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.