In influential work of the 70s and 80s, Segal and Waldhausen each construct a version of K-theory that produces spectra from certain types of categories. These constructions agree, in the sense that appropriately equivalent categories yield weakly equivalent spectra. In the 2000s, work of Elmendorf-Mandell and Blumberg-Mandell produced more structured versions of Segal and Waldhausen K-theory, respectively. These versions are "multiplicative," in the sense that appropriate notions of pairings of categories yield multiplication-type structure on their resulting spectra. In this talk, I will discuss joint work with Osorno in which we show that these constructions agree as multiplicative versions of K-theory. Consequently, we get comparisons of rings spectra built from these two constructions. Furthermore, the same result also allows for comparisons of related constructions of spectrally-enriched categories.

Let G be a finite group. A N_∞ G-operad is an equivariant generalization of an E_∞ operad. Such operads govern the natural algebraic structure on spectra over incomplete universes and on localizations of genuine commutative ring G-spectra. Since Blumberg and Hill’s pioneering work, it has been known that the homotopy theory of N_∞ operads is essentially algebraic. They proved that the homotopy type of a N_∞ operad is completely determined by a single combinatorial invariant, and subsequent work has revealed that the homotopy theory of N_∞ operads may be modeled entirely with discrete operads in the category of G-sets. On the other hand, there are natural classes of geometrically defined N_∞ operads, which generalize the classical linear isometries and infinite little discs operads. Such operads encode real representation-theoretic properties of G, in addition to purely algebraic data. In this talk, I will explain how to reduce the homotopy theory of N_∞ operads to combinatorics, and then I will discuss how the peculiarities of equivariant linear isometry and infinite little discs operads are encoded in algebra.

Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. For a C_n-equivariant ring spectrum, one can define C_n-relative THH. This leads to the question: What is the algebraic analogue of C_n-relative THH? In this talk, I will define twisted Hochschild homology for Green functors, which allows us to describe this algebraic analogue. This also leads to a theory of Witt vectors for Green functors, as well as an algebraic analogue of TR-theory. This is joint work with Andrew Blumberg, Mike Hill, and Tyler Lawson.

The Barratt-Eccles operad is an E_∞ operad given simply by taking the sequence of spaces EΣ_n. It can be constructed by first taking an operad P in categories and then taking its classifying space. Since algebras over P are precisely permutative categories, this operad is a key ingredient in the operadic infinite loop space machine that takes permutative categories to spectra. The categorical equivariant Barratt-Eccles operad P_G defined by Guillou-May-Merling plays a similar role in equivariant infinite loop space theory. In this talk I will give some results about the combinatorics of P_G. I will show that it is not finitely generated when |G| > 1, but that there is a finitely generated suboperad Q_G that is appropriately equivalent to P_G. In the cases of G=C_2 and C_3, I will moreover give an explicit finite presentation of Q_G. This finite presentation allows us to describe algebras over Q_G in a compact way. This is joint work with K. Bangs, S. Binegar, Y. Kim, K. Ormsby, D. Tamas-Parris and L. Xu.

The additivity of the Euler characteristic is one of its essential properties. Additivity holds and plays a similar role for the generalizations to the Lefschetz number and Reidemeister trace. For the Euler characteristic, additivity is captured by the observation that the Euler characteristic descends to a map out of algebraic K-theory. The corresponding statement for the Lefschetz number and Reidemister trace would realize a vision of Klein, McCarthy, and Williams and could provide a new approach to invariants for periodic points. I’ll describe some first steps toward this goal that connect the cyclotomic trace with the bicategorical trace. This is joint work with Jonathan Campbell.

TBA

The level p congruence subgroup of SL_n(Z) is defined to be the subgroup of matrices congruent to the identity matrix mod p. These groups have finite cohomological dimension. In the 1970s, Lee and Szczarba gave a conjectural description of the top dimensional cohomology groups of these congruence subgroups. In joint work in progress with Patzt and Putman, we show that this conjecture is false for p>5. In particular, these congruence subgroups have extra cohomology classes in their top degree cohomology coming from the first homology group of the associated compactified modular curve. I will also discuss joint work with Patzt and Nagpal on a stability pattern in the high dimensional cohomology of congruence subgroups.

Let G be a very nice p-adic analytic group; I have in mind examples such as Gl_n(Z_p) or the Morava stabilizer group. The category of continuous G-modules has a very elegant theory of duality reflecting Poincaré duality for G. We would very much like to extend this to stable homotopy theory where, in various contexts, it would help explain some deep structure we have seen so far only through computations. It is easy enough to define the dualizing objects, but then we are left with understanding them. It turns out that if we are only interested in finite subgroups of G (which would be a serious start) we can get away with classical computations with characteristic classes. This is an on-going project with Agnès Beaudry, Mike Hopkins, and Vesna Stojanoska.

It is a classical result that the rational homotopy groups, π_*(X)⊗\mathbbQ, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on X. The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its spectral Lie algebra". Unfortunately both input and differentials are hard to get at. We therefore simplify the homotopy groups by taking the unstable v_h-periodic homotopy groups, v_h⁻¹π_*( ) (note h=0 recovers rational homotopy groups). For h=1 we are able to compute the K-theory based v₁-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute v₁⁻¹π_*SU(d) in a very different way from the original computation by Davis.

Topological Hochschild homology (THH) is a genuine S¹ spectrum introduced in order to approximate algebraic K-theory. Its fixed points, known as topological Restriction homology (TR), are given by (a periodicization of) the de Rham-Witt complex of arithmetic geometry; the equivariant structure is reflected in the F, V, R, and d operators on the de Rham-Witt complex. However, equivariant stable homotopy theory provides yet further structure, which has hitherto been little exploited and remains arithmetically mysterious: we can consider homotopy groups graded by real representations of G, rather than just by \mathbbZ. These have been calculated for G=S¹ by Gerhardt and Angeltveit. This subsumes the case where G is cyclic of odd order; however, 2-primary cyclic groups have representations which are not restricted from an S¹-representation, namely the sign representations. We will present computations of these sign-graded TR groups, and briefly discuss applications to the slice filtration.

We will talk about the slice spectral sequence of a C₄-equivariant spectrum. This spectrum is a variant of the detection spectrum that Hill-Hopkins-Ravenel used in their proof of the Kervaire invariant problem. After periodization and K(4)-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory E₄ with C₄-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that E₄^hC₁2 is 384-periodic. This is joint work with Mike Hill, Guozhen Wang, and Zhouli Xu.

A beautiful theorem of Kelly classifies when a functor F:A → X identifies X as presheaves on A, up to equivalence: when F is full, faithful, and its essential image is a strong generator of small projectives. I will discuss a version of this theorem for V-enriched model categories which produces Quillen equivalences between a V-model category X and the projective model structure on V-enriched presheaves on A using homotopical analogues of the notions of strong generator and small projective. This gives a proof of the Reversed Straightening-Unstraightening Theorem of Stevenson avoiding any use of localization. This is joint work with Kim Nguyen and Daniel Schaeppi.

The Hopkins-Mahowald higher real K-theory spectra E_n^G are generalizations of real K-theory; they are ring spectra which give some insight into higher chromatic levels while also being computable. This will be a talk based on joint work with Drew Heard and XiaoLin Danny Shi, in which we compute that higher real K-theory spectra with group G=C₂ at prime 2 and height n are Gross-Hopkins self duals with a shift 4+n. This will allow us to detect exotic invertible K(n)-local spectra.

In joint work with M. Hoyois, we established (the beginnings of) a theory of "normed motivic spectra." These are motivic spectra with some extra structure, enhancing the standard notion of a motivic E_oo-ring spectrum (this is similar to the notion of G-commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting *power operations*. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any (homotopy) ring spectrum with 2=0 is generalized Eilenberg-MacLane.

In this talk, we discuss some recent applications of involutive Heegaard Floer homology (defined by Hendricks and Manolescu) to the homology cobordism group. We establish some non-torsion results and show that the homology cobordism group admits an infinite-rank summand. This is joint work with Jennifer Hom, Matthew Stoffregen, and Linh Truong.

The Tate construction is a powerful tool in classical homotopy theory. I will begin by reviewing the Tate construction and surveying some of its applications. I will then describe an enhancement of the Tate construction to genuine equivariant homotopy theory called the parametrized Tate construction and discuss some of its applications, including C_2-equivariant versions of Lin's Theorem and the Mahowald invariant, blueshift for Real Johnson-Wilson spectra (joint work with Guchuan Li and Vitaly Lorman), and trace methods for Real algebraic K-theory (work-in-progress with Jay Shah).

We calculate K_*(D; Z_p) where D is a central division algebra over a complete discrete valuation field K whose residue field has characteristic p. Suslin and Yufryakov have shown that if l is a prime other than p, K_*(D, Z_l) is isomorphic to K_*(K, Z_l) for all *>0 via a reduced trace map. We use a reduced trace map approach as well, comparing to K_*(K, Z_p) which has been calculated by Hesselholt and Madsen. The analogy to matrix algebras shows what a reduced trace map should be like, but it is not a natural construction nor indeed something that can always be done. Our calculation uses trace methods and Hochschild-type invariants for a maximal order A in D. Any central division algebra of degree d (that is, dimension d^2) over a field K as above contains a maximal unramified field extension L of K, which will be of dimension d. If we call K's ring of integers S and L's ring of integers T, then A tensored over S with T is close enough to being a matrix algebra that it has a reduced trace from its invariants to those of T. Comparison to that easier ring along with the action of the Galois group of L over K lets us define a reduced trace from the Hochschild-type invariants of A to those of S. (Joint with Lars Hesselholt and Michael Larsen.)

In classical algebra, the Frobenius provides a natural endomorphism of every ring in which p=0; this determines an action of the circle on the category of F_p algebras. In higher algebra, one can do away with the characteristic p assumption and define a Frobenius map for every E-infinity ring spectrum. In this talk, I will explain the analog of the circle action. The construction of the action features equivariant homotopy theory and a calculation of a non-group-complete variant of the K theory of F_p.

There is a canonical pairing on the Brauer group of a surface over a finite field, and an old conjecture of Tate predicts that this pairing is alternating. I will present a resolution to Tate’s conjecture, whose key ingredient is a surprising connection to Steenrod operations.

A motivic measure with values in an abelian group A is a function μ: {varieties}→ A which is additive, in the sense that for any closed embedding Y\hookrightarrow X we have μ(X) = μ(Y)+μ(X\Y). Many such measures, such as point counting, Euler characteristics, or the local zeta function actually take values in a K₀-group. In this talk we will give a description of how to lift such measures to maps between spectra and show how to use these to find nontrivi elements in higher K-groups of varieties.

Pretalk abstract: I will give a short introduction to algebraic K-theory, focusing mostly on exact categories and the Q-construction.

Elliptic cohomology is a natural big brother to ordinary cohomology and K-theory. In contrast to the geometric objects that provide representatives for cohomology and K-theory classes (which lead to many applications), as yet there is no such geometric description of elliptic cohomology. This talk will explain a step forward, in joint work with Arnav Tripathy, for the case of equivariant elliptic cohomology over the complex numbers. The geometric objects of interest are inspired by supersymmetric gauge theory. No prior knowledge of either elliptic cohomology or gauge theories will be assumed.