Seminars and Colloquia Schedule

Equilibrium quasi-periodic configurations in quasi-periodic media

Series
CDSNS Colloquium
Time
Monday, February 16, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lei Zhang – Georgia Institute of Technology
We consider an atomic model of deposition of materials over a quasi-periodic medium. The atoms of the deposited material interact with the medium (a quasi-periodic interaction) and with their nearest neighbors (a harmonic interaction). This is a quasi-periodic version of the well known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type. The iterative method is not based on transformation theory as the most usual KAM theory, but it is based on a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of the equilibrium equilibrium equation.

Nonnegative Inverse Eigenvalue and Singular Value Problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 16, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Matthew Lin – National Chung Cheng University, Georgia Tech –

Reference[1] Moody T. Chu<br />
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, Nonnegative Inverse Eigenvalue and Singular Value Problems, SIAM J. Numer. Anal (1992).[2] Wei Ma and Zheng-J. Bai, A regularized directional derivative-based Newton method for inverse singular value problems, Inverse Problems (2012).

Nonnegative inverse eigenvalue and singular value problems have been a research focus for decades. It is true that an inverse problem is trivial if the desired matrix is not restricted to any structure. This talk is to present two numerical procedures, based on a conquering procedure and an alternating projection process, to solve inverse eigenvalue and singular value problems for nonnegative matrices, respectively. In theory, we also discuss the existence of nonnegative matrices subject to prescribed eigenvalues and singular values. Though the focus of this talk is on inverse eigenvalue and singular value problems with nonnegative entries, the entire procedure can be straightforwardly applied to other types of structure with no difficulty.

Analyzing Related Switching Systems: Two Interesting Examples

Series
AMS Club Seminar
Time
Monday, February 16, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Tobias Hurth – graduate of Georgia Tech School of Math

Dr. Hurth is a recent graduate of the Georgia Tech School of Mathematics. After his talk, the AMS Graduate Chapter is taking Dr. Hurth to dinner at Gordon Biersch. Graduate students and others interested in speaking to Dr. Hurth are invited to join us. If interested, please RSVP to JD Walsh (in person or at <a href="mailto:walsh@math.gatech.edu">walsh@math.gatech.edu</a>).

Dr. Hurth will talk about two relatively simple, related switching systems: one in 1D, the other in 2D. For both systems, he will sketch how to analyse the density of the associated invariant measure. This is straightforward for the 1D-example, but proves somewhat unexpectedly difficult for the 2D one.

Graph Fourientations and the Tutte Polynomial

Series
Combinatorics Seminar
Time
Monday, February 16, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Spencer Backman – University of Rome
A fourientation of a graph is a choice for each edge of whether to orient it in either direction, bidirect it, or leave it unoriented. I will present joint work with Sam Hopkins where we describe classes of fourientations defined by properties of cuts and cycles whose cardinalities are given by generalized Tutte polynomial evaluations of the form: (k+l)^{n-1}(k+m)^g T (\frac{\alpha k + \beta l +m}{k+l}, \frac{\gamma k +l + \delta m}{k+m}) for \alpha,\gamma \in {0,1,2} and \beta, \delta \in {0,1}. We also investigate classes of 4-edge colorings defined via generalized notions of internal and external activity, and we show that their enumerations agree with those of the fourientation classes. We put forth the problem of finding a bijection between fourientations and 4-edge-colorings which respects all of the given classes. Our work unifies and extends earlier results for fourientations due to myself, Gessel and Sagan, and Hopkins and Perkinson, as well as classical results for full orientations due to Stanley, Las Vergnas, Greene and Zaslavsky, Gioan, Bernardi and others.

Random reflections, symmetrizations, and foldings on the sphere

Series
Math Physics Seminar
Time
Tuesday, February 17, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Almut Burchard – University of Toronto
Two-point symmetrizations are simple rearrangementsthat have been used to prove isoperimetric inequalitieson the sphere. For each unit vector u, there is atwo-point symmetrization that pushes mass towardsu across the normal hyperplane.How can full rotational symmetry be recovered from partialinformation? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d);in particular, a continuous function that is symmetric under thesereflections must be radial. How many two-point symmetrizationsare needed to verify that a function which increases under thesesymmetrizations is radial? I will show that d+1 such symmetrizationssuffice, and will discuss the ergodicity of the randomwalk generated by the corresponding folding maps on the sphere.(Joint work with G. R. Chambers and Anne Dranovski).

Optimizing the Combined Treatment of Tumor Growth using Mixed-Effect ODE Modeling

Series
Mathematical Biology Seminar
Time
Wednesday, February 18, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shelby Wilson – Morehouse College –
An array of powerful mathematical tools can be used to identify the key underlying components and interactions that determine the mechanics of biological systems such as cancer and its interaction with various treatments. In this talk, we describe a mathematical model of tumor growth and the effectiveness of combined chemotherapy and anti-angiogenic therapy (drugs that prevent blood vessel growth). An array of mathematical tools is used in these studies including dynamical systems, linear stability analysis, numerical differential equations, SAEM (Stochastic Approximation of the Expectation Maximization) parameter estimation, and optimal control. We will develop the model using preclinical mouse data and discuss the optimal combination of these cancer treatments. The hope being that accurate modeling/understanding of experimental data will thus help in the development of evidence-based treatment protocols designed to optimize the effectiveness of combined cancer therapies.

Stability of Three-dimensional Prandtl Boundary Layers

Series
PDE Seminar
Time
Wednesday, February 18, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 170 (Special)
Speaker
Wang, Yaguang – Shanghai Jiaotong University
In this talk, we shall study the stability of the Prandtl boundary layer equations in three space variables. First, we obtain a well-posedness result of the three-dimensional Prandtl equations under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. Second, we give an instability criterion for the Prandtl equations in three space variables. Both of linear and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, which is an exact complement to the above well-posedness result for a special flow. This is a joint work with Chengjie Liu and Tong Yang.

Braid Theory: Burau and Gassner Representations

Series
Geometry Topology Student Seminar
Time
Wednesday, February 18, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jamie Conway – Georgia Tech
We will describe the Burau representation of the braid group and the related Gassner representation of the pure braid group. We will explore how the Burau representation is related to the Alexander polynomial.

Conformal mapping and optimal meshes

Series
Analysis Seminar
Time
Wednesday, February 18, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris Bishop – SUNY Stony Brook –
The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry. If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all angles \leq 120 degrees and all new angles \geq 60 degrees (small angles in the original polygon must remain).

Uniform bounds on rational points on curves of low Mordell-Weil rank

Series
Algebra Seminar
Time
Wednesday, February 18, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eric Katz – University of Waterloo
In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves. I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform. These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.

Two combinatorial applications of smooth numbers

Series
Combinatorics Seminar
Time
Wednesday, February 18, 2015 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
We look at two combinatorial problems which can be solvedusing careful estimates for the distribution of smooth numbers. Thefirst is the Ramsey-theoretic problem to determine the maximal size ofa subset of of integers containing no 3-term geometric progressions.This problem was first considered by Rankin, who constructed such asubset with density about 0.719. By considering progressions among thesmooth numbers, we demonstrate a method to effectively compute thegreatest possible upper density of a geometric-progression-free set.Second, we consider the problem of determining which prime numberoccurs most frequently as the largest prime divisor on the interval[2,x], as well as the set prime numbers which ever have this propertyfor some value of x, a problem closely related to the analysis offactoring algorithms.

Harmonic analysis and the geometry of fractals

Series
School of Mathematics Colloquium
Time
Thursday, February 19, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Izabella Laba – University of British Columbia
Singular and oscillatory integral estimates, such as maximal theorems and restriction estimates for measures on hypersurfaces, have long been a central topic in harmonic analysis. We discuss the recent work by the speaker and her collaborators on the analogues of such results for singular measures supported on fractal sets. The common thread is the use of ideas from additive combinatorics. In particular, the additive-combinatorial notion of "pseudorandomness" for fractals turns out to be an appropriate substitute for the curvature of manifolds.

On models of short pulse type in continuous media

Series
CDSNS Colloquium
Time
Thursday, February 19, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yannan Shen – Univ. of Texas at Dallas
We develop a mathematical model for ultra-short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. The fundamental equation in the model is the short-pulse equation (SPE) which will be derived in frequency band gaps. We use a multi-scale ansatz to relate the SPE to the nonlinear Schroedinger equation, thereby characterizing the change of width of the pulse from the ultra short regime to the classical slow varying envelope approximation. We will discuss families of solutions of the SPE in characteristic coordinates, as well as discussing the global wellposedness of generalizations of the model that describe uni- and bi-directional nonlinear waves.

Conformal mapping and optimal meshes

Series
Analysis Seminar
Time
Thursday, February 19, 2015 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris Bishop – SUNY Stony Brook
The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry. If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all angles \leq 120 degrees and all new angles \geq 60 degrees (small angles in the original polygon must remain).

Vector Fields on Spheres

Series
Geometry Topology Student Seminar
Time
Friday, February 20, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta Kolay – Georgia Tech

This is a project for Prof. Wickelgren's course on Stable Homotopy Theory.

In this talk, I will show using Clifford algebras that there are ρ(n)-1 linearly independent vector fields on the unit sphere in the n dimensional Euclidean space, where ρ(n) is the Radon-Hurwitz number.

Introduction to regularity theory of second order Hamilton-Jacobi-Bellman equations

Series
PDE Working Seminar
Time
Friday, February 20, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Andrzej Sweich – GeorgiaTech
I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.