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

Location and Time

For the Fall 2024 semester, we meet Mondays 2-3pm in room 143 of Altgeld Hall.

Talk Schedule

Day	Speaker	Title and Abstract
9/9/24	Yigal Kamel	Transfer systems in equivariant homotopy theory An operad is an object that encodes a type of algebraic structure in homotopical contexts. For instance, the E_∞-operad encodes the homotopical notion of commutative monoid. In equivariant homotopy theory, G-commutative monoids are encoded by the N_∞-operads of Blumberg—Hill. In contrast to E_∞-operads, the category of N_∞-operads is far from trivial, and its homotopy category is equivalent to a category of combinatorial objects called G-transfer systems. I will introduce transfer systems from scratch and discuss various examples, such as those coming from little disks and linear isometries operads, the G-operads which canonically model “additive” and “multiplicative” structures, respectively. Time permitting, I will discuss ongoing joint work with DeMark, Hill, Niu, Stoeckl, Van Niel, and Yan, studying which pairs of G-transfer systems are sufficiently "compatible" to encode the additive and multiplicative structures of a “ring”.
9/16/24	Langwen Hui	Vector fields on spheres We will examine a classical problem in topology: How many linearly independent vector fields can exist on an n-dimensional sphere? This problem has been widely studied and was definitively answered by Adams in his 1962 paper. In this talk, we will discuss the key ideas behind the solution, offering an introduction to some important concepts and techniques in classical homotopy theory.
9/23/24	Anthony D'Arienzo	The fundamental group of a topos, and its applications to memory management The Galois correspondence between π₁(X)-representations and covering spaces has been extended to sites and topoi. The fundamental group of a topos describes the isotropy around a topos point, and is the first homotopy group of the shape of the topos. When building memory models in a programming language, memory is organized into a Grothendieck topology, and the fundamental group of the associated topos is an invariant describing how regions in computer memory may be stitched together. The first half of this talk will construct the fundamental group of a topos. The second half will describe its applications to memory management. This talk is based on conversations about “A Nominal Approach to Probabilistic Separation Logic” (2024) by John Li, et al.
9/30/24	Doron Grossman-Naples	A Deligne-Mumford stratification of elliptic cohomology with level structure Level structure is an essential aspect of the classical theory of modular forms and elliptic curves, and it plays a similarly important role in elliptic cohomology and its applications to physics and stable homotopy theory. The classical theory doesn’t carry over cleanly to the spectral case, however, because level structures and their associated isogenies can fail to be étale. This talk concerns a new approach I have developed for understanding elliptic cohomology with (possibly ramified) level structure. By studying the effect of isogenies on the dualizing line, I obtain a tractable invariant characterizing “how ramified” an isogeny is. This invariant can be used to filter the abstract moduli stack of isogenies between oriented elliptic curves, ultimately producing a stratification by Deligne-Mumford stacks.
10/7/24	Levi Poon	Orientations: Classical and Motivic The notion of complex orientations on ring spectra played a fundamental role in the development of chromatic homotopy theory. Complex orientations are closely connected to the theory of Chern classes and formal group laws, and it is a classical result that the complex cobordism spectrum MU carries a universal complex orientation that induces the universal formal group law. There is a completely analogous notion of orientations of motivic ring spectra, and Voevodsky conjectured that a similar statement is true for the algebraic cobordism spectrum MGL over any regular local base scheme. This hasn't been proven in its full generality but some partial results are known, inlcuding a result by Hoyois (building on works from Levine, Hopkins and Morel) that this is true over fields once you invert the exponential characteristic and a later extension due to Spitzweck to motivic spectra over local rings pro-smooth over a DVR of mixed characteristic, again after inverting the characteristic. In this talk, we will discuss the classical theory of complex orientations and its motivic analogue. If time permits, we will discuss the proof of these motivic results and the difficulties at the characteristic.
10/28/24	Vivasvat Vatatmaja	The moment map and equivariant cohomology I will talk about Atiyah and Bott’s paper “the moment map and equivariant cohomology”. If time permits, I will talk about the more general perspective of Goresky, Kottwitz, MacPherson that the cohomology of equivariantly formal spaces can be computed from the structure of the zero and one dimensional orbits of a maximal torus.
11/11/24	Daniel Teixeira	Cohomology of Eilenberg-MacLane spaces Spaces are built first out of their homotopy groups, then out of H^k(K(G,n),H)'s. With that motivation, we follow Serre's Cohomologie modulo 2 des complexes d'Eilenberg-MacLane and calculate this for G = any abelian group and H = 𝕫/2. With that information we will try to say something smart, like building super-2-vector spaces, or calculate some π_n(S^k).
11/18/24	Ea E	An Introduction to Formal (Infinity-)Category Theory via Modules At the heart of formal category theory is the construction of higher categories. However, structures such as 2-categories, or more generally bicategories (and their higher dimensional analogues), are often insufficient for encoding a number of important abstract constructions prevalent in category theory. Although notions such as adjunctions and equivalences can be defined purely algebraically internal to any bicategory, the naive definition of fully-faithfulness and of Kan extensions often fails to encapsulate the right properties in enriched settings. This failure stems from the inability to discuss representability, or equivalently yoneda constructions, internal to a general bicategory. In this talk we will provide a brief introduction to the framework of formal category theory via pro-arrow equipments which fixes this issue. After this brief introduction we will explore how this work generalizes to the construction of a formal theory of ∞-categories due to Riehl and Verity (with further recent work by Ruit) which provides a suitable definition of Kan extensions internal to an arbitrary ∞-cosmos (and hence any well-behaved model of ∞-categories).
12/2/24	Sterling Ebel
12/9/24	Johnson Tan
12/16/24	Ruoxi Li

Talks from previous semesters may be found in the archive.