Columbia Algebraic Topology Seminar

Spring 2023

The seminar will meet Fridays 10:00-11:00 AM in person in room 507. Please contact Inbar Klang or Allen Yuan with any inquiries regarding the seminar.

February 24: Elden Elmanto (Harvard)

Voevodsky’s slice conjecture via prismatic cohomology

Abstract: We prove a conjecture of Voevodsky that the zero-th slice of the motivic sphere is always motivic cohomology using prismatic cohomology. This offers a universal property of algebraic cycles for arbitrary schemes. Scary words aside, the teleological proof is quite easy and I will explain that. Joint with Tom Bachmann and Matthew Morrow.

March 3: Maru Sarazola (JHU)

Fibrantly-induced model structures

Abstract: Model structures are robust categorical structures that provide an abstract framework to do homotopy theory. Unfortunately, in practice it is often very hard to prove that something satisfies the requirements of a model structure. To this end, there are several results in the literature that explore techniques for constructing model structures on a given category C. A natural way to do this is to right-induce it through a right adjoint $R: C\to M$ to a known model structure M. This process defines the fibrations and weak equivalences in C as the morphisms whose image under R is one such map in M. Depending on the context, however, this may prove too restrictive. For example, one may be working in a setting where there is a desired class of “fibrant objects” in mind for C, and where the best one can hope for is for these well-behaved classes of fibrations and weak equivalences to hold only between fibrant objects. In this talk, I will present a generalization of the classical right transfer theorem that allows us to do this, and time permitting, some applications. Based on joint work with Leonard Guetta, Lyne Moser and Paula Verdugo.

April 14: Carissa Slone (Rochester) — NOTE: Talk will be virtual, at the link

Two-slices over cyclic groups of prime power order

Abstract: The slice filtration focuses on producing certain spectra, called slices, from a genuine G-spectrum X over a finite group G. We have a complete characterization of all 1-, 0-, and (-1)-slices for any such G, and a characterization for 2-slices over C_2 and Klein-4. We will characterize 2-slices over C_p (p odd) and expand this characterization to C_{p^n}.

April 21: Tomer Schlank (Hebrew University / MIT)

Ambidexterity in Chromatic Homotopy Theory

Abstract: The monochromatic layers of the chromatic filtration on spectra, that is- the K(n)-local (stable 00-)categories Sp_{K(n)} enjoy many remarkable properties. One example is the vanishing of the Tate construction due to Hovey-Greenlees-Sadofsky. The vanishing of Tate construction can be considered as a natural equivalence between the colimits and limits in Sp_{K(n)} parametrized by finite groupoids. Hopkins and Lurie proved a generalization of this result where finite groupoids are replaced by arbitrary \pi-finite 00-groupoids. They named this phenomenon “ambidexterity” or “higher semi-additivity”. I shall describe this phenomenon and attempt to demonstrate that it creates a surprising amount of properties and structure in the heart of chromatic homotopy. In particular, higher semi-additivity can be used as a tool to study the somewhat less approachable version of “monochromatic layers”, namely the T(n)-local category as well as T(n)-localized algebraic K-theory.

April 28: Kate Ponto (Kentucky)

Homotopical invariants for lifting and extensions

Abstract: Motivated by question in fixed point theory, Klein and Williams defined a homotopical obstruction to lifts and extensions that is not the familiar cohomological one. While they considered only a topological setting, their approach is remarkably flexible. I’ll describe how to make sense of it using a model category and a Blakers-Massey theorem.

May 5: Peter Haine (Berkeley / IAS)

Spectral weight filtrations

Abstract: This talk is a report on joint work in progress with Piotr Pstrągowski. Pstrągowski defined a left adjoint SH(ℂ)cell → Synev from cellular ℂ-motivic spectra to even (MU-based) synthetic spectra. This functor refines the Betti realization of a cellular motivic spectrum, and for any prime p, restricts to an equivalence on p-complete objects. We’ll explain how to further refine the Betti realization functor SH(ℂ) → Sp to a left adjoint SH(ℂ) → Syn to all synthetic spectra. To do this, we’ll give a description of motivic spectra as sheaves on a subcategory of compact pure motives. This description also lets us show that for a complex-orientable connective ring spectrum A, the A-linear Betti realization SH(ℂ) → ModA refines to a left adjoint functor landing in filtered A-modules. We’ll also explain the relationship between this story and weight filtrations on the Betti cohomology of a complex variety.

May 12: Sander Kupers (Toronto)

The Disc-structure space of a manifold

Abstract: Surgery theory attempts to understand the category of manifolds by mapping it to the category of topological spaces, with the goal of understanding the target and fiber of this map using homotopy-theoretic methods. In this talk I’ll propose an alternative inspired by embedding calculus, mapping the category of manifolds to the category of presheaves on the category of discs, and explore its surprising features. This is joint work with Manuel Krannich.

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! Link: )

Higher real K-theories in chromatic homotopy theory

Abstract: Chromatic homotopy theory is a powerful tool to study periodic phenomena in the stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the homotopy fixed points of Lubin–Tate theories $E_h$. These fixed points are generalizations of real K-theories with periodic homotopy groups that can be computed via homotopy fixed points spectral sequences. In this talk, we prove that at the prime 2, for all heights $h$ and all finite subgroups $G$ of the Morava stabilizer group, the $G$-homotopy fixed point spectral sequence of $E_h$ collapses after the $N(h,G)$-page and admits a horizontal vanishing line of filtration $N(h,G)$ where $N(h,G)$ are specific numbers. Our proof uses new equivariant techniques developed by Hill–Hopkins–Ravenel in their solution of the Kervaire invariant one problem. If time allows, I will present a computation of topological modular forms based on this vanishing result. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.

November 18: Adela Zhang (MIT)

Universal differentials in the bar spectral sequence

Abstract: Encoding spectral sequences with filtered objects is a powerful tool in understanding the structure of spectral sequences. I will explain how this approach helps one identify universal differentials in the bar spectral sequence associated to an algebra over an operad. This is mostly based on joint work with Andrew Senger.

December 2: Lucy Yang (Harvard)

Categorical dynamics on stable module categories

Abstract: Topological entropy is a measure of the complexity of a continuous self-map of a compact topological space. Categorical entropy generalizes this measure to exact endofunctors of triangulated categories. In this work we ask: How can categorical entropy serve to quantify growth in stable homotopy theory, and how can the methods of stable homotopy theory aid in understanding and computing categorical entropy? We show that the categorical polynomial entropy of a twist functor on the stable module category of a finite connected graded Hopf algebra over a field recovers one less than the Krull dimension of its cohomology, generalizing a computation of Fan, Fu, and Ouchi. Along the way, we will see how a stable homotopy theoretic-perspective permits us to make this refinement.

December 9: Foling Zou (UMich)

The C_p-equivariant dual Steenrod algebra for odd prime p

Abstract: Non-equivariantly, the dual Steenrod algebra spectrum HZ/p \smash HZ/p is a wedge of suspensions of HZ/p. I will talk about the computation of the Hopf algebroid (H_*, (H \smash H)_*), where H = HZ/p is the equivariant Eilenberg–MacLane spectra for G = C_p, the cyclic group of order p. It turns out that when p is odd, H \smash H is a wedge of suspensions of H and another spectrum, which we call HT. We found these suspension generators can be obtained using the Steenrod coaction on the G-spaces B_G S^1 and B_G Z/p. This is joint work with Po Hu, Igor Kriz, and Petr Somberg.

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.