# Columbia Algebraic Topology Seminar

## Fall 2022

The seminar will meet Fridays 10:30-11:30 AM in person in room 507. Please contact Allen Yuan with any inquiries regarding the seminar.

September 9: Maximilien Peroux (UPenn)

Topological Hochschild homology for twisted G-algebras

Abstract: Topological Hochschild homology (THH) is an important variant for ring spectra. It is built as a geometric realization of a cyclic bar construction. It is endowed with an action of circle. This is because it is a geometric realization of a cyclic object. The simplex category factors through Connes’ category Λ. Similarly, real topological Hochschild homology (THR) for ring spectra with anti-involution is endowed with a O(2)-action. Here instead of the cyclic category Λ, we use the dihedral category Ξ. From work in progress with Gabe Angelini-Knoll and Mona Merling, I present a generalization of Λ and Ξ called crossed simplicial groups, introduced by Fiedorwicz and Loday. To each crossed simplical group G, I define THG, an equivariant analogue of THH. Its input is a ring spectrum with a twisted group action. THG is an algebraic invariant endowed with different action and cyclotomic structure, and generalizes THH and THR.

September 16: Ningchuan Zhang (UPenn)

A Quillen-Lichtenbaum Conjecture for Dirichlet $L$-functions

Abstract: The original version of the Quillen-Lichtenbaum Conjecture, proved by Voevodsky and Rost, connects special values of the Dedekind zeta function of a number field with its algebraic $K$-groups. In this talk, I will discuss a generalization of this conjecture to Dirichlet $L$-functions. The key idea is to twist algebraic $K$-theory spectra of number fields with equivariant Moore spectra associated to Dirichlet characters. Rationally, we obtain a Quillen-Borel type theorem for Artin $L$-functions. This is joint work in progress with Elden Elmanto.

October 7: Guchuan Li (UMich) (VIRTUAL!)

December 2: Lucy Yang (Harvard)

December 9: Foling Zou (UMich)

## Spring 2022

The seminar will meet on Fridays 11:00am – noon in person in Mathematics 507 (**NOTE THE NEW ROOM**). Please contact Inbar Klang or Allen Yuan with any inquiries regarding the seminar.

April 1: Inna Zakharevich (Cornell)

Detecting non-permutative elements of K_1(Var) using point counting

Abstract: The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field k) modulo the relation that for a closed immersion Y –> X there is a relation that [X] = [Y] + [X \ Y]. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field k is finite for a variety X we can consider the “almost-finite” set X(\bar k), which represents the local zeta function of X. In this talk we will discuss how to detect interesting elements in K_1(\Var) (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on X.

April 8: Arpon Raksit (MIT)

Motivic filtrations on topological Hochschild homology

Abstract: Topological Hochschild homology (THH) is an invariant of associative ring spectra, closely related to algebraic K-theory. Bhatt–Morrow–Scholze defined a “motivic filtration” on the THH of discrete commutative rings, after completion at a prime number p, relating it to invariants (new and old) in p-adic Hodge theory. In this talk, I will discuss a proposal for a motivic filtration on the THH of certain non-discrete commutative ring spectra; this is part of work in progress with Jeremy Hahn and Dylan Wilson.

April 15: Hana Jia Kong (IAS)

Motivic image-of-J spectrum via the effective slice spectral sequence

Abstract: In classical homotopy theory, the J-homomorphism connects the homotopy groups of the orthogonal groups and spheres. It was defined geometrically, and its image detects an important family of classes in the stable homotopy groups. There is a spectrum j realizing the image of J-homomorphism, defined using K-theory and the Adams operations. In the motivic stable homotopy category, there is an analogous spectrum, the motivic image-of-J defined by Bachmann–Hopkins. I will talk about this motivic analog and how to calculate its bigraded motivic homotopy groups using the effective slice spectral sequence. Over real numbers, the result captures a regular pattern in the bigraded homotopy groups of the motivic sphere. This is joint work with Eva Belmont and Dan Isaksen.

April 22: Ishan Levy (MIT)

The K-theoretic telescope conjecture away from p

I will explain work joint with Robert Burklund on understanding algebraic K-theory away from the characteristic. One of our main results is the K-theoretic telescope conjecture away from p: namely that the K-theory of compact T(n) local spectra is the same as that of compact K(n) local spectra after inverting p. The main idea is that T(n)-locally, MU behaves like a p-pro-Galois extension of the sphere, allowing us to reduce to MU, for which the telescope conjecture is true. We moreover compute the p-inverted K-theory of many things including the E_n-local sphere, BP, Morava E theory, Johnson—Wilson theory. There are two main ingredients here: one is a formality result about p-inverted K-theory which uses trace methods and devissage, and the other is a canonical action of K(F_p) on the p-inverted K-theory of any E_1 ring for which a power of p is 0.

April 29: Mona Merling (UPenn)

Scissors congruence for manifolds via K-theory

Abstract: The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic.

May 6: Aleksandar Milivojevic (Bonn)

Realization of simply connected rational homotopy types by closed almost complex manifolds

Abstract: I will discuss the problem of realizing simply connected rational homotopy types, together with choices of putative rational Chern classes, by closed almost complex manifolds. Integrally this problem starts to become intractable in dimensions ten and higher; rationally, a satisfactory general statement can be made in all dimensions by adapting machinery introduced by Sullivan. Realizing particular rational homotopy types comes down to solving Diophantine systems of equations. As an example I will show that there is a closed almost complex manifold rationally equivalent to quaternionic projective three-space. From the general construction one sees that the realization problem for a given simply connected rational homotopy type is insensitive to higher cohomological operations, in expected stark contrast to the case of compact complex ddbar-manifolds. To make use of this flexibility in varying the rational homotopy type, I will describe a simple, partially natural, construction to extend any given cdga to a rational Poincare duality algebra.

May 13: Cary Malkiewich (Binghamton)

Periodic orbits and topological Frobenius homology

Abstract: Suppose f: X -> X is a self-map of a finite complex, considered up to continuous homotopy. It is an insight of Kate Ponto that the most refined fixed point invariants of f naturally lie in topological Hochschild homology (THH), in other words the stable homotopy type of the free loop space of X. In previous work with Ponto we showed the same for the periodic points of f and topological restriction homology (TR), using a formal “unwinding” argument for traces of norm maps. In this talk, I will describe a project that uses the same formal insight to give a new invariant for periodic orbits of a continuous flow f: X x R -> X, up to continuous homotopy. This time, the invariant lives in topological Frobenius homology (TF), and lifts earlier invariants constructed by Fuller and by Geoghegan and Nicas.