Department Colloquium

Abstract : There is a constant stream of claims about the ability of AI tools to "do math," often interpreted as the ability to provide natural language proofs to well-specified questions. The methods by which such proofs are produced are often opaque, because the systems used to produce them…

We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any…

Abstract: Three years ago, the best AI models were benchmarked on middle school mathematics. Now, they are regularly solving research math problems (albeit relatively simple ones ... so far). It seems inevitable that AI will redefine the mathematics profession. I will survey the development of…

Title: Cubic Weyl sums in 2-D from the Fourier restriction perspective

Abstract: We consider cubic Weyl sums of the form \sum_{n=N}^{2N} e(x\cdot(n,n^3)), where x lies in the unit square [0,1]^2. It is expected that for many x, such sums exhibit square root cancellation behavior, but…

A central theme of arithmetic Ramsey theory is that every finite coloring of the natural numbers should contain monochromatic solutions to certain algebraic equations. While Rado's theorem gives a complete understanding in the linear setting, the nonlinear case remains largely mysterious. A…

Title: A survey of Stein's restriction conjecture

Abstract: Stein's Restriction conjecture concerns functions whose Fourier transform is supported on the unit sphere in R^n. Over the decades, progress on this problem has drawn on tools from combinatorics, real algebraic geometry, and other…

It is conjectured that many models of statistical mechanics have a rich, fractal-like behaviour at and near their points of phase transition, with power-law scaling governed by critical exponents that are expected to depend on the dimension but not on the small-scale details of the model such as…

Abstract: This talk presents recent work on understanding certain solutions of PDE by combining modern mathematics with classical analysis. Machine learning, particularly Physics-Informed Neural Networks (PINNs), is being applied to discover new solutions to nonlinear PDEs with high accuracy. A…

I will explain a proof of the BCFW triangulation conjecture which states that the cells appearing in the Britto–Cachazo–Feng–Witten (BCFW) recursion triangulate the amplituhedron (in full generality at all loop levels). The key ingredient is a relation to …

Abstract: Over the past year, Aristotle, a new system combining formal methods and language modeling, has achieved gold-medal level performance at the IMO, solved open conjectures, and opened up a strange new way of working with math. This talk will explain the technology behind Aristotle and…