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Series: PDE Seminar

In this talk, we present the asymptotics of exit problem for controlled Markov diffusion processes with random jumps and vanishing diffusion terms, where the random jumps are introduced in order to modify the evolution of the controlled diffusions by switching from one mode of dynamics to another. That is, depending on the state-position and state-transition information, the dynamics of the controlled diffusions randomly switches between the different drift and diffusion terms. Here, we specifically investigate the asymptotic exit problem concerning such controlled Markov diffusion processes in two steps: (i) First, for each controlled diffusion model, we look for an admissible Markov control process that minimizes the principal eigenvalue for the corresponding infinitesimal generator with zero Dirichlet boundary conditions -- where such an admissible control process also forces the controlled diffusion process to remain in a given bounded open domain for a longer duration. (ii) Then, using large deviations theory, we determine the exit place and the type of distribution at the exit time for the controlled Markov diffusion processes coupled with random jumps and vanishing diffusion terms. Moreover, the asymptotic results at the exit time also allow us to determine the limiting behavior of the Dirichlet problem for the corresponding system of elliptic PDEs containing a small vanishing parameter. Finally, we briefly discuss the implication of our results.

Series: Job Candidate Talk

I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the "method of interlacing polynomials") and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.

Series: Algebra Seminar

I will explain how to explicitly compute the syntomic regulator for varieties over $p$-adic fields, recently developed by Nekovar and Niziol, in terms of Vologodsky integration. The formulas are the same as in the good reduction case that I found almost 20 years ago. The two key ingrediants are the understanding of Vologodsky integration in terms of Coleman integration developed in my work with Zerbes and techniques for understanding the log-syntomic regulators for curves with semi-stable reduction in terms of the smooth locus.

Series: Geometry Topology Seminar

I will talk about the long standing analogy between the mapping class group of a hyperbolic surface and the outer automorphism group of a free group. Particular emphasis will be on the dynamics of individual elements and applications of these results to structure theorems for subgroups of these groups.

Monday, January 29, 2018 - 13:55 ,
Location: Skiles 005 ,
Prof. Lou, Yifei ,
University of Texas, Dallas ,
Organizer: Sung Ha Kang

A fundamental problem in compressive sensing (CS) is to reconstruct a sparse signal under a few linear measurements far less than the physical dimension of the signal. Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, in which case conventional methods, such as L1 minimization, do not work well. In this talk, I will present a novel non-convex approach, which is to minimize the difference of L1 and L2 norms, denoted as L1-L2, in order to promote sparsity. In addition to theoretical aspects of the L1-L2 approach, I will discuss two minimization algorithms. One is the difference of convex (DC) function methodology, and the other is based on a proximal operator, which makes some L1 algorithms (e.g. ADMM) applicable for L1-L2. Experiments demonstrate that L1-L2 improves L1 consistently and it outperforms Lp (p between 0 and 1) for highly coherent matrices. Some applications will be discussed, including super-resolution, image processing, and low-rank approximation.

Series: CDSNS Colloquium

We will consider the nonlinear elliptic PDEs driven by the fractional Laplacian with superlinear or asymptotically linear terms or combined nonlinearities. An L^infinity regularity result is given using the De Giorgi-Stampacchia iteration method. By the Mountain Pass Theorem and other nonlinear analysis methods, the local and global existence and multiplicity of non-trivial solutions for these equations are established. This is joint work with Yuanhong Wei.

Series: Combinatorics Seminar

We study the number of random permutations needed to invariably generate the symmetric group, S_n, when the distribution of cycle counts has the strong \alpha-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of k-cycles relates to a conditioned Poisson random variable with mean \alpha/k. The special case \alpha=1 corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed.For strong \alpha-logarithmic measures, and almost every \alpha, we show that precisely $\lceil( 1- \alpha \log 2 )^{-1} \rceil$ permutations are needed to invariably generate S_n. A corollary is that for many other probability measures on S_n no bounded number of permutations will invariably generate S_n with positive probability. Along the way we generalize classic theorems of Erdos, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula.

Series: Math Physics Seminar

Quantum theory includes many well-developed bounds for wave-functions, which can cast light on where they can be localized and where they are largely excluded by the tunneling effect. These include semiclassical estimates, especially the technique of Agmon, the use of "landscape functions," and some bounds from the theory of ordinary differential equations. With A. Maltsev of Queen Mary University I have been studying how these estimates of wave functions can be adapted to quantum graphs, which are by definition networks of one-dimensional Schrödinger equations joined at vertices.

Series: ACO Student Seminar

Stochastic
programming is concerned with decision making under uncertainty,
seeking an optimal policy with respect to a set of possible future
scenarios.
While the value of Stochastic Programming is obvious to many
practitioners, in reality uncertainty in decision making is oftentimes
neglected.
For
deterministic optimisation problems, a coherent chain of modelling and
solving exists. Employing standard modelling languages and solvers for
stochastic
programs is however difficult. First, they have (with exceptions) no
native support to formulate Stochastic Programs. Secondly solving
stochastic programs with standard solvers (e.g. MIP solvers)
is often computationally intractable.
David
will be talking about his research that aims to make Stochastic
Programming more accessible. First, he will be talking about modelling
deterministic
and stochastic programs in the Constraint Programming language MiniZinc - a modelling paradigm that retains the structure of a problem much more strongly than MIP formulations. Secondly,
he will be talking about decomposition algorithms he has been working on to solve combinatorial Stochastic Programs.

Friday, January 26, 2018 - 10:00 ,
Location: Skiles 254 ,
Trevor Gunn ,
Georgia Tech ,
tgunn@gatech.edu ,
Organizer: Kisun Lee

We will first give a quick introduction to automatic sequences. We will then outine an algebro-geometric proof of Christol's theorem discovered by David Speyer. Christol's theorem states that a formal power series f(t) over GF(p) is algebraic over GF(p)(t) if and only if there is some finite state automaton such that the n-th coefficent of f(t) is obtained by feeding in the base-p representation of n into the automaton. Time permitting, we will explain how to use the Riemann-Roch theorem to obtain bounds on the number of states in the automaton in terms of the degree, height and genus of f(t).