# Paul Larson (Miami University)

September 30, 2021 at Harvard

Title: Geometric set theory

Abstract: The field of Geometric Set Theory studies structures on sets of countable objects (typically Polish spaces) by considering virtual objects, typically uncountable sets representing members of the space under consideration in some larger model of set theory. This approach can be used to study analytic equivalence relations on Polish spaces, where the the virtual objects represent equivalence classes. The representatives of the virtual classes can be used for instance to prove non-reducibility results between such equivalence relations. Another set of applications involves separating forms of the Axiom of Choice, specifically forms asserting the existence of a set of reals with certain first order properties. Typical examples include Vitali sets, Hamel bases, discontinuous homorphisms on the real line or countable colorings of various graphs on Euclidean space. We will give a brief tour of the some of the landmarks in the area, and discuss some directions for further research.

# Lisa Sauermann (MIT)

October 7, 2021 at Harvard (Science Center Hall A)

Title: Finding distinct-variable solutions to linear equations in F_p^n

Abstract: For a fixed prime p and large n, what is the largest size of a subset of F_p^n which does not contain a non-trivial solution to some given linear equation or system of linear equations? This is a fundamental question in additive combinatorics with a long history. While there are various different notions of “non-trivial solutions”, in this talk we say that a solution is non-trivial if all of its variables are distinct.

A particularly famous and important instance of the question above is the problem of bounding the largest size of a subset of F_p^n which does not contain a three-term arithmetic progression. In 2016, Ellenberg and Gijswijt made a breakthrough on this problem and the approach in their proof was later generalized by Tao yielding what is now called the slice rank polynomial method. Unfortunately, for other instances of the question above, the slice rank polynomial method cannot handle the condition for the variables in a non-trivial solution to be distinct.

In this talk, we will first give a brief survey on the slice rank polynomial method and some of its applications, and we will then discuss two different results concerning the question above. These results combine the slice rank polynomial method with additional combinatorial ideas in order to handle the distinctness condition. We also discuss an application of one of these results to the Erdös-Ginzburg-Ziv problem in discrete geometry.

# Siu-Cheong Lau (Boston University)

October 14, 2021 at Brandeis

Title: Noncommutative deformations of crepant resolutions via mirror symmetry

Abstract: Noncommutative crepant resolutions of singularities formulated by Van den Bergh admit interesting quantization deformations. On the other hand, nc deformations of crepant resolutions can also be constructed via a local-to-global approach using the notion of an algebroid stack. In this talk, I will explain a mirror method of constructing explicit nc deformations of crepant resolutions, and a Fourier-Mukai transform between these two notions of nc deformations. An important ingredient is a certain class of Lagrangian objects in the mirror manifold, whose (higher) morphisms can be found via a 3d enhancement of the corresponding objects in Riemann surfaces.

# John Pardon (Princeton)

October 21, 2021 at MIT

Title: Enough vector bundles on orbispaces

Abstract: An orbispace is a “space” which is locally the quotient of a topological space by a continuous action of a finite group. Familiar examples of orbispaces include orbifolds, (the analytifications of) Deligne–Mumford stacks over C, and moduli spaces of solutions to elliptic partial differential equations, as they appear in low-dimensional and symplectic topology. The fibers of a vector bundle over an orbispace are representations of its stabilizer groups. When do there exist vector bundles all of whose fiber representations are faithful? This condition is called having “enough” vector bundles, and plays an important role in the stable homotopy theory of orbispaces. In particular, it implies a Spanier–Whitehead duality functor in a certain stable homotopy category of orbispaces and a Pontryagin–Thom isomorphism for orbifold bordism groups.

# Evita Nestoridi (Princeton)

October 22, 2021 at MIT (note unusual date)

Title: Spectral techniques in Markov chain mixing

Abstract: How many steps does it take to shuffle a deck of $n$ cards, if at each step we pick two cards uniformly at random and swap them? Diaconis and Shahshahani proved that $\frac{1}{2} n log n$ steps are necessary and sufficient to mix the deck. Using the representation theory of the symmetric group, they proved that this random transpositions card shuffle exhibits a sharp transition from being unshuffled to being very well shuffled. This is called the cutoff phenomenon. In this talk, I will explain how to use the spectral information of a Markov chain to study cutoff. As an application, I will briefly discuss the random to random card shuffle (joint with M. Bernstein) and the non-backtracking random walk on Ramanujan graphs (joint with P. Sarnak).

# Benjamin Bakker (UIC)

November 4, 2021 at Northeastern

Title: Period integrals of algebraic varieties

Abstract: A period integral of a complex algebraic variety is the integral of an algebraic differential form along a topological cycle. While such integrals are transcendental in nature, they have had pervasive importance in algebraic and arithmetic geometry going back to the foundational work of Abel and Jacobi. The functions one obtains from taking period integrals in families of algebraic varieties moreover recur in many areas of mathematics. In this talk I will first survey how algebraic information can be extracted from period integrals, as well as their connection to various conjectural frameworks. I will then discuss recent progress on understanding the algebraic properties of period integrals in families and recent applications to Hodge theory, algebraic geometry, arithmetic geometry, and logic.

The colloquium meets (by default) on Thursdays at 4:30 PM Eastern in varying modalities (contact institutional organizers for details). The organizers include Bong Lian at Brandeis; Fabian Gundlach, Myrto Mavraki, and Assaf Shani at Harvard; Scott Sheffield at MIT; and Matthew Hogancamp, Ben Knudsen, Gabor Lippner, and Jonathan Weitsman at Northeastern. This website is maintained by Ben Knudsen. The image of Boston is the property of Wikimedia user King of Hearts and is reproduced here under Creative Commons license CC BY-SA 4.0. Images of speakers are their own property and are reproduced here with permission.