Abstract: This talk will introduce the ideas of homological stability and highlight some important examples. The second part of the talk will focus on Quillen's spectral sequence argument for the homological stability of symmetric groups. 

References: These mini-course notes by Sander Kupers, this write-up by Nathalie Wahl, and these talk notes by Jenny Wilson. The preliminary syllabus for Talbot 2025 also has more reference suggestions (see in particular Talks 1-4 and Talk 9).

Abstract: This talk will elaborate on the connectivity arguments that were swept under the rug during pre-talk 1. In particular, we will focus on tools and techniques coming from discrete Morse theory. After introducing the basics of discrete Morse theory, we will use it to prove the Solomon-Tits Theorem. 

References: For discrete Morse theory, see the user's guide by Robert Forman and these mini-course notes by Mladen Bestvina. For connectivity arguments, see Section 2 of this paper by Hatcher-Vogtmann and also these notes by Jenny Wilson. See also the worksheets for this class taught by Wilson, particularly #18-20. 

Abstract: TBA.

Abstract: TBA.

Abstract: This talk will introduce the basic definitions of motivic homotopy theory, such as Zariski and Nisnevich descent, along with key examples of motivic spaces including Eilenberg–Maclane spaces.

References: These course notes by Thomas Brazelton.

Abstract: This talk will introduce the construction and properties of Postnikov towers, the basic ideas of obstruction theory, and connections to characteristic classes.

References: Chapter 4 of Hatcher(!) and Chapter 2 of this article by Antieau-Elmanto.

Abstract: We will motivate and introduce the basic ideas of Hermitian K-theory, highlighting the similarities and differences with algebraic K-theory of rings. This will be approached from two perspectives: one being a stripped-down low-tech version of the (very high-tech) story of Poincare infinity categories, and the other being the classical picture of the K-theory of exact categories with duality. Finally we’ll discuss how Hermitian K-theory can be promoted to a motivic spectrum, which is used as a crucial tool to access stable and unstable motivic homotopy groups of spheres.

Notes from the talk are available here.

Abstract: Given a finite CW complex, it is a folklore result in obstruction theory that it admits only finitely many isomorphism classes of complex vector bundles with some prescribed Chern classes. Given a smooth affine variety of finite Krull dimension over a field, we can ask an analogous question for algebraic bundles, and the answer is mostly unknown. However Morel's "affine representability" theorem indicates that these sorts of algebraic questions are amenable to techniques from motivic obstruction theory. Following work of Mohan Kumar, Murthy and others in the 20th century, as well as the contemporary research program of Asok and Fasel, we understand that the complexity of these sorts of questions are governed by two factors: (1) the 2-cohomological dimension of the base field, and (2) the corank (dimension of base minus rank of bundle). In joint work with Opie and Syed we explore corank two in the first interesting setting. We explore to what extent Chow-valued Chern classes and Chow-Witt-valued Euler classes uniquely classify algebraic vector bundles over smooth affine fourfolds over an algebraically closed field.

Abstract: This talk will review the definition of dualizable categories, and explain (but not prove!) the characterization in terms of compactly exhaustible objects. We will use this to recall the dualizability of Shv(X,Sp) and (time permitting) sketch a proof.

References: Section 1.8 of Efimov's paper, pp. 1-3 of Hoyois' talk notes, Section 2 of these course notes by Krause-Nikolaus-Püzstück, particularly Theorem 2.2.15 and Corollary 2.2.20. 

Abstract: In this talk, we will review Verdier duality, construct the six operations, and define local rigidity and rigidity in connection with Verdier duality.

References: Section 4.6 of these course notes by Krause-Nikolaus-Püzstück, particularly 4.6.4 and Remark 4.6.1.

Abstract: Algebraic K-theory, topological Hochschild homology (THH), topological cyclic homology (TC), and topological restriction homology (TR), are all examples of localizing invariants. In this talk, I will sample some of the ways Efimov’s theory of dualizable categories and the category of localizing motives interact with these specific invariants. In particular, I will discuss (1) the interaction of algebraic K-theory and infinite products of categories (originally studied by Kasprowski—Winges) and (2) how to express THH of any dualizable category completely algebraically. Time permitting, I will discuss some other applications and pose a question or two. 

Abstract: Let X be a locally compact Hausdorff space, C a dualizable stable infinity-category, and denote by Shv(X, C) the infinity category of C-valued sheaves on X. In this talk, we explain (in as much detail as time permits) how to compute the K-theory of Shv(X, C) in terms of the compactly supported cohomology of X with coefficients in the spectrum K(C). The strategy we adopt, which was suggested by Dustin Clausen, consists in providing an axiomatic characterization of sheaf cohomology on compact Hausdorff spaces. If time permits, we will also outline applications to simple homotopy theory and functoriality of Becker-Gottlieb transfers.

In this talk, we will review the definition of the stable and unstable categories of genuine G-spectra for a finite group G, using the perspective of spectral Mackey functors. We will also define the Hill-Hopkins-Ravenel norm on genuine G-spectra and (if time permits) the isotropy separation sequence.

References: Sections 1-2 of these notes by Ramzi, Section 9 of this paper by Bachmann-Hoyois, Part 1 of this paper by Mathew-Naumann-Noel, and Appendix A of Nardin's thesis.

In this talk, we will explain how the ∞-categories of G-spaces and G-spectra upgrade to the parametrized ∞-categorical setting. We will define genuine C_p-E_∞-operads after Nardin-Shah and discuss how being a genuinely G-commutative monoid gives rise to norm maps.

References: Section 2 of this paper by Yang, Sections 2.1 and 4.1 of this paper by Cnossen, and Sections 2.1-2.4 of this paper by Nardin-Shah.

Abstract: In this talk we will explain how one can adapt (semi)-recent techniques from parametrized homotopy theory to the setting of sheaves on equivariant manifolds. More specifically, one can recover the category of spaces via the category of sheaves on manifolds. An interesting question is what commutative monoids are in this setting. One guess is that it’s those that possess a covariant functoriality known as a transfer with respect to bundles with compact manifold fiber, but in fact we need much less: it suffices to have transfers along finite covering maps. Quillen conjectured that this was the case and equipped with the tools of Bachmann-Hoyois, the proof of this statement is relatively formal. Recent advances in parametrized equivariant homotopy theory as well as the properties of equivariant sheaves on G-manifolds allows one to assemble these proofs together equivariantly. This talk aims to tell this story in a way that makes a case for the formalism of parametrized equivariant homotopy theory. 🙂

Abstract: Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an E_∞-algebra. Our work concerns the C_p-E_∞-algebras of Nardin–Shah with respect to a cyclic group C_p of prime order. We show that many of the higher coherences inherent to the definition of parametrized algebras collapse; in particular, they may be described more simply and conceptually in terms of ordinary E_∞-algebras as a diagram category which we call normed algebras. Our main result provides a relatively straightforward criterion for identifying C_p-E_∞-algebra structures. We visit some applications of our result to real motivic invariants.