Georgia Institute of Technology College of Sciences

A Springer fiber is the set of complete flags in Cn which are fixed by a given nilpotent matrix. It is a fundamental object of study in geometric representation theory and algebraic combinatorics. The irreducible components of a Springer fiber are indexed by combinatorial objects called standard Young tableaux. It is an open problem to describe geometric properties of these components (such as their singular loci and cohomology classes) in terms of the combinatorics of tableaux. We initiate a new approach to this problem by characterizing which irreducible components are equal to Richardson varieties, which are comparatively much better understood. Another motivation comes from Lusztig's recent study of the cell decomposition of the totally nonnegative part of a Springer fiber into totally positive Richardson cells. This is joint work in progress with Martha Precup.

Neural networks have become powerful tools for solving Partial Differential Equations (PDEs), with wide-ranging applications in engineering, physics, and biology. In this talk, we explore the performance of deep neural networks in solving PDEs, focusing on two primary sources of error: approximation error, and generalization error. The approximation error captures the gap between the exact PDE solution and the neural network’s hypothesis space. Generalization error arises from the challenges of learning from finite samples. We begin by analyzing the approximation capabilities of deep neural networks, particularly under Sobolev norms, and discuss strategies to overcome the curse of dimensionality. We then present generalization error bounds, offering insight into when and why deep networks can outperform shallow ones in solving PDEs.

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

TBD

We consider the strong parity edge coloring problem, which aims to color the edges of a graph G so that in each open walk on G, some color appears an odd number of times.  We show that this problem is equivalent to the problem of embedding a graph in a vector space over F2 so that the number of difference vectors attained at the edges is minimized. Using this equivalence, we achieve the following:

1. We characterize graphs on n vertices that can be embedded with ceil(log_2 n) difference vectors, answering a question of Bunde, Milans, West, and Wu.

2. We show that the number of colors needed for a strong parity edge coloring of K_{s,t} is given by the Hopf-Stiefel function, confirming a conjecture of Bunde, Milans, West, and Wu.

3. We find an asymptotically optimal embedding for the power of a path.

This talk is based on joint work with Sergey Norin and Doug West.

Work of Levin and Przytycki shows that if two non-special rational
functions f and g of degree $> 1 $over $\mathbb{C}$ share the same set of
preperiodic points, there are $m$, $n$, and $r$ such that $f^m g^n = f^r$.
In other words, $f$ and $g$ nearly commute.  One might ask if there are
other sorts of relations non-special rational functions $f$ and $g$ over $\mathbb{C}$
might satisfy when they do not share the same set of preperiodic
points.  We will present a recent proof of Beaumont that shows that
they may not, that if f and g do not share the same set of preperiodic
points, then they generate a free semi-group under composition.  The
proof builds on work of Bell, Huang, Peng, and the speaker, and uses a
ping-pong lemma similar to the one used by Tits in his proof of the
Tits alternative for finitely generated linear groups.

Given a graph $F$, a graph $G$ is weakly $F$-saturated if all non-edges of $G$ can be added in some order so that each new edge introduces a copy of $F$. The weak saturation number $wsat(n,F)$ is the minimum number of edges in a weakly $F$-saturated graph on $n$ vertices. Bollobás initiated the study of weak saturation in 1968 to study percolation processes, which originated in biology and have applications in physics and computer science. It was shown by Alon that for each $F$, there is a constant $w_F$ such that $wsat(n,F) = w_F n + o(n)$. We characterize all possible rational values of $w_F$, proving in particular that $w_F$ can equal any rational number at least $3/2$. The techniques involve a combination of random and deterministic constructions and structural methods. Joint work with Xiaoyu He.