Calendar

In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

Given an arithmetic subgroup Γ in a semi-simple Lie group G, the multiplicity of an irreducible representation of G in L^2(Γ\G) is unknown in general.
We observe the multiplicity of any discrete series representation pi of SL (2, R) in L^2 (Γ(n)\SL (2, R)) is close to the von Neumann dimension of pi over the group algebra of Γ(n).
We extend this result to other Lie groups and bounded families of irreducible representations of them.
By applying the trace formulas, we show the multiplicities are exactly the von Neumann dimensions if we take certain towers of descending lattices in some Lie groups.

Please see website for more details: www.math.harvard.edu/~ctm/sem.

Deligne connects the weight-zero compactly supported cohomology of a moduli space to the combinatorics of its compactifications. This gives a method for using combinatorics to compute a piece of the cohomology of a moduli space. In this talk, we discuss this method for the cases of the moduli space of abelian varieties and the moduli space of n-marked hyperelliptic curves.

===============================

For more info, see https://math.mit.edu/combin/

The theory of “random surfaces” has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics. Just as “Brownian motion” is a special kind of random path, there is a similarly special kind of random surface.

Random surfaces are often motivated by physics: statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory. But even without that motivation, one may be drawn to wonder what a “typical” two-dimensional manifold looks like, or how one can make sense of that question.

I will give an overview of what this theory is about, including many computer simulations and illustrations. In particular, I will discuss the so-called Liouville quantum gravity surfaces, and explain how they are approximated by discrete random surfaces called random planar maps.

===============================

https://people.math.harvard.edu/~gammage/ons/

A central goal of condensed matter physics is to understand which quantum phases of matter can emerge in a quantum material. For this purpose, one should be able to not only describe the quantum phases using some effective field theories, but also capture the important microscopic information of the material via mathematical formulation. In this talk, I will present a framework to classify quantum phases in quantum materials, where the microscopic information of a material is encoded in its quantum anomaly. I will talk about the application of this framework to classify various exotic quantum phases of matter in different lattice systems. Using our framework, we have obtained many results unexpected from the previous literature.

Zoom: https://harvard.zoom.us/j/977347126
Password: cmsa

Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a “translational tile”, without any positive measure overlaps.
Can we determine whether a given set is a translational tile? Does any translational tile admit a periodic tiling? A well known argument shows that these two questions are closely related. In the talk, we will discuss this relation and present some new developments, joint with Terence Tao, establishing answers to both questions.

===============================

For more info, see https://math.mit.edu/combin/