Seminars and Colloquia Schedule

Pairings between periodic orbits in hyperbolic coupled map lattices.

Series
CDSNS Colloquium
Time
Monday, October 17, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Boris Gutkin – Georgia Tech (School of Physics)
Upon quantization, hyperbolic Hamiltonian systems generically exhibit universal spectral properties effectively described by Random Matrix Theory. Semiclassically this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to single-particle systems. I will discuss an extension of this program to hyperbolic coupled map lattices with a large number of sites (i.e., particles). The crucial ingredient is a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such a symbolic dynamics can be constructed explicitly.

Backward SDE method for nonlinear filtering problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 17, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Yanzhao Cao – Auburn University Mathematics
A nonlinear filtering problem can be classified as a stochastic Bayesian optimization problem of identifying the state of a stochastic dynamical system based on noisy observations of the system. Well known numerical simulation methods include unscented Kalman filters and particle filters. In this talk, we consider a class of efficient numerical methods based on forward backward stochastic differential equations. The backward SDEs for nonlinear filtering problems are similar to the Fokker-Planck equations for SDEs. We will describe the process of deriving such backward SDEs as well as high order numerical algorithms to solve them, which in turn solve nonlinear filtering problems.

Genome-scale estimation of the Tree of Life

Series
IMPACT Distinguished Lecture
Time
Monday, October 17, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tandy Warnow – The University of Illinois at Urbana-Champaign
Estimating the Tree of Life is one of the grand computational challenges in Science, and has applications to many areas of science and biomedical research. Despite intensive research over the last several decades, many problems remain inadequately solved. In this talk I will discuss species tree estimation from genome-scale datasets. I will describe the current state of the art for these problems, what is understood about these problems from a mathematical perspective, and identify some of the open problems in this area where mathematical research, drawing from graph theory, combinatorial optimization, and probability and statistics, is needed. This talk will be accessible to mathematicians, computer scientists, probabilists and statisticians, and does not require any knowledge of biology. (Refreshments will be served after the talk.)

Constrained exact optimization in Phylogenetics

Series
Mathematical Biology Seminar
Time
Tuesday, October 18, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tandy Warnow – The University of Illinois at Urbana-Champaign
The estimation of phylogenetic trees from molecular sequences (e.g., DNA, RNA, or amino acid sequences) is a major step in many biological research studies, and is typically approached using heuristics for NP-hard optimization problems. In this talk, I will describe a new approach for computing large trees: constrained exact optimization. In a constrained exact optimization, we implicitly constrain the search space by providing a set X of allowed bipartitions on the species set, and then use dynamic programming to find a globally optimal solution within that constrained space. For many optimization problems, the dynamic programming algorithms can complete in polynomial time in the input size. Simulation studies show that constrained exact optimization also provides highly accurate estimates of the true species tree, and analyses of both biological and simulated datasets shows that constrained exact optimization provides improved solutions to the optimization criteria efficiently. We end with some discussion of future research in this topic. (Refreshments will be served before the talk at 10:30.)

Some Properties of Effective Hamiltonians

Series
PDE Seminar
Time
Tuesday, October 18, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Yifeng Yu – Universtiy of California, Irvine –
A major open problem in periodic homogenization of Hamilton-Jacobi equations is to understand deep properties of the effective Hamiltonian. In this talk, I will present some related works in both convex and non-convex situations. If time permits, relevant problems from applications in turbulent combustion and traffic flow will also be discussed.

PDE models for collective dynamics

Series
Research Horizons Seminar
Time
Wednesday, October 19, 2016 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Yao Yao – Department of Mathematics, Georgia Institute of Technology –

Refreshments will be provided before the seminar.

Collective behavior can be seen in many animal species, such as flocking birds, herding mammals, and swarming bacteria. In the continuum limit, these phenomena can be modeled by nonlocal PDEs. In this talk, after discussing some PDE models for collective dynamics, I will focus on the analysis of the Keller-Segel equation, which models bacterial chemotaxis. Mathematically, this equation exhibits an intriguing "critical mass phenomenon": namely, solutions exist globally in time for all initial data whose mass is below some certain constant, whereas finite-time blow-up always happen if the initial mass is above this constant. I will introduce some useful analysis tools that lead to this result, and discuss some active areas of current research.

Intersection forms and homotopy equivalence

Series
Geometry Topology Student Seminar
Time
Wednesday, October 19, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew McCullough – Georgia Institute of Technology –
We will discuss some facts about intersection forms on closed, oriented 4-manifolds. We will also sketch the proof that for two closed, oriented, simply connected manifolds, they are homotopy equivalent if and only if they have isomorphic intersection forms.

Fractional Calculus, Reproducing Kernel Hilbert Spaces, and Approximation Theory

Series
Analysis Seminar
Time
Wednesday, October 19, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joel Rosenfeld – University of Florida –
I will present results on numerical methods for fractional order operators, including the Caputo Fractional Derivative and the Fractional Laplacian. Fractional order systems have been of growing interest over the past ten years, with applications to hydrology, geophysics, physics, and engineering. Despite the large interest in fractional order systems, there are few results utilizing collocation methods. The numerical methods I will present rely heavily on reproducing kernel Hilbert spaces (RKHSs) as a means of discretizing fractional order operators. For the estimation of a function's Caputo fractional derivative we utilize a new RKHS, which can be seen as a generalization of the Fock space, called the Mittag-Leffler RKHS. For the fractional Laplacian, the Wendland radial basis functions are utilized.

The Kelmans-Seymour conjecture on subdivisions of $K_5$

Series
School of Mathematics Colloquium
Time
Thursday, October 20, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xingxing Yu – Georgia Tech
A well-known theorem of Kuratowski (1930) in graph theory states that a graph is planar if, and only if, it does not contain a subdivision of $K_5$ or $K_{3,3}$. Wagner (1937) gave a structural characterization of graphs containing no subdivision of $K_{3,3}$. Seymour in 1977 and, independently, Kelmans in 1979 conjectured that if a graph does not contain a subdivision of $K_5$ then it must be planar or contain a set of at most 4 vertices whose removal results in a disconnected graph. In this talk, I will discuss additional background on this conjecture (including connection to the Four Color Theorem), and outline our recent proof of this conjecture (joint work with Dawei He and Yan Wang). I will also mention several problems that are related to this conjecture or related to our approach.

Mechanical response of three-dimensional tensegrity lattices

Series
GT-MAP Seminar
Time
Friday, October 21, 2016 - 15:00 for 2 hours
Location
Skiles 006
Speaker
Prof. Julian Rimoli – GT AE
Most available techniques for the design of tensegrity structures can be grouped in two categories. On the one hand, methods that rely on the systematic application of topological and geometric rules to regular polyhedrons have been applied to the generation of tensegrity elementary cells. On the other hand, efforts have been made to either combine elementary cells or apply rules of self-similarity in order to generate complex structures of engineering interest, for example, columns, beams and plates. However, perhaps due to the lack of adequate symmetries on traditional tensegrity elementary cells, the design of three-dimensional tensegrity lattices has remained an elusive goal. In this work, we first develop a method to construct three-dimensional tensegrity lattices from truncated octahedron elementary cells. The required space-tiling translational symmetry is achieved by performing recursive reflection operations on the elementary cells. We then analyze the mechanical response of the resulting lattices in the fully nonlinear regime via two distinctive approaches: we first adopt a discrete reduced-order model that explicitly accounts for the deformation of individual tensegrity members, and we then utilize this model as the basis for the development of a continuum approximation for the tensegrity lattices. Using this homogenization method, we study tensegrity lattices under a wide range of loading conditions and prestressed configurations. We present Ashby charts for yield strength to density ratio to illustrate how our tensegrity lattices can potentially achieve superior performance when compared to other lattices available in the literature. Finally, using the discrete model, we analyze wave propagation on a finite tensegrity lattice impacting a rigid wall.

On the k-SUM problem

Series
Combinatorics Seminar
Time
Friday, October 21, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Esther Ezra – Georgia Tech

Joint work with Micha Sharir (Tel-Aviv University).

Following a recent improvement of Cardinal etal. on the complexity of a linear decision tree for k-SUM, resulting in O(n^3 \log^3{n}) linear queries, we present a further improvement to O(n^2 \log^2{n}) such queries. Our approach exploits a point-location mechanism in arrangements of hyperplanes in high dimensions, and, in fact, brings a new view to such mechanisms. In this talk I will first present a background on the k-SUM problem, and then discuss bottom-vertex triangulation and vertical decomposition of arrangements of hyperplanes and how they serve our analysis.