Home

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

Location and Time

For the Spring 2025 semester, we meet Mondays 1-2pm in room 143 of Altgeld Hall.

Talk Schedule

Day	Speaker	Title and Abstract
02/10/25	Langwen Hui	Computing homotopy groups of tmf Topological modular forms (tmf) is a spectrum that interacts in interesting ways with geometry and arithmetic. In this talk I will examine this object from the angle of stable homotopy theory, and sketch a computation of its homotopy groups, which is a blend of classical modular forms and the stable homotopy groups of spheres.
02/24/25	Samuel Hsu	Opetopic types in homotopy type theory Homotopy type theorists have celebrated some major successes in formalizing (and often generalizing) results from classical homotopy theory. For many results from modern homotopy theory however, we must confront the issue of defining ∞-categories inside HoTT. Very roughly, the problem is that HoTT is designed to internalize the infinite coherences of only a very particular class of higher structure, namely (globular) ∞-groupoids, and not any other higher structures. This issue has plagued HoTT since the very start. By now there are several proposed approaches, each adding on top of HoTT with various advantages and drawbacks. In this talk we will give an overview of the solution put forth by Finster, Allioux, and Sozeau using opetopic types. Some familiarity with type theory will be helpful, although the emphasis will be such that prerequisites can be kept to a minimum, and we will quickly review some at the start. Note: this talk will be on zoom.
03/03/25	Doron Grossman-Naples	Computing Pullbacks of ∞-Topoi; or, How I Learned to Stop Worrying and Love Gabriel-Ulmer Duality Fiber products are a pretty fundamental construction in geometry, so it's not surprising that pullbacks of ∞-topoi arise quite naturally when studying spectral algebraic geometry. Fortunately, Lurie has shown that the category of ∞-topoi has all (small) limits. Actually computing them is another matter, however, as I discovered when doing so became necessary for my research. There isn't quite a complete recipe, and even the constructive aspects depend on some surprisingly abstract categorical machinery. Join me in this talk as I walk you through this journey: why you might want to undertake it, the category-theoretic structure that makes it possible, and what such a computation actually looks like.
03/31/25	Timmy Feng	Non-representable ambidexterity In this talk, we will introduce the NisLoc topology. It turns out that every algebraic stack is Nis-loc. We will then discuss how to extend the six functor formalisms in Motivic Homotopy Theory to algebraic stacks using the NisLoc topology.
04/07/25	Ea E	The Universality of Infinite Loop Space Machines Infinite loop space machines are certain functors which produce spectra from algebraic and space-level data. These machines have a rich history dating back to the early 70s, with applications including the construction of multiplicative structures on K-theory and the construction of $$E_\infty$$-spectra from $$E_\infty$$-spaces. In 1977 Thomason and May proved a uniqueness theorem for the variety of infinite loop space machines that were being defined at the time. In this talk we provide an overview of Gepner, Groth, and Nikolaus' paper "Universality of Multiplicative Infinite Loop Space Machines" which provides an alternative approach to this uniqueness using universality at the level of quasi-categories. The perspective of Lawvere theories and base-change theorems will underlie the main points of the talk, aiming to provide an emphasis on the algebraic content of infinite loop space machines.
04/14/25	Jiantong Liu	Descent Properties in Algebraic K-Theory Thomason–Trobaugh K-theory has served as a foundational definition of algebraic K-theory, further characterized by a universal property in the language of ∞-categories by Blumberg–Gepner–Tabuada. Thomason and Trobaugh also extended their construction to a non-connective spectrum via Bass delooping, resulting in a K-theory that satisfies certain representability conditions, as well as Zariski and Nisnevich descent. This naturally raises the question of finding a notion of homotopy-invariant K-theory that supports the desired representability and satisfies suitable descent properties. We will provide an overview of Cisinski’s approach to this problem, demonstrating that several such notions coincide, possess the expected representability, and, in fact, satisfy even stronger descent properties.
04/21/25	Levi Poon	Synthetic spectra and deformation theory In recent years, the synthetic philosophy has proven itself to be very useful in understanding classical homotopy theory and uncovered a somewhat surprising connection between classical and motivic homotopy theory. In this talk, I will introduce Pstragowski's notion of synthetic spectra and its relationship to the deformation theory of categories. If time permits, I'll also survey some applications of this circle of ideas.
04/28/25	Yigal Kamel	Unifying the orientations of topological K-theory, and a nonexistence theorem I will introduce a C_2-commutative ring spectrum, called Real spin bordism, that simultaneously refines the Real orientation of Atiyah's KR-theory, the spin orientation of KO, and the spin^c orientation of KU. I will also discuss why the fixed point spectrum of Real spin bordism is not equivalent to spin bordism, which stems from an incompatibility between the spin and spin^c bordism spectra. This talk will cover similar material to a talk given in the UIUC topology seminar in October. Joint work with Zach Hallady and Hassan Abdallah.
05/05/25	Juhan Kim	Squares Categories and the K-theory of Scissors Congruence In 2018, Campbell and Zakharevich introduced the notion of the CGW-category whose K-theory unifies the known theory of varieties and Quillen's Q-construction. In this talk, I'll introduce their more recent work of Squares categories which generalizes Waldhausen's S-construction and the CGW category. As an application, I'll show how it's used to define the K-theory of the scissors congruence of manifolds and how it induces the derived euler characteristics. If time permits, I'll also introduce the application to the scissors congruence of polytopes and the relation to their construction of Goncharov's map.

Talks from previous semesters may be found in the archive.