Georgia Institute of Technology College of Sciences

We consider one of the most basic high-dimensional testing problems: that of detecting the presence of a rank-1 "spike" in a random Gaussian (GOE) matrix. When the spike has structure such as sparsity, inherent statistical-computational tradeoffs are expected. I will discuss some precise results about the computational complexity, arguing that the so-called "linear spectral statistics" achieve the best possible tradeoff between type I & II errors among all polynomial-time algorithms, even though an exponential-time algorithm can do better. This is based on https://arxiv.org/abs/2311.00289 with Ankur Moitra which uses a version of the low-degree polynomial heuristic, as well as forthcoming work with Ansh Nagda which gives a stronger form of reduction-based hardness.

This thesis introduces a modular framework written in Macaulay2 designed to solve nonlinear algebra problems.  First, we will introduce the background for the framework, covering gates, circuits, and straight-line programs, and then we will define the gates used in the framework.  The remainder of the talk will include well-known algorithms such as Newton's method and Runge-Kutta for solving nonlinear algebra problems, their implementation in the framework, and explicit conic problems with a comparison between different methods.

The classical Laudenbach-Poénaru theorem states that any diffeomorphism of $\#_n S^1 \times S^2$ extends over the boundary connect sum of $n$ $S^1 \times B^3$'s. This implies the familiar fact that in Kirby diagrams for closed 4 manifolds, you do not need to specify the attaching spheres for 3 handles; it is also the backbone result of trisection theory, which allows one to describe a closed 4 manifold by three cut systems of curves on a surface. We extend this result to the case of infinite 4-dimensional 1-handlebodies, with an eye towards developing trisections for noncompact 4 manifolds. The proof is geometric and based on extending the recent proof of Laudenbach-Poenaru due to Meier and Scott.

 In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach based on the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations.