Fall 2023 Talks

Day	Speaker	Title and Abstract
09/01	Sam	Reconciling two ways to generalize counting In this talk, we will look at two ways to generalize counting from finite sets to structures that allow subtraction and division, respectively, in a topologically meaningful way. At first glance they seem to be orthogonal generalizations, but some intuitions involving divergent sums suggest they can be combined. This will indeed be the case given some restrictions, as we shall see. One application is to a question that can be stated in group theory. Furthermore, some accounts – Lurie and Yanovski in particular – lead to structures from chromatic homotopy theory e.g. generalizations of character theory. This is intended to be a fun introductory talk, and in particular very little knowledge of anything in homotopy theory is assumed until towards the end.
09/08	Sam	What are these lax limits and actions doing in my traces? Trace methods play a key role in both computations of and celebrated theorems on algebraic K-theory. One might wonder if there is a geometric picture that can be sketched relating K, THH, and TC. This has a positive answer thanks to a series of papers by Ayala, Mazel-Gee, and Rozenblyum, where THH of a "space" X may be thought of as functions on loops on X and TC as the ones whose values are traces of the monodromy of vector bundles on X, in an appropriate sense. The goal of this talk is to draw some pictures to help (me) intuit why lax behavior comes up in making such heuristics precise and orient us towards AMGR's stratified noncommutative geometry. Willingness to blackbox a few facts about (∞,2)-categories will be assumed, but minimally.
09/15	Anthony	Homotopy Groups of First-Order Theories This summer Campion and Ye released a paper calculating the second homotopy group on the classifying space of the first-order theory of dense cyclic orders with two points. This expounds on their earlier work on classifying spaces for theories and their model-theoretic properties. Campion and Ye’s construction is a surprising application of Quillen’s Theorem A. I will outline Campion and Ye’s work, and I will show how it fits into a network of papers building a bridge between first-order logic and topology.
09/22	CANCELLED
09/29	Johnson	Borel Equivariance in Homotopy Theory In equivariant homotopy theory, we are interested in studying homotopical objects with the additional requirement that they come with a notion of a group acting on them. The most natural examples are spaces or spectra with a coherent action of a group G. While these examples could be considered the initial reasons to develop such a theory, it is well known by this point that it is beneficial to work in the larger setting of "genuine" G-spaces and G-spectra. In this talk our goal will be to introduce the two notions as well as how to move between the two. In particular we will discuss how one might try to extend these ideas into the setting of motivic homotopy theory.
10/06	Timmy	Gerbe Fixed Points in Motivic Homotopy Theory In classical equivariant homotopy theory, fixed point functors play a crucial role in identifying genuine G-spectra. However, in motivic homotopy theory, this identification is not as straightforward. This discrepancy arises from the fact that the category of motivic G-spectra is not solely generated by objects resembling G-orbits. In this talk, we will introduce the concept of gerbes and explore how gerbe fixed point functors form a conservative family. We will delve into the reasons behind this conservativity and its implications. If time permits, we will also provide a brief overview of how this conservative family, in conjunction with other theorems, leads to the development of motivic spectral Mackey functors.
10/13	CANCELLED
10/20	Doron	Complex Orientations and Strict Elements The theory of complex orientations is typically phrased in geometric language, but they are crucial to the algebraic project of understanding the category of spectra. In this talk, I give an algebraic description of complex orientations and propose a generalization thereof.
10/27	Gabby	Ambidexterity in Chromatic Homotopy Theory In this talk, we will introduce the theory of higher semiadditivity with an emphasis on proving that the category of T(n)-local spectra is infinitely semiadditive, following the paper by Carmelli, Schlank and Yanovski. We will go over key notions in ambidexterity, construct algebraic tools such as additive p-derivation, and talk about their application in chromatic homotopy theory. Time permitting, we will talk about the application to more general localization of spectra. No background in ambidexterity is assumed, but some knowledge in Bousfield localization and Sp_{T(n)} might be helpful.
11/03	Langwen	Algebraic models of spaces It is known that E_∞ cochains completely capture localizations of simply connected spaces subject to certain finiteness conditions. This is due to Quillen-Sullivan in the rational case, and Mandell in the p-adic case. However, the embedding into Z-valued cochains turns out to be faithful but not full. The subject of finding an algebraic model for integral homotopy types has seen significant progress in the past few years. We now have several fully faithful embeddings of simply connected finite complexes into "algebraic" ∞-categories. In this talk, I will introduce two approaches due independently to Yuan and Horel.
11/10	CANCELLED
11/17	Levi	An Introduction to Framed Correspondences In classical homotopy theory, the category of spectra provides a nice way to study cohomology theories. In a similar vein, motivic homotopy theory is at least in part conceived to study cohomology theories in algebraic geometry. Classically, in some special cases, cohomology theories admit certain covariant functoriality known as transfers. In the early 2000s, Voevodsky introduced the theory of framed correspondences encoding certain transfers in motivic homotopy theory, in hope that it will provide a more computationally tractable characterization of the theory of motivic spectra. This idea has since been developed further by Ananyevskiy, Garkusha, Neshitov, and Panin in terms of framed motives. More recently, using this theory, Elmanto-Hoyois-Khan-Sosnilo-Yakerson proved a recognition principle of motivic inifinte P^1 loop spaces over a perfect field. In this talk, I will introduce the theory of framed correspondences and outline their results. Time permitting, I will also discuss some further applications of the theory.
12/01	Connor	The Stringor 2-Bundle The Atiyah-Singer index theorem famously connects the index of Dirac operators on spinor bundles to push-forwards in K-theory. Ever since Witten's celebrated paper "The index of the Dirac operator on loop space" in 1987, much work in geometry and topology has been focused on constructing a "higher" version of this story. While the higher replacement for K-theory was constructed in the1994 by Hopkins, Mahowald, and Miller, the construction of a higher Dirac operator has remained elusive. In this talk I will discuss the notion of a "2-vector bundle", and the first construction of the stringor 2-vector bundle by Kristel, Luedwig, and Waldorf in 2022. If there is time, I will also discuss some recent work by myself and collaborators towards producing Dirac operators on stringor 2-bundles.