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Series: Algebra Seminar

Problems from enumerative geometry have Galois groups. Like those from field extensions, these Galois groups reflect the internal structure of the original problem. The Schubert calculus is a class of problems in enumerative geometry that is very well understood, and may be used as a laboratory to study new phenomena in enumerative geometry.I will discuss this background, and sketch a picture that is emerging from a sustained study of Schubert problems from the perspective of Galois theory. This includes a conjecture concerning the possible Schubert Galois groups, a partial solution of the inverse Galois problem, as well as glimpses of the outline of a possible classification of Schubert problems for their Galois groups.

Friday, March 15, 2019 - 12:00 ,
Location: Skiles 006 ,
Trevor Gunn ,
Georgia Tech ,
Organizer: Trevor Gunn

I will introduce briefly the notion of Berkovich analytic spaces and certain metric graphs associated to them called the skeleton. Then we will describe divisors on metric graphs and a lifting theorem that allows us to find tropicalizations of curves in P^3. This is joint work with Philipp Jell.

Series: ACO Alumni Lecture

Hadwiger (Hajos and Gerards and Seymour, respectively) conjectured that the vertices of every graph with no K_{t+1} minor (topological minor and odd minor, respectively) can be colored with t colors such that any pair of adjacent vertices receive different colors. These conjectures are stronger than the Four Color Theorem and are either wide open or false in general. A weakening of these conjectures is to consider clustered coloring which only requires every monochromatic component to have bounded size instead of size 1. It is known that t colors are still necessary for the clustered coloring version of those three conjectures. Joint with David Wood, we prove a series of tight results about clustered coloring on graphs with no subgraph isomorphic to a fixed complete bipartite graph. These results have a number of applications. In particular, they imply that the clustered coloring version of Hajos' conjecture is true for bounded treewidth graphs in a stronger sense: K_{t+1} topological minor free graphs of bounded treewidth are clustered t-list-colorable. They also lead to the first linear upper bound for the clustered coloring version of Hajos' conjecture and the currently best upper bound for the clustered coloring version of the Gerards-Seymour conjecture.

Series: Other Talks

Chun-Hung will discuss his employment experience as an ACO alummus. The conversations will take place over coffee.

Series: Stochastics Seminar

Series: Graph Theory Working Seminar

Erdős
and Nešetřil conjectured in 1985 that every graph with maximum degree Δ
can be strong edge colored using at most 5/4 Δ^2 colors. The conjecture
is still open for Δ=4. We show the conjecture is true when an edge cut
of size 1 or 2 exists, and in certain cases when an edge cut of size 4
or 3 exists.

Series: High Dimensional Seminar

We will try to address a few universality questions for the behavior of large random matrices over finite fields, and then present a minimal progress on one of these questions.

Series: Analysis Seminar

The Bishop-Phelps-Bolloba ́s property for numerical radius says that if we have a point in the Banach space and an operator that almost attains its numerical radius at this point, then there exist another point close to the original point and another operator close to the original operator, such that the new operator attains its numerical radius at this new point. We will show that the set of bounded linear operators from a Banach space X to X has a Bishop-Phelps-Bolloba ́s property for numerical radius whenever X is l1 or c0. We will also discuss some constructive versions of the Bishop-Phelps- Bolloba ́s theorem for l1(C), which are an essential tool for the proof of this result.

Series: PDE Seminar

The Euler-Alignment system arises as a macroscopic representation of the Cucker-Smale model, which describes the flocking phenomenon in animal swarms. The nonlinear and nonlocal nature of the system bring challenges in studying global regularity and long time behaviors. In this talk, I will discuss the global wellposedness of the Euler-Alignment system with three types of nonlocal alignment interactions: bounded, strongly singular, and weakly singular interactions. Different choices of interactions will lead to different global behaviors. I will also discuss interesting connections to some fluid dynamics systems, including the fractional Burgers equation, and the aggregation equation.

Series: Dissertation Defense

The length LC_n of the longest common subsequences of two strings X = (X_1, ... , X_n) and Y = (Y_1, ... , Y_n) is a way to measure the similarity between X and Y. We study the asymptotic behavior of LC_n when the two strings are generated by a hidden Markov model (Z, (X, Y)) and we build upon asymptotic results for LC_n obtained for sequences of i.i.d. random variables. Under some standard assumptions regarding the model we first prove convergence results with rates for E[LC_n]. Then, versions of concentration inequalities for the transversal fluctuations of LC_n are obtained. Finally, we outline a proof for a central limit theorem by building upon previous work and adapting a Stein's method estimate.