dimensional Seminar
Time
Wednesday, November 14, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben Cousins – Columbia University –

The following is a well-known and difficult problem in rare event simulation: given a set and a Gaussian distribution, estimate the probability that a sample from the Gaussian distribution falls outside the set. Previous approaches to this question are generally inefficient in high dimensions. One key challenge with this problem is that the probability of interest is normally extremely small. I'll discuss a new, provably efficient method to solve this problem for a general polytope and general Gaussian distribution. Moreover, in practice, the algorithm seems to substantially outperform our theoretical guarantees and we conjecture that our analysis is not tight. Proving the desired efficiency relies on a careful analysis of (highly) correlated functions of a Gaussian random vector.Joint work with Ton Dieker.

Some combinatorial enumeration problems: results and techniques

Series
Research Horizons Seminar
Time
Wednesday, November 14, 2018 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prasad Tetali – Georgia Tech
There has been much interest in the past couple of decades in identifying (extremal) regular graphs that maximize the number of independent sets, matchings, colorings etc. There have been many advances using techniques such as the fractional subaddtivity of entropy (a.k.a. Shearer's inequality), the occupancy method etc. I will review some of these and mention some open problems on hypergraphs.

Scaling down the laws of thermodynamics

Series
Other Talks
Time
Wednesday, November 14, 2018 - 04:00 for 1 hour (actually 50 minutes)
Location
Molecular Science and Engineering Building, Classroom G011
Speaker
Christopher Jarzynski – Director, Institute for Physical Science and Technology University of Maryland
Thermodynamics provides a robust conceptual framework and set of laws that govern the exchange of energy and matter. Although these laws were originally articulated for macroscopic objects, it is hard to deny that nanoscale systems, as well, often exhibit “thermodynamic-like” behavior. To what extent can the venerable laws of thermodynamics be scaled down to apply to individual microscopic systems, and what new features emerge at the nanoscale? I will review recent progress toward answering these questions, with a focus on the second law of thermodynamics. I will argue that the inequalities ordinarily used to express the second law can be replaced by stronger equalities, known as fluctuation relations, which relate equilibrium properties to far-from-equilibrium fluctuations. The discovery and experimental validation of these relations has stimulated interest in the feedback control of small systems, the closely related Maxwell demon paradox, and the interpretation of the thermodynamic arrow of time. These developments have led to new tools for the analysis of non-equilibrium experiments and simulations, and they have refined our understanding of irreversibility and the second law. Bio Chris Jarzynski received an AB degree in physics from Princeton University in 1987, and a PhD in physics from the University of California, Berkeley in 1994. After postdoctoral positions at the University of Washington in Seattle and at Los Alamos National Laboratory in New Mexico, he became a staff member in the Theoretical Division at Los Alamos. In 2006, he moved to the University of Maryland, College Park, where he is now a Distinguished University Professor in the Department of Chemistry and Biochemistry, with joint appointments in the Institute for Physical Science and Technology and the Department of Physics. His research is primarily in the area of nonequilibrium statistical physics, where he has contributed to an understanding of how the laws of thermodynamics apply to nanoscale systems. He has been the recipient of a Fulbright Fellowship, the 2005 Sackler Prize in the Physical Sciences, and the 2019 Lars Onsager Prize in Theoretical Statistical Physics. He is a Fellow of the American Physical Society and the American Academy of Arts and Sciences. Contact: Professor Jennifer Curtis Email: jennifer.curtis@physics.gatech.edu

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