Mathematics & Statistics Colloquium

Keith Merrill (Brandeis University): Formal privacy definitions in official statistics
February 14, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: The concept of privacy has undergone significant changes in the last century, particularly in light of the preponderance of data collected about individuals in modern society. In this talk, we will discuss some famous examples of "privacy violations," how they have shaped modern notions of privacy, and how official statistical releases must try to balance the inherent trade-off between accuracy of data and the privacy guarantees made to data subjects. We will recount the path towards a formal mathematical definition of what it means for a data release to be "private." We will also see how such definitions interact with numerous laws and other obligations by data curators. The result is an emerging subfield of computer science which is a fascinating mix of mathematics, statistics, psychology, and law.

This talk should be accessible to a wide audience, no prior knowledge of the field is assumed.
Keith Merrill's Website

Adam Wagner (Worcester Polytechnic Institute): Reinforcement learning and pattern finding in combinatorics
February 28, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: We will look at two ways we can use tools from machine learning to help us with research in combinatorics. First we discuss reinforcement learning, a method that gives us a way to check conjectures for counterexamples efficiently. While it usually does not perform as well as other simpler methods, there have been several examples of projects in the past few years where RL was crucial for success. In the second half of the talk we will consider the following question of Ellenberg: at most how many points can we pick in the N by N grid, without creating an isosceles triangle? The best known constructions, found by computer searches for small values of N, clearly follow a pattern which we do not yet understand. We will discuss how one can train transformers to understand this pattern, and use this trained transformer to help us find a bit better constructions for various N.

This is joint work with Jordan Ellenberg, Marijn Heule, and Geordie Williamson.
Adam Wagner's Website

Jeff Schenker (Michigan State University): Theory of Ergodic Quantum Processes
March 13, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: An "open" quantum system has non-trivial interactions with its environment. In an open system, states are no longer represented by vectors in a Hilbert space and the nature of the dynamical evolution of the changes dramatically. In place of the unitary dynamics of a closed system, the evolution of an open quantum system is described by completely positive, trace preserving maps on the mixed states, represented by density matrices. In this talk, we will review the background of open quantum systems and their description and introduce the notion of a "quantum process," as a sequence of quantum channels. If the process is given by repetition of a single channel with decoherence, it drives the system to a unique equilibrium state, a result which follows from a generalization of the Perron-Frobenius theorem. Finally, we will "ergodic quantum processes" composed of stochastic channels with arbitrary correlations and non-negligible decoherence and present a recent theorem which shows that such a process drives the system to a unique fluctuating equilibrium with ergodic time dependence. Applications and extensions to repeated measurement of quantum systems will be discussed if time permits.
Jeff Schenker's Website

Lior Alon (Massachusetts Institute of Technology): Fourier Quasicrystals via Lee-Yang Polynomials
March 27, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: The concept of "quasi-periodic" sets, functions, and measures is prevalent in diverse mathematical fields such as Mathematical Physics, Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for periodicity of discrete sets, and a Fourier Quasicrystals (FQ) generalizes this notion of periodicity: a counting measure of a discrete set is called a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition.
Recently Kurasov and Sarnak provided a method for constructing one-dimensional FQs as the intersections of an irrational line in the torus with the zero set of a multivariate Lee-Yang polynomial. In this talk, I will show that the Kurasov-Sarnak construction generates all one-dimensional FQs. I will also discuss the distribution of gaps between atoms in such FQs, showing that the countably many gaps equidistribute on an interval, with a distribution given explicitly in terms of ergodic dynamical systems on tori. In the last part, I will present a generalization of the Kurasov-Sarnak construction to any dimension, by introducing Lee-Yang varieties.
The talk is aimed at a broad audience, no prior knowledge in the field is assumed. Based on joint works with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov.
Lior Alon's Website

Dominique Maldague (Massachusetts Institute of Technology): Sharp square function estimates in Fourier restriction theory
April 10, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: This talk will provide an overview of recent developments in Fourier restriction theory, which is the study of exponential sums over restricted frequency sets with geometric structure, typically arising in pde or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums. I will highlight recent work which uses the latest tools developed in decoupling theory to prove much more delicate sharp square function estimates for frequencies lying in the cone in R^3 (Guth-Wang-Zhang) and moment curves (t,t^2,...,t^n) in all dimensions (Guth-Maldague).
Dominique Maldague's website