# Craig Westerland (University of Minnesota)

January 20, 2022 at Northeastern (virtual)

Title: Braids and Hopf algebras

Abstract: The Milnor–Moore theorem identifies a large class of Hopf algebras as enveloping algebras of the Lie algebras of their primitives. If we broaden our definition of a Hopf algebra to that of a braided Hopf algebra, much of this structure theory falls apart. The most obvious reason is that the primitives in a braided Hopf algebra no longer form a Lie algebra. In this talk, we will discuss recent work to understand what precisely is the algebraic structure of the primitives in a braided Hopf algebra in order to “repair” the Milnor–Moore theorem in this setting. It turns out that this structure is closely related to the dualizing module for the braid groups, which implements dualities in the (co)homology of the braid groups.

# Eric Sommers (UMass Amherst)

March 17, 2022 at Brandeis

Title: Hessenberg varieties and the geometric modular law

Abstract: The chromatic symmetric functions of Stanley that are associated to Dyck paths satisfy a modular law introduced by Guay–Paquet. This law was used in one proof of the Shareshian–Wachs conjecture: the graded version of the chromatic symmetric function characterizes Tymoczko’s symmetric group representation on the cohomology of a Hessenberg variety. The latter object can be generalized to any simple Lie algebra. This talk concerns various topics surrounding Hessenberg varieties in the general Lie theory setting, including a geometric version of the modular law and the decomposition of certain perverse sheaves into their simple constituents. Along the way, we explain a proof of a conjecture of Brosnan. This is joint work with Martha Precup.

# Alexander Goncharov (Yale)

March 31, 2022 at Brandeis

Title: Arithmetic analysis

Abstract: Historically, a large part of Algebraic Geometry was developed by Abel – Riemann – Jacobi – … to understand the properties of integrals. I will explain new methods which allow to make precise predictions about a wide class of integrals without calculating them. These methods, which sound quite elementary, are based on deep ideas of the arithmetic theory of motives, such as motivic Galois groups, originated by Grothendieck, Deligne, Beilinson, … Application of arithmetic analysis to the simplest kinds of iterated integrals revealed surprising connections to the geometry of modular curves and more generally locally symmetric spaces. During the last 10+ years, arithmetic analysis was successfully used to understand/calculate scattering amplitudes in Quantum Field Theory. It is likely that it will become a part of the mathematical structure describing QFT’s.

# Jennifer Hom (Georgia Tech)

April 21, 2022 at Northeastern (virtual)

Title: Homology cobordism and Heegaard Floer homology

Abstract: Under the operation of connected sum, three-manifolds form a monoid. Modulo an equivalence relation called homology cobordism, this monoid (of homology spheres) becomes a group. What is the structure of this group? What families of three-manifolds generate (or don’t generate) this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with (various subsets of) I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.

# Colin Guillarmou (Université Paris-Saclay)

April 26, 2022 at MIT

Title: Mathematical resolution of the Liouville conformal field theory

Abstract: Conformal field theory is a field theory with conformal symmetries. This theory was developed in the eighties in relation to statistical physics and to string theory. It inspired a lot of interesting mathematical questions, going from vertex operator algebras and representation theory to the theory of random geometries. In this talk, I will explain several aspects of conformal field theory (CFT) in dimension 2, and we will review recent results we obtained in a particular case called Liouville CFT with Kupiainen, Rhodes, Vargas and Baverez. For this model, involving random Riemann surfaces, we are able to perform a mathematically rigorous proof of the conformal bootstrap that allows us to obtain formulas for all correlation functions and all surfaces in terms of conformal blocks and 3-point correlation functions. This work is based on the use of probability, scattering theory, and an explicit representation of the Virasoro algebra involving the theory of Gaussian multiplicative chaos. I will aim for the talk to be accessible to non-specialists.

# Matt Kerr (Washington University)

May 5, 2022 at Harvard

Title: Higher normal functions and irrationality proofs

Abstract: R. Apéry’s 1978 proof of the irrationality of ζ(3) relied upon two sequences of rational numbers whose ratio limits to ζ(3) very quickly. Beukers and Peters discovered in 1984 that the generating function of the first sequence was a period of a family of K3 surfaces. The corresponding algebro-geometric interpretations for the second generating function and the limit, however, have been missing until recently. Normal functions are certain “well-behaved’’ sections of complex torus bundles, first studied by Poincaré and Lefschetz. They arise in particular from algebraic cycles (formal sums of subvarieties) on families of complex algebraic manifolds. A more general notion of cycles, due to Bloch and Beilinson and closely related to algebraic K-theory and motivic cohomology, leads to generalizations called “higher normal functions”. Both sorts of cycles are found lurking beneath many an arithmetic or functional property of periods. In this talk, we offer a brief tour of their unexpected role in Apéry’s proof, and in a more general circle of objects surrounding it, including motivic Gamma functions, Feynman integrals, and Fano/LG model mirror symmetry. (No knowledge of algebraic cycles will be assumed.)

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.