Georgia Institute of Technology College of Sciences

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.

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.

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.

Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs.  In work in progress with Robert Chang (Rhodes College), we introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with positive Hurst index s -1/2.  For a special Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams.  The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp.  The global Gaussian fluctuations depend on s and, if we continue s to negative values, coincides with the process in Kerov's CLT for s = - 1/2.  Although it might be possible to give a direct explanation for this coincidence by regularization, in this talk we give an indirect dynamical explanation by combining (i) results of Eliashberg and Dubrovin for a specific Hamiltonian QFT and (ii) the fact that in Hamiltonian systems, at short time scales, the quantum evolution of pure Gaussian wavepacket initial data agrees statistically with the classical evolution of mixed Gaussian random initial data.