Seminars and Colloquia by Series

Wednesday, January 31, 2018 - 13:55 , Location: Skiles 006 , Sudipta Kolay , GaTech , Organizer: Anubhav Mukherjee
The Jordan curve theorem states that any simple closed curve decomposes the 2-sphere into two connected components and is their common boundary. Schönflies strengthened this result by showing that the closure of either connected component in the 2-sphere is a 2-cell. While the first statement is true in higher dimensions, the latter is not. However under the additional hypothesis of locally flatness, the closure of either connected component is an n-cell. This result is called the Generalized Schönflies theorem, and was proved independently by Morton Brown and Barry Mazur. In this talk, I will describe the proof of due to Morton Brown.
Wednesday, January 31, 2018 - 12:10 , Location: Skiles 006 , Greg Blekherman , GA Tech , Organizer: Timothy Duff
In recent years the problem of low-rank matrix completion received a tremendous amount of attention. I will consider the problem of exact low-rank matrix completion for generic data. Concretely, we start with a partially-filled matrix M, with real or complex entries, with the goal of finding the unspecified entries (completing M) in such a way that the completed matrix has the lowest possible rank, called the completion rank of M. We will be interested in how this minimal completion rank depends on the known entries, while keeping the locations of specified and unspecified entries fixed. Generic data means that we only consider partial fillings of M where a small perturbation of the entries does not change the completion rank of M.
Wednesday, January 31, 2018 - 11:00 , Location: Skiles 006 , Prof. Mansoor Haider , North Carolina State University, Department of Mathematics & Biomathematics , Organizer: Sung Ha Kang
Many biological soft tissues exhibit complex interactions between passive biophysical or biomechanical mechanisms, and active physiological responses.  These interactions affect the ability of the tissue to remodel in order to maintain homeostasis, or govern alterations in tissue properties with aging or disease. In tissue engineering applications, such interactions also influence the relationship between design parameters and functional outcomes.   In this talk, I will discuss two mathematical modeling problems in this general area.  The first problem addresses  biosynthesis and linking of articular cartilage extracellular matrix in cell-seeded scaffolds. A mixture approach is employed to, inherently, capture effects of evolving porosity in the tissue-engineered construct.  We develop a hybrid model in which cells are represented, individually, as inclusions within a continuum reaction-diffusion model formulated on a representative domain.  The second problem addresses structural remodeling of cardiovascular vessel walls in the presence of pulmonary hypertension (PH).  As PH advances, the relative composition of collagen, elastin and smooth muscle cells in the cardiovascular network becomes altered.  The ensuing wall stiffening increases blood pressure which, in turn, can induce further vessel wall remodeling.  Yet, the manner in which these alterations occur is not well understood.  I will discuss structural continuum mechanics models that incorporate PH-induced remodeling of the vessel wall into 1D fluid-structure models of pulmonary cardiovascular networks. A Holzapfel-Gasser-Ogden (HGO)-type hyperelastic constitutive law for combined bending, inflation, extension and torsion of a nonlinear elastic tube is employed. Specifically, we are interested in formulating new, nonlinear relations between blood pressure and vessel wall cross-sectional area that reflect structural alterations with advancing PH.
Series: PDE Seminar
Tuesday, January 30, 2018 - 15:00 , Location: Skiles 006 , Getachew K. Befekadu , University of Florida , Organizer: Yao Yao
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.
Monday, January 29, 2018 - 15:00 , Location: Skyles006 , Amnon Besser , Georgia Tech/Ben-Gurion University , , Organizer: Amnon Besser
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.
Monday, January 29, 2018 - 14:00 , Location: Skiles 006 , Caglar Uyanik , Vanderbilt University , Organizer: Dan Margalit
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.
Monday, January 29, 2018 - 11:15 , Location: skiles 005 , Xifeng Su , Beijing Normal University , Organizer: Livia Corsi
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.
Friday, January 26, 2018 - 15:00 , Location: Skiles 202 , Evans Harrell , Georgia Tech , , Organizer: Michael Loss
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.
Friday, January 26, 2018 - 15:00 , Location: Skiles 005 , Gerandy Brito , Georgia Tech , Organizer: Lutz Warnke
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.