Monday, November 12, 2018 - 13:00 , Location: Skiles 006 , Tom Bachmann , MIT , Organizer: Kirsten Wickelgren
I will review various ways of modeling the homotopy theory of spaces: several model categories of simplicial sheaves and simplicial presheaves, and related infinity categorical constructions.
Monday, November 12, 2018 - 13:55 , Location: Skiles 005 , Prof. Xiaoliang Wan , Louisiana State University , Organizer: Molei Tao
In this talk, we will discuss some computational issues when applying the large deviation theory to study small-noise-induced rare events in differential equations. We focus on two specific problems: the most probable transition path for an ordinary differential equation and the asymptotically efficient simulation of rare events for an elliptic problem. Both problems are related to the large deviation theory. From a computational point of view, the former one is a variational problem while the latter one is a sampling problem. For the first problem, we have developed an hp adaptive minimum action method, and for the second problem, we will present an importance sampling estimator subject to a sufficient and necessary condition for its asymptotic efficiency.
Monday, November 12, 2018 - 14:00 , Location: Skiles 006 , Tom Bachmann , MIT , Organizer: Kirsten Wickelgren
It is a classical theorem in algebraic topology that the loop space of a suitable Lie group can be modeled by an infinite dimensional variety, called the loop Grassmannian. It is also well known that there is an algebraic analog of loop Grassmannians, known as the affine Grassmannian of an algebraic groop (this is an ind-variety). I will explain how in motivic homotopy theory, the topological result has the "expected" analog: the Gm-loop space of a suitable algebraic group is A^1-equivalent to the affine Grassmannian.
Series: PDE Seminar
We discuss bilateral obstacle problems for fully nonlinear second order uniformly elliptic partial differential equations (PDE for short) with merely continuous obstacles. Obstacle problems arise not only in minimization of energy functionals under restriction by obstacles but also stopping time problems in stochastic optimal control theory. When the main PDE part is of divergence type, huge amount of works have been done. However, less is known when it is of non-divergence type. Recently, Duque showed that the Holder continuity of viscosity solutions of bilateral obstacle problems, whose PDE part is of non-divergence type, and obstacles are supposed to be Holder continuous. Our purpose is to extend his result to enable us to apply a much wider class of PDE. This is a joint work with Shota Tateyama (Tohoku University).
Series: Other Talks
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: firstname.lastname@example.org
Series: Research Horizons Seminar
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.
Series: High Dimensional Seminar
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.
Series: Analysis Seminar
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.
Wednesday, November 14, 2018 - 14:00 , Location: Skiles 006 , Hyunki Min , Georgia Tech , Organizer: Hyun Ki Min
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.
Series: Math Physics Seminar
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.
Series: Graph Theory Working Seminar
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.
Series: Stochastics Seminar
(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.
Series: School of Mathematics Colloquium
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.
Series: ACO Student Seminar
Consider a linear combination of independent identically distributed random variables $X_1, . . . , X_n$ with fixed weights $a_1, . . . a_n$. If the random variablesare continuous, the sum is almost surely non-zero. However, for discrete random variables an exact cancelation may occur with a positive probability. Thisprobability depends on the arithmetic nature of the sequence $a_1, . . . a_n$. We will discuss how to measure the relevant arithmetic properties and how to evaluate the probability of the exact and approximate cancelation.
Series: Algebra Seminar
Let K be a non-trivially valued non-Archimedean field, R its valuation subring. A formal Gubler model is a formal R-scheme that comes from a polyhedral decomposition of a tropical variety. In this talk, I will present joint work with Sam Payne in which we show that any formal model of any compact analytic domain V inside a (not necessarily projective) K-variety X can be dominated by a formal Gubler model that extends to a model of X. This result plays a central role in our work on "structure sheaves" on tropicalizations and our work on adic tropicalization. If time permits I will explain some of this work.
Friday, November 16, 2018 - 14:00 , Location: Skiles 006 , Stavros Garoufalidis , Georgia Tech and MPI , Organizer: John Etnyre
I will explain some connections between the counting of incompressible surfaces in hyperbolic 3-manifolds with boundary and the 3Dindex of Dimofte-Gaiotto-Gukov. Joint work with N. Dunfield, C. Hodgson and H. Rubinstein, and, as usual, with lots of examples and patterns.
Series: Combinatorics Seminar
Let P be a system of unique shortest paths through a graph with real edge weights (i.e. a finite metric). An obvious fact is that P is "consistent," meaning that no two of these paths can intersect each other, split apart, and then intersect again later. But is that all? Can any consistent path system be realized as unique shortest paths in some graph? Or are there more forbidden combinatorial intersection patterns out there to be found? In this talk, we will characterize exactly which path systems can or can't be realized as unique shortest paths in some graph by giving a complete list of new forbidden intersection patterns along these lines. Our characterization theorem is based on a new connection between graph metrics and certain boundary operators used in some recent graph homology theories. This connection also leads to a principled topological understanding of some of the popular algebraic tricks currently used in the literature on shortest paths. We will also discuss some applications in theoretical computer science.
Friday, November 16, 2018 - 15:05 , Location: Skiles 156 , Sergio Mayorga , Georgia Tech , Organizer: Jiaqi Yang
In this talk I will begin by discussing the main ideas of mean-field games and then I will introduce one specific model, driven by a smooth hamiltonian with a regularizing potential and no stochastic noise. I will explain what type of solutions can be obtained, and the connection with a notion of Nash equilibrium for a game played by a continuum of players.