All talks will take place in Stevenson Center 1308.
(This is on the ground level of the Mathematics Building which is called Stevenson Center 1)
Reception and coffee breaks will take place in Stevenson Center 1425
Saturday, April 13:
9:00-10:00: Registration; coffee and bagels
10:00-11:00: Dan Freed
11:00-11:30: Coffee break
11:30-12:30: Kirsten Wickelgren
2:30-3:30: Bert Guillou
3:30-4:00: Coffee break
4:00-5:00: Inna Zakharevich
Sunday, April 14:
9:30-10:30: Emily Riehl
10:30-11:00: Coffee break
11:00-12:00: Craig Westerland
Dan Freed (University of Texas)
Title: The two-dimensional Ising model and topological field theory
Abstract: I will describe joint work with Constantin Teleman, in which we cast topological eyes on a well-studied system in condensed matter physics. In particular, we use the symmetry in a strong form and apply the technology of extended topological field theory to construct proofs and make predictions.
Kirsten Wickelgren (Georgia Tech University)
Title: Wild Ramification in an enriched Riemann–Hurwitz formula
Abstract: For a map f: X -> Y between curves, the Riemann–Hurwitz formula relates the Euler characteristics of X and Y to the degree of f and information about the ramification of f. In A^1-homotopy theory, the Euler characteristic takes values in the Grothendieck–Witt group of quadratic forms. M. Levine proved an enrichment of the classical Riemann–Hurwitz formula to an equality in the Grothendieck–Witt group. In its strongest form,
Levine’s theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann–Hurwitz formula, and consider an example suggested by S. Saito. This is joint work with Candace Bethea and Jesse Kass.
Bert Guillou (University of Kentucky)
Title: From structured G-categories to ring and module G-spectra
Abstract: Several machines were built in the 1970’s for producing a stable homotopy type, or spectrum, K(C) from a category C with a sufficiently nice product. A second product on C gives K(C) the extra structure of a ring spectrum. I will describe a variation on this construction that extends to the equivariant world, producing ring and module G-spectra from appropriately structured G-categories (for finite G). This is joint work with J. P. May, M. Merling, and A. Osorno.
Inna Zakharevich Cornell University
Title: Quillen’s devissage in geometry
Abstract: In this talk we discuss a new perspective on Quillen’s devissage theorem. Originally, Quillen proved devissage for algebraic K-theory of abelian categories. The theorem showed that given a full abelian subcategory A of an abelian category B, K(A) is equivalen to K(B) if every object of B has a finite filtration with quotients lying in A. This allows us, for example, to relate the K-theory of torsion Z-modules to the K-theories of F_p-modules for all p. Generalizations of this theorem to more general contexts for K-theory, such as Walhdausen categories, have been notoriously difficult; although some such theorems exist they are generally much more complicated to state and prove than Quillen’s original. In this talk we show how to translate Quillen’s algebraic approach to a geometric context. This translation allows us to construct a devissage theorem in geometry, and prove it using Quillen’s original insights.
Emily Riehl (Johns Hopkins University)
Title: The synthetic theory of ∞-categories
Abstract: The pioneering work of Joyal, Lurie, et al to extend ordinary category theory to the setting of ∞-categories is “analytic,” with the precise statements of theorems given in reference to a particular model (quasi-categories) and proofs drawing on the combinatorics of simplicial sets. This talk will describe joint work with Dominic Verity that reveals that much of that theory can be redeveloped “synthetically” in an axiomatic framework that is natively “model-independent,” casting new light on the theory of quasi-categories while simultaneously generalizing it to other models. At the conclusion, we consider the question of the model-invariance of ∞-category theory, proving that ∞-categorical structures are preserved, reflected, and created by a number of “change-of-model” functors. As we explain, it follows that even the “analytically-proven” theorems that exploit the combinatorics of one particular model remain valid in the other “biequivalent” models.
Craig Westerland (University of Minnesota)
Title: Topology and arithmetic statistics
Abstract: here are many questions in number theory and arithmetic geometry of the sort “Does the following situation ever occur?” For instance, the inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rationals. Similarly, one might ask whether one expects the rank of elliptic curves to be unbounded.
Arithmetic statistics, broadly speaking, pursues the more quantitative question of how often such situations occur. The extension of the inverse Galois problem to this setting is a conjecture of Malle’s, which predicts an asymptotic formula for the number of occurrences of a given finite group G as the Galois group of a number field, as a function of the discriminant. There are analogous statistical conjectures regarding the distribution of class groups ordered by discriminant (e.g., the Cohen-Lenstra heuristics), or the rank of elliptic curves ordered by height (Katz-Sarnak).
In this talk, we will give an introduction to these sort of questions, focusing on Malle’s conjecture. Additionally, we will explain how to formulate function field analogues of this conjecture and transform this conjecture into a problem in algebraic topology (about the homology of certain moduli spaces of branched covers of P^1). In joint work with Ellenberg and Tran, we partially solved this problem, giving the upper bound in Malle’s conjecture.