Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. For example, Hochschild-Pirashvili homology for a wedge of circles or Jacobi diagrams in handlebodies give rise to interesting functors on gr. Some of these natural examples satisfy further properties: they are analytic and/or exponential. Pirashvili proves that the category of exponential contravariant functors from gr to the category k-Mod of k-modules is equivalent to the category of cocommutative Hopf algebras over k. Powell proves an equivalence between the category of analytic contravariant functors from gr to k-Mod, and the category of linear functors on the linear PROP associated to the Lie operad when k is a field of characteristic 0. In this talk, after explaining these two equivalences of categories, I will explain how they interact with each other. (This is a joint work with Minkyu Kim).[-]

Hilbert's Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry. In this talk, I will discuss a version of Hilbert's Nullstellensatz in chromatic homotopy theory, where Lubin-Tate theories play the role of algebraically closed fields. Time permitting, I will then indicate some of the applications of the chromatic nullstellensatz including to redshift for the algebraic K-theory of commutative algebras. This is joint work with Tomer Schlank and Allen Yuan.[-]

Since the early 70s, Mackey and Green functors have been successfully used to model the induction and restriction maps which are ubiquitous in the representation theory of finite groups. In concrete examples, the latter maps are typically distilled, in some way, from induction and restriction functors between additive (abelian, triangulated...) categories. In order to better capture this richer layer of equivariant information with a (light!) set of axioms, we are naturally led to the notions of Mackey and Green 2-functors. Many such structures have been in use for a long time in algebra, geometry and topology. We survey examples and applications of this young—yet arguably overdue—theory. This is partially joint work with Paul Balmer and Jun Maillard.[-]

I will explain a construction of the motivic filtration on the topological cyclic homology of ring spectra, generalizing work of Bhatt–Morrow–Scholze and Bhatt–Lurie on the topological cyclic homology of discrete rings. This is joint with Arpon Raksit and Dylan Wilson. As time permits, I will discuss works in progress using the motivic spectral sequence to obtain new calculations in algebraic $K$-theory and prove higher chromatic variants of local Tate duality.[-]

Topological Hochschild homology is a fundamental invariant of rings and ring spectra, related to algebraic $K$-theory via the celebrated Dennis-Bökstedt trace map $K \rightarrow T H H$. Blumberg, Gepner and Tabuada showed that algebraic $K$-theory becomes especially well-behaved when considered as an invariant of stable $\infty$-categories, rather than just ring spectra: in that setting it can be described as the free additive invariant generated by the unit, that is, the initial additive functor under the the core $\infty$-groupoid functor, corepresented by the unit of $\mathrm{Cat}^{\mathrm{ex}}$. In this talk I will describe joint work with Thomas Nikolaus and Victor Saunier showing that $T H H$ similarly acquires a universal property when extended to stable $\infty$-categories, when one allows in addition to take coefficients in an arbitrary bimodule. In particular, we view $\mathrm{THH}$ as a functor on the category TCat ${ }^{\mathrm{ex}}$ whose objects are pairs $(C, M)$ where $C$ is a stable $\infty$-category and $M$ is a bimodule, that is, a biexact functor $C^{\mathrm{op}} \times C \rightarrow$ Spectra. We define a notion of being a trace-like invariant on TCat ${ }^{\mathrm{ex}}$, which amounts to sending certain maps in TCat ${ }^{\mathrm{ex}}$ to equivalences. We then show that $\mathrm{THH}$ is the free exact trace-like invariant generated from the unit of $\mathrm{TCat}^{\mathrm{ex}}$, where exact means exact in the bimodule entry. At the same time, algebraic $K$-theory can also be extended to to $\mathrm{TCat}^{\mathrm{ex}}$, in the form of endomorphism $K$-theory. Comparing universal properties we then get that $T H H$ is universally obtained from endomorphism $K$-theory by forcing exactness. This yields a conceptual proof that $T H H$ is the first Goodwillie derivative of endomorphism $K$-theory, and can be used to extend the Dundas-Goodwillie-McCarthy theorem to the setting of stable $\infty$-categories.[-]

Inspired by the triad of rational, trigonometric and elliptic functions appearing in representation theory, Grojnowski defined in 1995 a higher analogue of equivariant ordinary cohomology and equivariant K-theory: equivariant elliptic cohomology. However, his approach only works over the complex numbers. Based on ideas of Lurie, David Gepner and I have recently defined equivariant elliptic cohomology without these restrictions. This allows in particular to refine topological modular forms to a genuine equivariant theory.[-]

In joint work with Gabriel Angelini-Knoll, Christian Ausoni, Dominic Leon Culver and Eva Höning, we calculate the $\bmod \left(p, v_1, v_2\right)$ homotopy $V(2)_* T C(B P\langle 2\rangle)$ of the topological cyclic homology of the truncated Brown-Peterson spectrum $B P\langle 2\rangle$, at all primes $p \geq 7$, and show that it is a finitely generated and free $\mathbb{F}_p\left[v_3\right]$-module on $12 p+4$ generators in explicit degrees within the range $-1 \leq * \leq 2 p^3+2 p^2+2 p-3$. Our computation is the first that exhibits chromatic redshift from pure $v_2$-periodicity to pure $v_3$-periodicity in a precise quantitative manner.[-]