Fall 2023 Algebraic topology seminar

In the 1990s, Kontsevich introduced certain rational chain complexes by an explicit presentation, known as graph complexes. The come in a few flavors and have shown up in different parts of mathematics. I will recall these chain complexes and explain how they showed up in joint work with Chan and Payne on the cohomology of moduli spaces of complex curves.

Abstract: A smooth, oriented n-manifold is called a homotopy sphere if it is homeomorphic, but not necessarily diffeomorphic, to the standard n-sphere. In dimensions n>4, one often studies the group $\Theta_n$ of homotopy spheres up to orientation-preserving diffeomorphism, with group operation given by connected sum. I will give a leisurely introduction to the telescope conjecture in stable homotopy theory, and explain how its failure gives new lower bounds on the complexity of $\Theta_n$. To disprove the telescope conjecture, we construct invariants capable of distinguishing many diffeomorphism classes of exotic spheres: interestingly, key finiteness properties of these invariants are proved in part using intuitions and ideas from prismatic cohomology in p-adic algebraic geometry. The talk is based on joint projects with Burklund, Carmeli, Levy, Raksit, Schlank, Wilson, and Yanovski.

Abstract: It has been known for a while that NK, the obstruction of K-theory being $\mathbb{A}^1$ homotopy invariant, and regularity are closely connected. One of the first results Quillen proved after defining higher algebraic K-theory of exact categories was that NK of regular rings vanishes. This has lead many people to use NK and related objects to measure how bad singularities are, such as the notion of $K_n$ regularity for all integers n. In this talk I will review some of these ideas and definitions and talk about work, joint with Elden Elmanto, which gives new proofs of results of Cortiñas-Haesemeyer-Weibel, Davis, and Vorst and generalizes them to derived qcqs schemes.

Abstract: In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture on the enumeration of function fields by studying the homology of configuration spaces with certain exponential coefficients. In this talk, we will extend their framework to study the twisted homology of various configuration spaces. As an application, we study character sums of the resultant of monic squarefree polynomials over finite fields, answering and generalizing a question of Ellenberg and Shusterman, and Malle's conjecture with prescribed ramification.

Abstract: The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings and can be effectively calculated through resolutions of modules, as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of E_2-rings

Abstract: An exotic sphere is a smooth manifold which is homeomorphic, but not diffeomorphic, to a sphere with its standard smooth structure. There are many exotic spheres, but even after decades of study, many simple-sounding questions about their geometry remain unanswered. Can you rotate an exotic sphere? Are exotic spheres round? In this talk, I will summarize what is known about the geometry of exotic spheres. I will then discuss how the Mahowald invariant, a construction from stable homotopy theory, can be used to detect smooth effective U(1)- and Sp(1)-actions on exotic spheres. This is joint work with Boris Botvinnik.

Abstract: Hopf algebras arise naturally from the homology of spaces with multiplications (i.e. H-spaces or "Hopf" spaces). When spaces have additional structure, this is reflected in homology. For example, the spaces classifying homology theories have a structure mimicking that of a graded ring. In turn, their homology has the structure of a Hopf ring, which is a graded ring in the category of coalgebras. This talk will survey some of the ways Hopf rings lead to computational techniques and elegant descriptions of answers in algebraic topology. It will also describe work in progress and early results in a program focusing on Hopf rings and computations in equivariant homotopy theory.

Abstract: Over the past few years, there has been enormous progress in our understanding of the interaction between chromatic homotopy theory and trace methods. I will survey some of these recent developments and explain applications to topological restriction homology. All original results are joint with Jonas McCandless.

Abstract: The trace map from K to THH can be enriched to the real trace map from KR to THR. After taking the $\mathbb{Z}/2$-fixed point functor, this induces the Hermitian trace map from GW to $\mathrm{THR}^{\mathbb{Z}/2}$. Through the real trace map, THR should carry some information about KR and GW as THH does about K. In this talk, I will explain that THR satisfies the isovariant etale base change property, which leads to defining THR of schemes. As an application, I will explain a natural complete filtration on real Hochschild homology of smooth algebra. This work is joint with Jens Hornbostel.