My mathematics seminar, Summer 2023

Overview

Schedule

• Jun. 27: I will give a gentle introduction to the 2-category which corepresents adjunctions and describe some of the surprising features which it exhibits upon stabilization. I will also discuss the Ayala-Francis calculation of its trace, and relations to the 2-category of spherical adjunctions.
• Jul. 11: Nick Addington and I have begun exploring the notion of a Quillen-Lichtenbaum (QL) dimension of a complex variety and its implications to birational geometry. For example, the QL dimension of projective space/rational variety is zero, the Barlow surface is QL dim zero but not rational, the QL dimension of a rationally connected 3-fold is 2 and a host of other examples and the cubic 4 fold is zero (so we failed to prove irrationality of the generic cubic 4-fold). This dimension is also defined in characteristic p > 0. I want to outsource some problems around this area to the interested audience.
• Jul. 18: Good moduli spaces!
• Jul. 25: Recently, Marc Levine gave an algebraic construction of orientation data for the Hilbert scheme of 0-dimensional subschemes of a smooth projective threefold, dependent on a chosen square root of the canonical sheaf. Using this orientation and recent work of Kass—Levine—Solomon—Wickelgren on A^1-degrees of finite flat maps, Levine is able to construct an “enriched” Donaldson—Thomas invariant. In this talk, I will give a survey of these results and the relevant technical background.
• Aug 1:Algebraic solutions to the classical Painleve VI differential equations turn out to classify 2-dimensional representations of a 4-punctured sphere whose conjugacy class has finite orbit under the diffeomorphism group of the punctured sphere. After a gargantuan effort, these solutions have been classified. A natural generalization is to ask for a classification of 2-dimensional representations of n-punctured with the same property. These correspond to "Garnier integrable systems." We will explain some joint work with Daniel Litt and Josh Lam, where we classify representations such that one of the loops around the punctures has infinite order under the representation.
• Aug 8:Kronecker once wrote that "God made the integers; all else is the work of man". Lars Hesselholt has proclaimed that the integers are a "twenty-thousand-year old mistake", and that the sphere spectrum should instead encode the act of counting. Taking these two at face value, one is led to the belief that homotopy theory might help us organize the avalanche of ideas resulting from that one fatal "error" twenty-thousand years ago in an interesting/unexpected way. Motivated by this, I would like to tell you about two (somewhat disjoint) ideas I have recently been thinking about: the utility of the chromatic elements p, v_1, v_2, ... in mod p Hodge theory; and chromatic probings of reductive algebraic groups.