Topology Seminar
Past Talks
Listed below are details about past talks of the 20232024 UChicago Algebraic Topology Seminar.
Spring 2024

 Mar 262024
Elden Elmanto (University of Toronto)
Motivic cohomology of schemes (and stacks?)
In the 80's Beilinson and Lichtenbaum conjectured the existence of motivic cohomology, complexes which form the graded pieces of a filtration on algebraic Ktheory. I will speak on recent development on motivic cohomology of equicharacteristic schemes (joint with Morrow) and an extension to stacks in characteristic zero (with Kubrak and Sosnilo). I will focus on one local and one global computation  of the Milnor range and of ``zero cycles'' on cones respectively.

 Mar 192024
Inna Zakharevich (Cornell University)
Cyclic objects and Ktheory
The Dennis trace has been one of the most important computational tools for algebraic Ktheory. It is constructed by linearizing and adding a cyclic structure to Ktheory. In this talk we will discuss a different type of cyclic structure on Ktheory and its relation to the Dennis trace. This is joint work with Jonathan Campbell and Kate Ponto.
Winter 2024

 Mar 052024
William Balderrama (University of Virginia)
Some chromatic and computational aspects of equivariant Ktheory and equivariant cobordism
Chromatic homotopy theory is an approach to finding regular patterns in stable homotopy theory. This approach filters spectra by a form of complexity called height, with each layer exhibiting periodic behavior of varying wavelengths. The subject has its origins in work of Adams and others at what we now call height 1: topics surrounding topological Ktheory and the Jhomomorphism. Higher heights arise when using complex cobordism and related theories to connect homotopical phenomena to the geometry of 1dimensional formal group laws. This talk will be about some things you find when trying to ``do chromatic homotopy theory'' in the context of equivariant stable homotopy theory. The focus will be on some particular examples and computations, rather than grand overarching theorems. At height 1, I'll describe some phenomena detected by equivariant Ktheory. At higher heights, I'll say a little about equivariant complex cobordism, and describe a computation connecting equivariant Real bordism ``$MR_G$'' to the moduli of equivariant formal group laws.

 Feb 202024
Naruki Masuda (Johns Hopkins)
Categorical spectra and their tensor product
Categorical spectra are a higher categorical generalization of spectra. Namely, it is a sequence $(X_n)$ of pointed $(\infty, \infty)$categories with the identification $X_n=\Omega X_{n+1}$, where the latter means the endomorphisms of the basepoint. I will give some examples and explain why this is something worth exploring. I will define a (nonsymmetric) tensor product of categorical spectra extending one of spectra and give some TQFTrelated applications.

 Feb 132024
Thomas Brazelton (Harvard University)
Metalinear vector bundles
Throughout the literature, from motivic homotopy theory to real enumerative geometry, one cares about making a compatible choice of charts on real manifolds and their vector bundles in order to produce welldefined characteristic classes. A natural class of socalled ''metalinear'' bundles which satisfy this condition support a theory of Euler classes in motivic homotopy theory which are computationally accessible. In this talk we will compute the ChowWitt theory of the classifying space of metalinear vector bundles and discuss connections to the literature. This is joint work in progress with Matthias Wendt.

 Jan 302024
Liam Keenan (University of Minnesota)
A chromatic vanishing result for TR
Over the past few years, there has been enormous progress in our understanding of the interaction between chromatic homotopy theory and trace methods, motivated by Rognes' chromatic redshift philosophy. In this talk, I will survey some of these recent results and explain some joint work with Jonas McCandless in which we apply some of these ideas to topological restriction homology.

 Jan 232024
Sanath Devalapurkar (Harvard University)
Equivariant homotopy theory and geometric Langlands duality
Equivariant cohomology has played a pivotal role in the development of both modern homotopy theory and geometric Langlands duality. Some basic manifestations of the relationship between these two fields can already be seen in classical work of Chevalley and Borel (among others). In this talk, I will describe some of the ways in which equivariant homotopy theory  via generalized cohomology theories such as complex Ktheory and elliptic cohomology  can inform and extend parts of the story of geometric Langlands. In particular, I will describe some of my recent work on a ''geometrization'' of the theory of spherical harmonics (which is a special case of recent conjectures of BenZvi, Sakellaridis, and Venkatesh). In the pretalk, I will give some background on (generalized) equivariant cohomology theory.

 Jan 162024
Ishan Levy (MIT)
Telescopic stable homotopy theory
Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into $v_n$periodic layers corresponding to height $n$ formal groups. There are two natural ways to do this, via either the $K(n)$localizations which are computationally accessible, or via the $T(n)$localizations, which detect the $v_n$periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For $n$ at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the $T(n)$ and $K(n)$localizations of the algebraic Ktheory of a family of ring spectra, which in the case $n=2$ are certain finite Galois extensions of the $K(1)$local sphere. I will then explain that this can be used to obtain an infinite family of elements in the $v_n$periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the $T(n)$local category agrees with that of the $K(n)$local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.

 Jan 092024
Piotr Pstrągowski (Harvard University)
The even filtration and prismatic cohomology
The even filtration, introduced by HahnRaksitWilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the AdamsNovikov filtration of the sphere or the BhattMorrowScholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of $\mathbb{E}_2$rings.
Autumn 2023

 Nov 282023
Rhiannon Griffiths (Cornell University)
The homotopy types of higher categories
Grothendieck’s Homotopy Hypothesis states that homotopy ntypes are modeled by $n$groupoids, and by extension, that spaces are modeled by infinity groupoids. Moreover, this equivalence is induced by the homotopy groups of both constructions.
While there are some more geometric solutions to the HH, the most interesting solution would be to show that some completely algebraic $n$groupoids are equivalent to homotopy $n$types. Unfortunately, the HH is known to be false for strict higher groupoids, and fully weak ones are too complex to work with.
In this talk, I will present an operadic method for calculating the homotopy groups of algebraic higher groupoids. I will then use this to identity models algebraic $n$groupoid that are tractable enough to work with directly, but which escape the homotopy degeneracies that appear in strict $n$groupoids.

 Nov 212023
NO SPEAKER (Thanksgiving Break)

 Nov 142023
Candace Bethea (University of South Carolina)
Equivariant curve counts valued in the Burnside Ring of a finite group
Enumerative geometry asks for integral solutions to geometric questions, such as how many genus $g$, degree $d$ rational curves with $n$ marked points lie on a given surface. Recently, motivic and equivariant homotopy have been used to generalize classical enumerative results to nonclosed fields and under the presence of a group action respectively. In this talk, we will discuss the foundational ideas of equivariant enumerative geometry and results on the count of nodal orbits in an invariant pencil of plane curves, enriched in the Burnside Ring of a finite group. Time permitting, we will discuss joint work in progress with Kirsten Wickelgen on computing GromovWitten invariants equivariantly.

 Nov 092023
Tomer Schlank (Hebrew University of Jerusalem)
Stable homotopy theory and the telescope conjecture
Spectra are the homotopy theorist abelian groups, they have a fundamental place in algebraic topology but also appear in arithmetic geometry, differential topology, mathematical physics and symplectic geometry. In a similar vein to the way that abelian groups are the bedrock of algebra and algebraic geometry we can take a similar approach of spectra, I will discuss the picture that emerges and how one can use it to address classical questions about homotopy groups of spheres, algebraic Ktheory and cobordism classes.

 Nov 072023
Guoqi Yan (Notre Dame)
The generalized Tate diagram of the equivariant slice spectral sequence
The generalized Tate diagram developed by Greenlees and May is a fundamental tool in equivariant homotopy theory. In this talk, I will discuss an integration of the generalized Tate diagram with the equivariant slice filtration of HillHopkinsRavenel, resulting in a generalized Tate diagram for equivariant spectral sequences. This new diagram provides valuable insights into various equivariant spectral sequences and allows us to extract information about isomorphism regions between these equivariant filtrations.
As an application, we will begin by proving a stratification theorem for the negative cone of the slice spectral sequence. Building upon the work of MeierShiZeng, we will then utilize this stratification to establish shearing isomorphisms, explore transchromatic phenomena, and analyze vanishing lines within the negative cone of the slice spectral sequences associated with periodic HillHopkinsRavenel and LubinTate theories.
This is joint work with Yutao Liu and XiaoLin Danny Shi.

 Oct 312023
Bar Roytman (UCLA)
Highly Structured Real Orientations
The Fujii–Landweber Real bordism is a central object in equivariant approaches to chromatic homotopy at the prime 2. For example, one technique of recent interest is to employ Real orientations to help compute slice spectral sequences of spectra satisfying a Real analogue of Landweber exactness. However, even after passage to homotopy groups, Real orientations are not guaranteed to preserve all multiplicative structures available in Real bordism. We will discuss how this hindrance can be overcome with the theory of operads.

 Oct 242023
Connor Malin (Notre Dame)
Applications of Koszul duality to Goodwillie calculus and factorization homology
Goodwillie calculus is a homotopical technique to understand functors through successive approximation. There are surprising connections between Goodwillie calculus, operad theory, and Koszul duality. In the case of functors from the category of $\mathcal{O}$algebras to spectra, we use Koszul duality to show that the Goodwillie derivatives admit right $\mathcal{O}$module structures. As an application, we give a new construction of the Poincare/Koszul duality map in factorization homology.

 Oct 172023
Jeremy Hahn (MIT)
Telescopes, prismatization, and exotic spheres
A smooth, oriented $n$manifold is called a homotopy sphere if it is homeomorphic, but not necessarily diffeomorphic, to the standard $n$sphere. In dimensions $n>4$, one often studies the group $\Theta_n$ of homotopy spheres up to orientationpreserving diffeomorphism, with group operation given by connected sum. I will give a leisurely introduction to the telescope conjecture in stable homotopy theory, and explain how its failure gives new lower bounds on the complexity of $\Theta_n$. To disprove the telescope conjecture, we construct invariants capable of distinguishing many diffeomorphism classes of exotic spheres: interestingly, key finiteness properties of these invariants are proved in part using intuitions and ideas from prismatic cohomology in $p$adic algebraic geometry. The talk is based on joint projects with Burklund, Carmeli, Levy, Raksit, Schlank, Wilson, and Yanovski.
Note: The pretalk this week has its own title and abstract, as follows:
The motivic filtration
Say that a commutative ring spectrum is even if its homotopy groups are trivial in odd degrees. We can approximate an arbitrary ring $R$ by taking the limit, over all maps $R\to A$ with $A$ even, of $A$. Moreover, we can filter $R$ by declaring $\mathrm{fil}^n(R)$ to be the limit, over all maps $R\to A$ with A even, of the $(2n1)$connected cover of A. I will discuss how to compute $\mathrm{fil}^*(R)$ for various commutative ring spectra $R$. This is joint work with Arpon Raksit and Dylan Wilson.

 Oct 102023
Andrew Senger (Harvard University)
The $\mathrm{mod}$ $(p,v_1)$ Ktheory of $\mathbb{Z}/p^n$
While $\mathbb{Z}/p^n$ is a relatively simple ring, its algebraic Ktheory groups are very complicated, and their computation has been an open problem for many years. Recently, prismatic methods have made the computation of these groups a purely algebraic question, and AntieauKrauseNikolaus have leveraged this to (among other results) write a computer program which can compute $K_*(\mathbb{Z}/p^n)$ in a range of stems. In this talk, I will describe a different approach to the computation of $K_*(\mathbb{Z}/p^n)$. Instead of computing the groups in a range, we completely compute the mod $(p,v_1)$ algebraic Ktheory $\pi_* K (\mathbb{Z}/p^n)/(p,v_1)$, which is substantially simpler. Time permitting, I will share some hopes and results about the $v_1$Bockstein spectral sequence converging to $\pi_* K (\mathbb{Z}/p^n)/p$. This is joint work with Jeremy Hahn and Ishan Levy.

 Oct 032023
Josefien Kuijper (Stockholm University)
Sixfunctor formalisms are compactly supported
I will discuss a new infinitycategorical definition of abstract sixfunctor formalisms. Our definition is a variation on Mann's definition, with the additional requirement of having Grothendieck and Wirthmüller contexts, and recollements. Using Nagata's compactification theorem, we show that such a sixfunctor formalism can be given by just specifying adjoint triples on open immersions and on proper maps, satisfying certain compatibilities. Moreover, the existence of recollements is equivalent to a sheaf condition for a Grothendieck topology on the category of “varieties and spans with an open immersion and a proper map”. This brings to light an interesting analogy between abstract sixfunctor formalisms and compactly supported cohomology. We can show that sixfunctor formalisms, according to our definition, are uniquely determined by the restriction of the inverse image (upper star) to smooth and complete varieties. Moreover we can characterize which lax symmetric monoidal functors from the category of complete varieties to the category of stable infinitycategories and adjoint triples, extend to sixfunctor formalisms.

 Sep 262023
Anh Trong Nam Hoang (University of Minnesota)
FoxNeuwirth cells, quantum shuffle algebras, and twisted homology of surface braid groups
Braided vector spaces play an important role in the study of Hopf algebras. Tensor powers of a braided vector space $V$ form a family of braid group representations. When equipped with a shuffle product, they also form a noncommutative, noncocommutative braided Hopf algebra called the quantum shuffle algebra. Recently, Ellenberg, Tran, and Westerland identified the homology of braid groups with these coefficients with the cohomology of this algebra. In this talk, we will extend their techniques to prove a similar result for the homology of braid groups on punctured planes. Time permitting, we will discuss some computations in cases relevant to arithmetic applications.