Seminars and Colloquia by Series

Hodge theory for tropical varieties 1

Series
Algebra Seminar
Time
Wednesday, November 11, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Matthieu Piquerez

Please Note: Part 1 of 3-part series

The aim of these two talks is to give an overview of our work on tropical Hodge theory. We show that cohomology groups of smooth projective tropical varieties verify hard Lefschetz property and Hodge-Riemann relations. Providing a description of the Chow groups of matroids in terms of cohomology groups of specific smooth projective tropical varieties, these results can be regarded as a generalization of the work of Adiprasito-Huh-Katz to more general tropical varieties. We also prove that smooth projective tropical varieties verify the analogue in the tropical setting of the weight-monodromy conjecture, affirming a conjecture of Mikhalkin and Zharkov.

BlueJeans link: https://bluejeans.com/476849994

Universal graphs and planarity

Series
Graph Theory Seminar
Time
Tuesday, November 10, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Louis EsperetUniversité Grenoble Alpes

Please Note: Note the unusual time!

The following are two classical questions in the area of universal graphs.

1. What is the minimum number of vertices in a graph that contains all $n$-vertex planar graphs as induced subgraphs?

2. What is the minimum number of edges in a graph that contains all $n$-vertex planar graphs as subgraphs?

We give nearly optimal constructions for each problem, i.e. with $n^{1+o(1)}$ vertices for Question 1 and $n^{1+o(1)}$ edges for Question 2. The proofs combine a recent structure theorem for planar graphs (of independent interest) with techniques from data structures.

Joint work with V. Dujmovic, C. Gavoille, G. Joret, P. Micek, and P. Morin.

Marstrand's Theorem in general Banach spaces

Series
Analysis Seminar
Time
Tuesday, November 10, 2020 - 02:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Bobby WilsonUniversity of Washington

We will discuss Marstrand's classical theorem concerning the interplay between density of a measure and the Hausdorff dimension of the measure's support in the context of finite-dimensional Banach spaces. This is joint work with David Bate and Tatiana Toro.

Ranking from pairwise comparisons

Series
Undergraduate Seminar
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.

A Combinatorial Description of the knot concordance invariant epsilon

Series
Geometry Topology Seminar
Time
Monday, November 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Hakan DogaUniversity of Buffalo

Computing, understanding the behavior of concordance invariants obtained from knot Floer homology theories is quite central to the study of the concordance group and low-dimensional topology in general. In this talk, I will describe the method that allows us to compute the concordance invariant epsilon using combinatorial knot Floer homology and talk about some computational results. This is a joint work with S. Dey.

Counting integer partitions with the method of maximum entropy

Series
Combinatorics Seminar
Time
Friday, November 6, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Bluejeans link: https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Gwen McKinleyUniversity of California, San Diego, CA

We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.

A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

Automated Feature Extraction from Large Cardiac Electrophysiological Data Sets

Series
Mathematical Biology Seminar
Time
Friday, November 6, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Peter HinowUniversity of Wisconsin-Milwaukee

Please Note: https://bluejeans.com/819527897/5512

A multi-electrode array-based application for the long-term recording of action potentials from electrogenic cells makes possible exciting cardiac electrophysiology studies in health and disease. With hundreds of simultaneous electrode recordings being acquired over a period of days, the main challenge becomes achieving reliable signal identification and quantification. We set out to develop an algorithm capable of automatically extracting regions of high-quality action potentials from terabyte size experimental results and to map the trains of action potentials into a low-dimensional feature space for analysis. Our automatic segmentation algorithm finds regions of acceptable action potentials in large data sets of electrophysiological readings. We use spectral methods and support vector machines to classify our readings and to extract relevant features. We show that action potentials from the same cell site can be recorded over days without detrimental effects to the cell membrane. The variability between measurements 24 h apart is comparable to the natural variability of the features at a single time point. Our work contributes towards a non-invasive approach for cardiomyocyte functional maturation, as well as developmental, pathological, and pharmacological studies.

This is joint work with Dr. Viviana Zlochiver (Advocate Aurora Research Institute) and John Jurkiewicz (graduate student at UWM).

Meeting room: https://bluejeans.com/819527897/5512

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.
 

Hankel index of a projected of rational curves

Series
Student Algebraic Geometry Seminar
Time
Friday, November 6, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Jaewoo JungGeorgia Tech

Please Note: Teams meeting link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1604670786929?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

If we can write a (homogeneous) polynomial as a sum of squares(SOS), the polynomial is guaranteed to be a non-negative polynomial. However, every non-negative forms does not have to be written as sums of squares in general. This implies that set of sums of square is strictly contained in set of non-negative forms in general. We want to discuss about one way to describe the gaps between the two sets. Since the sets have cone structures, we can consider dual cones of each cones. In particular, the description of dual cone of non-negative polynomials is simple: convex hull of point evaluations. Therefore, we are interested in positive semi-definite quadratic forms that is not point evaluations. We call "Hankel index" the minimal rank of quadratic form (on extreme ray of the dual cone of SOS) which is not a point evaluation. In this talk, we introduce the Hankel index of variety and will discuss about a criterion to obtain the Hankel index of projected rational curves.

Pages