## Seminars and Colloquia by Series

### Invariant objects and Arnold diffusion. From theory to computation.

Series
Research Horizons Seminar
Time
Wednesday, March 4, 2020 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael de la LlaveGeorgia Tech

We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.

Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate (Arnold diffusion).

This can be understood by noting that the small perturbations generate some invariant objects in phase space that act as routes which allow accumulation of effects.

We will present some rigorous results about geometric objects lead to Arnold diffusion as well as some computational tools that allow to find them in concrete applications.

Thanks to the work of many people, an area which used to be very speculative, is becoming an applicable tool.

### The Shape of Things: Organizing space using algebra

Series
Research Horizons Seminar
Time
Wednesday, November 20, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Miriam Kuzbary

Determining when two objects have “the same shape” is difficult; this difficulty depends on the dimension we are working in. While many of the same techniques work to study things in dimensions 5 and higher, we can better understand dimensions 1, 2, and 3 using other methods. We can think of 4-dimensional space as the “bridge” between low-dimensional behavior and high-dimensional behavior.

One way to understand the possibilities in each dimension is to examine objects called cobordisms: if an (n+1)-dimensional space has an edge,” which is called a boundary, then that boundary is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the boundary of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form an algebraic structure called a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions.

In this talk, I will discuss different types of cobordism groups and how to study them using tools from a broad range of mathematical areas.

### Variational models, PDEs, numerical analysis and applications

Series
Research Horizons Seminar
Time
Wednesday, November 13, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sung Ha KangGeorgia Tech

Starting from mathematical approaches for image processing, we will discuss different models, analytic aspects of them, and numerical challenges.  If time permits we will consider numerical applications to data understanding. A few other applications may be presented.

### The 4x4 orthostochastic variety

Series
Research Horizons Seminar
Time
Wednesday, November 6, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin ChenGeorgia Tech

A real matrix is called orthostochastic if it is the entrywise square of an orthogonal matrix. These matrices have been shown to be deeply connected to determinantal representations of polynomials, and also arise naturally in physics. However, the equations defining the real variety are known only up to the 3x3 case. I will show how various techniques of numerical algebraic geometry give a way of finding (set-theoretic) defining equations for the 4x4 orthostochastic variety, which are smaller (both in number and degree) than the naive equations one might initially guess. Based on joint work with Papri Dey.

### Spectrum of quasi-periodic Schrodinger operators

Series
Research Horizons Seminar
Time
Wednesday, October 30, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rui HanGeorgia Tech

One of the simplest and, at the same time, most prominent models for the discrete quasi-periodic Schrodinger operator is the almost Mathieu operator (also called the Harper's model). This simple-looking operator is known to present exotic spectral properties. Three (out of fifteen) of Barry Simon's problems on Schrodinger operators in the 21st century concerns the almost Mathieu operator. In 2014, Artur Avila won a Fields Medal for work including the solutions to these three problems. In this talk, I will concentrate on the one concerning the Lebesgue measure of the spectrum. I will also talk about the difficulties in generalizing this result to the extended Harper's model. Students with background in numerics are especially welcome to attend!

### Models for DNA-based Tile Self-Assembly

Series
Research Horizons Seminar
Time
Wednesday, October 23, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel CruzGeorgia Tech
A set of elementary building blocks undergoes self-assembly if local interactions govern how this set forms intricate structures. Self-assembly has been widely observed in nature, ranging from the field of crystallography to the study of viruses and multicellular organisms. In this talk, we give an overview of tile assembly models (TAMs) whose elementary building blocks (i.e. tiles) are polygons which have been defined with rules for local interaction. In particular, we present the basic concepts associated with two of the most well-studied TAMs: the abstract Tile Assembly Model (aTAM) and the Two-Handed Assembly Model (2HAM). We show how TAMs are related to the problem of designing nanoscale structures with DNA. We also present some of the major results within this field of study.

### Geometric Approaches for Metastability in Stochastic Dynamical Systems with Applications

Series
Research Horizons Seminar
Time
Wednesday, October 9, 2019 - 13:10 for
Location
Skiles 005
Speaker
Larissa SerdukovaGeorgia Tech

Please Note: NOTE THE UNUSUAL TIME: This seminar takes place from 1:10-1:50 for THIS WEEK ONLY.

Basin of attraction for a stable equilibrium point is an effective concept for stability in deterministic systems. However, it does not contain information on the external perturbations that may affect it. The concept of stochastic basin of attraction (SBA) is introduced by incorporating a suitable probabilistic notion of basin. The criteria for the size of the SBA is based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small intensity but also with whose amplitude is proportional or in general is a function of an order parameter. The efficiency of the concept is presented through two applications.

### The isoperimetric inequality

Series
Research Horizons Seminar
Time
Wednesday, October 2, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker

The classical isoperimetric inequality states that in Euclidean space spheres form the least perimeter enclosures for any give volume. We will review the historic development of this result in mathematics, and various approaches to proving it. Then we will discuss how one of these approaches, which is a variational argument, may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, in order to generalize the isoperimetric inequality.

### An Introduction to Quantum Topology

Series
Research Horizons Seminar
Time
Wednesday, September 4, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker

We will explore some of the basic notions in quantum topology.  Our focus will be on introducing some of the foundations of diagrammatic algebra through the lens of the Temperley-Lieb algebra.  We will attempt to show how these diagrammatic techniques can be applied to low dimensional topology.  Every effort will be made to make this as self-contained as possible.  If time permits we will also discuss some applications to topological quantum computing.

### Random graph processes: results and techniques

Series
Research Horizons Seminar
Time
Wednesday, April 24, 2019 - 00:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lutz WarnkeGeorgia Tech

During the last 30 years there has been much interest in random graph processes, i.e., random graphs which grow by adding edges (or vertices) step-by-step in some random way. Part of the motivation stems from more realistic modeling, since many real world networks such as Facebook evolve over time. Further motivation stems from extremal combinatorics, where these processes lead to some of the best known bounds in Ramsey and Turan Theory (that go beyond textbook applications of the probabilistic method). I will review several random graph processes of interest, and (if time permits) illustrate one of the main proof techniques using a simple toy example.