Georgia Institute of Technology College of Sciences

Baker and Lorscheid have developed a theory of foundation that characterize the representability of matroids. Justin Chen and I are developing a computer program that computes representations of matroids based on the theory of foundation. In this talk, I will introduce backgrounds on matroids and the foundation, then I will talk about the key algorithms in computing the morphisms of pastures. If possible, I will also show some examples of the program.

Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1648750292956?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

Wave turbulence is the theory of nonequilibrium statistical mechanics for wave systems. Initially formulated in pioneering works of Peierls, Hasselman, and Zakharov early in the past century, wave turbulence is widely used across several areas of physics to describe the statistical behavior of various interacting wave systems. We shall be interested in the mathematical foundation of this theory, which for the longest time had not been established.

The central objects in this theory are: the "wave kinetic equation" (WKE), which stands as the wave analog of Boltzmann’s kinetic equation describing interacting particle systems, and the "propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacks mathematical justification. Mathematically, the aim is to provide a rigorous justification and derivation of those two central objects; This is Hilbert’s Sixth Problem for waves. The problem attracted considerable interest in the mathematical community over the past decade or so. This culminated in recent joint works with Yu Deng (University of Southern California), which provided the first rigorous derivation of the wave kinetic equation, and justified the propagation of chaos hypothesis in the same setting.

Meeting also available online: https://gatech.zoom.us/j/92742811112

Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer $s$, there exists a positive integer $n$ such that for every partition of the edges of the complete graph on $n$ vertices into two classes, one of the classes must contain a complete subgraph on $s$ vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest $n$ that satisfies this property.

For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that "pseudo-randomness" and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.

Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various camera models in computer vision. Considerable attention has been paid to multiview ideals, which collect all polynomial constraints on images that must be satisfied by a given camera arrangement. We extend past work on multiview ideals to settings where the camera arrangement is unknown. We characterize various "image formation ideals", which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out from our framework. We give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. (Preliminary report based on ongoing joint projects with Sameer Agarwal, Max Lieblich, Jessie Loucks Tavitas, and Rekha Thomas.)