- You are here:
- GT Home
- Home
- News & Events

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: 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.

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: 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: 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: 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).

Series: Analysis Seminar

The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.

Wednesday, January 24, 2018 - 13:55 ,
Location: Skiles 006 ,
Justin Lanier ,
GaTech ,
Organizer: Anubhav Mukherjee

Take a map from the interval [0,1] to itself. Such a map can be iterated, and many phenomena (such as periodic points) arise. An interval self-map is an example of a topological dynamical system that is simple enough to set up, but wildly complex to analyze. In the late 1970s, Milnor and Thurston developed a combinatorial framework for studying interval self-maps in their paper "Iterated maps of the interval". In this talk, we will give an introduction to the central questions in the study of iterated interval maps, share some illustrative examples, and lay out some of the techniques and results of Milnor and Thurston.