Seminars and Colloquia by Series

The lattice metric space and its applications

Series
Research Horizons Seminar
Time
Friday, November 13, 2020 - 11:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yuchen HeGeorgia Tech
Lattice patterns are commonly observed in material sciences where microscopic structural nuances induce distinct macroscopic physical or chemical properties. Provided with two lattices of the same dimension, how do we measure their differences in a visually consistent way? Mathematically, any n-D lattice is determined by a set of n independent vectors. Since such basis-representation is non-unique, a direct comparison among basis-representations in Euclidean space is highly ambiguous. In this talk, I will focus on 2-D lattices and introduce the lattice metric space proposed in my earlier work. This geometric space was constructed mainly based on integrating the Modular group theory and the Poincaré metric. In the lattice metric space, each point represents a unique lattice pattern, and the visual difference between two patterns is measured by the shortest path connecting them. Some applications of the lattice metric space will be presented. If time allows, I will briefly discuss potential extensions to 3D-lattices.

Paradoxical decompositions and graph theory

Series
Research Horizons Seminar
Time
Friday, November 6, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Anton Bernshteynanton.bernshteyn@math.gatech.edu

 

The Banach--Tarski paradox is one of the most counterintuitive facts in all of mathematics. It says that it is possible to divide the 3-dimensional unit ball into a finite number of pieces, move the pieces around (without changing their shape), and then put them back together to form two identical copies of the original ball. Many other, equally difficult to believe, equidecomposition statements are also true. For example, a ball of radius 1 can be split into finitely many pieces, which can then be rearranged to form a ball of radius 1000. It turns out that such statements are best understood through the lens of graph theory. I will explain this connection and discuss some recent breakthroughs it has led to.
 

Explorations in high-dimensional convexity

Series
Research Horizons Seminar
Time
Friday, October 30, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Galyna LivshytsGeorgia Tech

We will discuss a few beautiful questions in high-dimensional convexity, and path their connections to areas such as Analysis, Probability Theory and Differential Geometry. I shall mention some of my recent results too, in particular a new inequality about convex sets in high dimensions. I will describe its relations to one of the difficult problems in the area.

Symmetries of Surfaces

Series
Research Horizons Seminar
Time
Friday, October 16, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Marissa LovingGeorgia Tech

There are many ways to study surfaces: topologically, geometrically, dynamically, algebraically, and combinatorially, just to name a few. We will touch on some of the motivation for studying surfaces and their associated mapping class groups, which is the collection of symmetries of a surface. We will also describe a few of the ways that these different perspectives for studying surfaces come together in beautiful ways.

Learning latent combinatorial structures in noisy data

Series
Research Horizons Seminar
Time
Friday, October 2, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Cheng MaoGeorgia Tech

Learning latent structures in noisy data has been a central task in statistical and computational sciences. For applications such as ranking, matching and clustering, the structure of interest is non-convex and, furthermore, of combinatorial nature. This talk will be a gentle introduction to selected models and methods for statistical inference of such combinatorial structures. I will particularly discuss some of my recent research interests.

Taming the randomness of chaotic systems

Series
Research Horizons Seminar
Time
Friday, September 25, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Alex BlumenthalGeorgia Tech
All around us in the physical world are systems which evolve in chaotic, seemingly random ways: fire, smoke, turbulent fluids, the flow of gas around us. Over the last ~60 years, mathematicians have made tremendous progress in understanding these processes and how chaotic behavior can emerge and, remarkably, the extent to which chaotic systems emulate probabilistic randomness. This talk is a brief introduction to these ideas, with an emphasis on examples and pretty pictures. 

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.

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