Seminars and Colloquia by Series

Low Dimensional Topology and Cobordism Groups: Organizing spaces using algebra

Series
Time
Monday, November 23, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Miriam KuzbaryGeorgia Tech

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,”  then that edge is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the edge of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions and organize this information using algebra. 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.

PDE Models for Collective Behavior

Series
Time
Monday, November 16, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting: https://gatech.bluejeans.com/759112674
Speaker
Dr. Yao YaoGeorgia Institute of Technology

Self-organization is a common feature in the collective behavior of many animal species, such as flocking birds, herding mammals, and swarming bacteria. As the number of individuals gets large, instead of tracking the movement of each individual, it is more efficient to model the evolution of the whole population density using partial differential equations (PDEs). In this talk, I will introduce some PDE models for collective dynamics, and discuss the challenges in both the modeling part and the mathematical analysis.

Ranking from pairwise comparisons

Series
Time
Monday, November 9, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting: https://gatech.bluejeans.com/759112674
Speaker
Dr. Mao ChengGeorgia Institute of Technology

Ranking items from comparisons is a ubiquitous task in many real-world applications. For example, sports teams can be ranked based on outcomes of matches; students' homework solutions can be ranked based on peer grading. In this lecture, I will discuss: (1) how we can design mathematical models for the problem of ranking or rating a set of items from pairwise comparisons between them; (2) how to do statistical inference based on the models. The model we focus on is the Bradley-Terry model proposed in 1952, which is also related to the Elo rating system implemented for the US Chess Federation in 1960.

What is tropical convexity?

Series
Time
Monday, November 2, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/759112674
Speaker
Cvetelina HillGeorgia Tech

We say that a set is convex if for any two points in the set, the straight line segment connecting them is also contained in the set.  For example, a triangle, a square, a cube, a ball are all convex sets. We typically speak of convex sets in Euclidean space with the ordinary addition and multiplication operations. What happens if we replace addition with taking the minimum between two elements, and multiplication with ordinary addition? These are the tropical arithmetic operations and using these we can define tropical convexity. What does it mean for a set to be tropically convex? What does a tropical triangle look like? In this talk we will answer these questions and explore how ordinary and tropical convexity interact.

Synchronization of coupled pendulum clocks and metronomes

Series
Time
Monday, October 26, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Guillermo GoldszteinGeorgia Tech

In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they tend to synchronize in phase, not antiphase. Here, using a simple model of coupled clocks and metronomes, we calculate the regimes where antiphase and in-phase synchronization are stable. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and a three-time scale asymptotic analysis.

Random Growth Models

Series
Time
Monday, October 19, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Michael DamronGeorgia Tech

Random and irregular growth is all around us: tumor growth, fluid flow through porous media, and the spread of bacterial colonies. Simple models for these processes originated in the '50s with percolation theory and have since given rise to many new models and interesting mathematics. I will introduce a few models (percolation, invasion percolation, first-passage percolation, diffusion-limited aggregation, ...), along with some of their basic properties.

Mathematics of Soap Films

Series
Time
Monday, October 12, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Ben JayeGeorgia Tech

In this talk we shall give a brief introduction to the mathematics of soap films (aka minimal surfaces). These are the surfaces that, amongst all possible surfaces with prescribed boundary, have the least area. If one dips a wire mesh into soap solution, then the surface formed is a minimal surface. We shall see how minimal surfaces arise in science and engineering, look at the physical laws that a minimal surface should obey, and see how much mathematicians understand about them.

Regression of functions on a low-dimensional set by neural networks

Series
Time
Monday, October 5, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Wenjing LiaoGeorgia Tech

Many data set in image analysis and signal processing is in a high-dimensional space but exhibit low-dimensional structures. For example, data can be modeled as point clouds in a high-dimensional space but are concentrated on a low-dimensional set (or manifold in particular). Our goal is to estimate functions on the low-dimensional manifold from finite samples of data, for statistical inference and prediction. This talk introduces approximation theories of neural networks for functions supported on a low-dimensional manifold. When the function is estimated from finite samples, we give an estimate of the mean squared error for the approximation of these functions. The convergence rate depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that neural networks are adaptive to low-dimensional geometric structures of data.

Off the rails: Train tracks on surfaces

Series
Time
Monday, September 28, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Marissa LovingGeorgia Tech

Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at their computer's screen a bit at times.

Exploration of convex geometry in high dimension

Series
Time
Monday, September 21, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Han HuangGeorgia Tech

A ball and a cube looks so different, but in higher dimension, it turns out a high dimensional ball and a high dimensional cube could be hard to distinguish them. Our intuitions on 3 dimensional geometry often fails in higher dimension! In this talk, we will start from the basic mathematical definition of high dimensional spaces. Then we will explore some phenomenons of high dimensional convex geometry. In the end, we will show how these nice observations could be applied to speed up algorithms in computer science.