Weekly Bulletin

This talk proposes a shift in the implicit foundations of mathematics from set theory and logic to dependent type theory (where the primitive notion of “type” replaces both sets and propositions). While these new foundations are more complex, we contend they make it easier for students to learn to write correct proofs and for professionals to communicate precise mathematical ideas to other humans or to a computer.

<p>Let \(Q\) be a positive definite quadratic form in two variables. Associated to \(Q\) we can define an exponential sum, known as a Siegel theta sum. We show that if \(Q\) satisfies a particular diophantine condition, then we have a functional limit theorem for normalised theta sums, that is, convergence to a limiting stochastic process. </p>

Let G be a compact Lie group. Let n be an integer. Given n elements g₁, …, g_n in G, one can choose one of the g_i and replace it into its inverse, or into the product g_ig_j for some j distinct from i; these basic operations do not change the subgroup of G generated by the g_i. Doing so, one gets a group of transformations of Gⁿ (generated by all these basic operations). The problem I will discuss is: what can be said on the dynamics of this group action on Gⁿ? For instance, can we describe the closures of orbits, or the invariant probability measures? As we shall see, the answer is yes when n becomes large, in fact larger than an explicit constant that depends on G. This is based on a joint work with Christophe Dupont and Florestan Martin-Baillon, which complements previous works of Tsachik Gelander.

When does the behaviour of a single eigenfunction determine the shape of a domain? Schiffer’s conjecture predicts that a bounded, simply connected planar domain must be a disc if it admits a nonconstant Neumann eigenfunction that is constant on the boundary. I will present joint work with George Stepaniants disproving this conjecture by constructing a noncircular domain with real-analytic boundary. The same example disproves the planar Pompeiu conjecture, connecting spectral geometry to a problem about averages over translated and rotated domains dating back to 1929. The challenge is not merely to find a convincing numerical picture, but to prove that an exact domain and eigenfunction exist. I will explain how conformal mapping, spectral analysis and a contraction argument turn a numerical candidate into a theorem. The crucial step is to control everything a finite computation leaves out, including the infinitely many omitted modes, while rigorously accounting for numerical error. I will then draw on recent work on eigenvalue certification and the conjectures of Forsythe and Kenig to discuss the changing role of computers and AI in mathematical discovery. These examples range from assistance with code to the discovery of proof mechanisms and formal verification in Lean. They also expose persuasive arguments that fail. I will distinguish the roles of exploration, mathematical judgment and independent verification, and discuss how these tools are changing which problems are practical to pursue. No background in computer-assisted proof or AI will be assumed.

Recent breakthroughs in quantum computers have shown quantum advantage (aka quantum supremacy), that is, quantum computers outperform classical computers for solving specific problems. More importantly, there are already general-purpose programmable quantum computing devices available to the public, for example, IBM Quantum Experience, Microsoft Quantum, and Amazon Braket. Such quantum devices are commonly known as noisy intermediate-scale quantum (NISQ)devices. It has been reported that NISQ with 1,000 qubits has been released. Despite the success of quantum computers, the problems investigated are highly physics-oriented and may not necessarily appeal to researchers in other fields, such as statistics and data science. The key challenge in leveraging quantum advantages in these fields is identifying practical applications in which integrating statistical analysis and quantum computing can effectively overcome computational bottlenecks. In this talk, I will present some of our recent work on best model selection, which exhibits excellent performance and computational efficiency in applications, particularly in the analysis of single-cell multi-omics data. Bio: Professor Ping Ma is a Distinguished Research Professor at the University of Georgia and co-directs the big data analytics lab. His research interests include statistics, machine learning, and their applications in biology, physics, and engineering. He was a Beckman Fellow at the Center for Advanced Study at the University of Illinois at Urbana-Champaign, a Faculty Fellow at the US National Center for Supercomputing Applications, and a recipient of the National Science Foundation CAREER Award. His paper won the best paper award from the Canadian Journal of Statistics in 2011. He delivered the 2021 National Science Foundation Distinguished Lecture. Professor Ma serves on multiple editorial boards. He is a Fellow of the American Association for the Advancement of Science and the American Statistical Association.

We will discuss exponential-utility maximization in discrete-time Gaussian markets. The Gaussian structure reduces the problem to a matrix optimization problem and gives an explicit characterization of the optimal strategy and value. We will illustrate the approach with delayed trading and AR(1) models, and then focus on fractional Brownian motion. In the high-frequency limit, spectral methods lead to explicit asymptotics for the optimal utility and a scaling limit for the optimal strategy.