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This is the webpage for the Graduate Student Homotopy Theory Seminar (GSHTS) at UIUC.

Location and Time

For the Spring 2024 semester, we meet Fridays 3-4pm in room 347 of Altgeld Hall.

Talk Schedule

Day	Speaker	Title and Abstract
02/02	Yigal Kamel	The phony multiplication on Quillen’s K-theory There comes a point in every child’s journey through mathematics when they internalize the fact that the natural numbers can be turned into a ring by introducing so-called “negative numbers”. This observation is the idea lying at the root of a vast generalization and strengthening of this process that currently goes by the name algebraic K-theory. One procedure for constructing algebraic K-theory using the philosophy of “negative numbers” was introduced by Quillen via his “S^-1S-construction”. In a cautionary paper, Thomason observed that the typical way we learn to extend the multiplication of the natural numbers to the integers as children fails in the general context of K-theory. After a brief introduction to algebraic K-theory a la Quillen, I will explain how I stumbled into Thomason’s observation in my own work in topological K-theory and discuss various ways to avoid falling into this subtle trap.
02/16	Doron Grossman-Naples	Orientations of Formal Groups Abstract: In his 2018 monograph on elliptic cohomology, Lurie developed the theory of orientations of formal groups, a powerful new tool in chromatic homotopy theory that strengthens the classical parameterization of complex-oriented cohomology theories by formal groups and extends it to the 𝔼_∞ setting. The goal of this talk is to provide a conceptually accessible introduction to orientations: what they are, why they’re useful, and how they can be interpreted algebraically and geometrically.
02/23	Sam Hsu	Some (other) applications of higher commutativity Following the ambidextrous zeitgeist that's been floating around our department lately, we will sample some more concrete (— to me —) applications of higher semiadditivity that aren't about building up to the disproof of the telescope conjecture. We will focus on examples at height 1 following Carmeli and Yuan. This is intended to be an introductory talk, and in particular I won't assume the audience attends the anti-telescope telescope club or similar, so we will briefly review higher semiadditivity.
3/08	Johnson Tan	Cyclotomic Structure for THH One of the key features of topological hochschild homology is that it is canonically acted on by S¹. Nowadays we know that this S¹-action is part of the richer structure that makes THH into a cyclotomic spectra. In this talk we will go over a neat way of obtaining the cyclotomic structure in the case of commutative ring spectra as well as some sample calculations making use of this structure. If there is time at the end we will look at how one might do something similar for motivic spectra.
03/22	Anthony D'Arienzo	Stacky Cohomologies via L2-Theory On a compact smooth manifold, the ordinary cohomology can be computed using differential forms. Hodge’s theorem leverages this to calculate the topology of a smooth manifold using the L2 properties of a Riemannian metric on it. When singularities are introduced to the manifold, there are multiple replacements for the ordinary cohomology. I will discuss two examples, orbifolds and intersection cohomology, where the orbifold cohomology and the intersection cohomology can be computed using the L2 theory of a metric on the complement of the singular locus. Afterwards, I will sketch how this identifies a stacky description for stratified spaces.
03/29	Sam Hsu	How many layers of univalence is your universe (of ∞-categories) on? In this talk we revisit the straightening/unstraightening correspondence following Cisinski and Nguyen. To begin, we'll look at ordinary univalence from ∞-category theory, and some of the roles it plays. Next, we will recall some facts and constructions surrounding the universal coCartesian fibration. We will then define directed univalence and discuss how straightening/unstraightening follows. If there's time, we may tie this in with some related works concerning complete Segal objects in an ∞-topos, and perhaps something to do with (∞,2)-univalence as suggested by Rasekh. Time very permitting, we might even say a few words about "enhanced" ∞-equipments (enhanced in the sense of enhanced twisted arrow ∞-categories) induced by one conjectured notion of elementary (∞,2)-topos. The prerequisites will be kept to a minimum, but some familiarity with marked simplicial sets and the coCartesian model structure may be helpful. A willingness to blackbox some results is all that should be required though.
04/05	Gabrielle Li	Galois extensions in stable homotopy theory I will introduce Galois extension in stable homotopy theory following the work of Rognes and present examples including topological K-theory, the pro-G_n (the Morava stabilizer group) Galois extension from the K(n)-local sphere to the Lubin—Tate theory E_n, and cyclotomic extension. I will briefly discuss cyclotomic completeness and its trending application in distinguishing the K(n)-local and T(n)-local spectra.
04/12	Langwen Hui	Calculus of functors I will provide an overview of the calculus of functors, as pioneered by Goodwillie and expanded upon by subsequent researchers. Applications in unstable homotopy theory and the recent construction of derived rings will be highlighted.
04/19	Levi Poon	Norms in Homotopy Theory Starting from elementary algebraic topology, it is well known that the existence of transfer maps (or "wrong-way maps") is immensely useful, a notable example of that being the resolution of the Kervaire invariant 1 problem via the Hill-Hopkins-Ravenel norm on equivariant spectra. Bachmann-Hoyois adapted these notions to the motivic context by considering multiplicative transfers along finite étale maps and they showed that various motivic spectra admit a natural structure of a normed spectrum, restricting to norm maps previously constructed by hand on the corresponding cohomology theories. In this talk, I will discuss these results and if time permits, go into some of these examples in a bit more detail.

Talks from previous semesters may be found in the archive.