UCSD Topology Seminar

The UCSD Topology Seminar Fall 2022 meets on Tuesdays, in person at AP&M 7218, 4:30pm -- 5:30pm PDT.

See titles and abstracts below.

For any questions regarding the seminar, please email Eva Belmont, Yunze Lu, or me.

UCSD Topology Seminar

Titles and Abstracts -- Fall 2022

Research Talks

Title: Equivariant A-theory and spaces of equivariant h-cobordisms

Abstract: Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about the equivariant homotopical lift of the h-cobordism theorem.

Title: New perspectives on the étale homotopy type

Abstract: Étale homotopy theory was invented by Artin and Mazur in the 1960s as a way to associate to a scheme X, a homotopy type with fundamental group the étale fundamental group of X and whose cohomology captures the étale cohomology of X with locally constant constructible coefficients. In this talk we’ll explain how to construct a stratified refinement of the étale homotopy type that classifies constructible étale sheaves and gives rise to a new definition of the étale homotopy type. The stratified étale homotopy type also plays a role in the reconstruction of schemes: in nice cases, schemes can be completely reconstructed from their stratified étale homotopy types. This is joint work with Clark Barwick and Saul Glasman.

Title: Structures and computations in the motivic stable homotopy categories

Abstract: A fundamental question in classical stable homotopy theory is to understand the stable homotopy groups of the spheres. A relatively new method is via the motivic approach. Motivic stable homotopy theory has an algebro-geometric root and closely connects to questions in number theory. Besides, it relates to the classical and the equivariant theories. The motivic category has good properties and allows different computational tools. I will talk about some of these properties and computations, and will show how it relates to the classical and equivariant categories.

Title: Multiplicative uniqueness of rational equivariant K-theory

Abstract: Topological K-theory is one of the classical motivating examples of a commutative ring spectrum, and it has a natural equivariant generalization. The equivariant structure here has the strongest possible type of compatibility with the multiplication, making K-theory an example of a "genuine-commutative" ring spectrum. There's quite a lot of structure involved here, so in order to understand it, we employ a classic strategy and rationalize. After rationalizing, we can use algebraic models due to Barnes--Greenlees--Kedziorek and to Wimmer to show that all of the additional "norm" structure is determined by the equivariant homotopy groups and the underlying multiplication. This is joint work with Christy Hazel, Jocelyne Ishak, Magdalena Kedziorek, and Clover May.





Student Talks

  • Yueshi Hou

  • Yuchen Wu

  • Cheng Li

  • Maxwell Johnson

  • Scotty Tilton

  • Arseniy Kryazhev

  • Shangjie Zhang

  • Jordan Benson

Titles and Abstracts -- Spring 2022

Research Talks

Title: The motivic lambda algebra and Hopf invariant one problem

Abstract: Current best approaches to understanding the stable homotopy groups of spheres at the prime 2 make use of the Adams spectral sequence, which computes stable stems starting with information about the cohomology of the Steenrod algebra. The first major success of the Adams spectral sequence was in Adams' resolution of the Hopf invariant one problem. Later, J.S.P. Wang used a certain algebraic device, the lambda algebra, to give a more thorough computation of the cohomology of the Steenrod algebra, and used this to give a much simplified and almost entirely algebraic derivation of the Hopf invariant one differentials.

In this talk, I will go over some of the above history, and then describe work (joint with Dominic Culver and J.D. Quigley) on analogues in motivic stable homotopy theory. In particular, I will describe a mod 2 motivic lambda algebra, defined over any base field of characteristic not equal to 2, as well as some of what can be said about the 1-line of the motivic Adams spectral sequence for various base fields.

Title: Motivic cohomology and negative K-theory

Abstract: In joint work with Matthew Morrow, we construct a motivic cohomology of singular schemes. This captures a part of K-theory which sees singularities - negative K-theory. I will talk about the latter and talk about some observations. In particular, the talk will be hopefully elementary and accessible to algebraic topologists and algebraic geometers.

Title: Moduli of oriented formal groups and cellular motivic spectra

Abstract: The moduli stack of oriented formal groups embodies, in the world of spectral algebraic geometry, the fundamental chromatic connection between the stable homotopy category and formal groups. As such, it validates the folklore picture of Morava, Hopkins, et al. Somewhat surprisingly, it is also closely related to a more recent development: the "cofiber of tau philosophy" of Gheorghe-Isaksen-Wang-Xu.

In this talk, we will introduce the moduli stack of oriented formal groups, and explain how the algebro-geometric structure of its connective cover reflects and gives rise to the tau-deformation structure of cellular motivic spectra over C.

  • April 19: Dana Hunter, U of Oregon

Title: The Curtis-Wellington spectral sequence through cohomology

Abstract: In this talk, we will discuss an unstable approach to studying stable homotopy groups as pioneered by Curtis and Wellington. Using the Barratt-Priddy-Quillen theorem, we can identify the (co)homology of $BS_\infty$ with the (co)homology of the base point component of the loop space which represents stable homotopy. Using cohomology instead of homology to exploit the nice Hopf ring presentation of Giusti, Salvatore, and Sinha for the cohomology of symmetric groups, we find a width filtration, whose subquotients are simple quotients of Dickson algebras, which thus give a new filtration of stable homotopy. We make initial calculations and determine towers in the resulting width spectral sequence. We also make calculations related to the image of J and conjecture that it is captured exactly by the lowest filtration in the width spectral sequence.

  • April 26: Yang Hu, U of Oregon

Title: Metastable complex vector bundles over complex projective spaces

Abstract: We study unstable topological complex vector bundles over

complex projective spaces. It is a classical problem in algebraic

topology to count the number of rank r bundles over \mathbb{C}P^n (with

1 < r < n) having fixed Chern class data. A particular case is when the

Chern data is trivial, which we call the vanishing Chern enumeration. We

apply a modern tool, Weiss calculus, to produce the vanishing Chern

enumeration in the first two unstable cases (which belong to what we

call the “metastable” range, following Mark Mahowald), namely rank (n -

1) bundles over \mathbb{C}P^n for n > 2, and rank (n - 2) bundles over

\mathbb{C}P^n for n > 3.

  • May 3: Peter Marek, Indiana

Title: Computing with Synthetic Spectra

Abstract: In recent years, our understanding of stable homotopy groups of spheres at p=2 increased drastically due to work of Isaksen, Wang, and Xu. A primary method they used is the "cofiber-of-tau philosophy" in the stable infinity category of 2-complete C-motivic spectra. To a sufficiently nice spectrum E, Pstragowski produced an infinity-categorical deformation of spectra called "E-synthetic spectra," which exhibits and generalizes the cofiber-of-tau phenomena seen in C-motivic spectra. E-synthetic spectra are closely related to the E-Adams spectral sequence and this relation has had many applications in recent years for Adams spectral sequence calculations.

In this talk, we discuss some of the basic calculational features of synthetic spectra in the case of E=HF_2, including how to compute bigraded synthetic homotopy groups and their applications to classical Adams spectral sequence calculations for p=2. In particular, we discuss our computation of the bigraded synthetic homotopy groups of 2-complete tmf, the connective topological modular forms spectrum.

Title: 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 Dedekind zeta functions and algebraic K-groups of number fields. 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 with the equivariant Moore spectra introduced in my thesis. This is joint work in progress with Elden Elmanto.

Student Talks

  • March 29: Cheng Li

  • April 5: Arseniy Kryazhev

  • April 12: Arseniy Kryazhev

  • April 19: Shangjie Zhang

  • April 26: Shangjie Zhang

  • May 3: Shangjie Zhang

  • May 24: Maxwell Johnson

  • May 31: Maxwell Johnson

Titles and Abstracts -- Winter 2022

Research Talks

Title: The Dual Motivic Witt Cohomology Steenrod Algebra

Abstract: Over a field k, the zeroth homotopy group of the motivic sphere spectrum is given by the Grothendieck-Witt ring of symmetric bilinear forms GW(k). The Grothendieck-Witt ring GW(k) modulo the hyperbolic plane is isomorphic to the Witt ring of symmetric bilinear forms W(k) which further surjectively maps to Z/2. We may take motivic Eilenberg-Maclane spectra of Z/2. W(k) and GW(k). Voevodsky has computed the motivic Steenrod algebra of HZ/2 and solved the Bloch-Kato conjecture with its help. We move one step up in the above picture; we study the motivic Eilenberg-Maclane spectrum corresponding to the Witt ring and compute its dual Steenrod algebra.

Title: Generalizations of Hochschild homology for rings with anti-involution

Abstract: In the late 1980’s, Krasauskas and Fiedorowicz-Loday independently developed the theory of crossed simplicial groups, which generalize Connes’ cyclic category. Of particular interest is the Dihedral category, which has recently been used to develop the theory of Real topological Hochschild homology, a first approximation to Grothendieck-Witt groups.

In the first part of my talk, I will discuss ongoing joint work with Mona Merling and Maximilien Péroux on a topological analogue of the homology of crossed simplicial groups. As a special case, we recover the theory of Real topological Hochschild homology.

In the second part of my talk, I will discuss joint work with Teena Gerhardt and Mike Hill. We provide a norm model for Real topological Hochschild homology, prove a multiplicative double coset formula for Real topological Hochschild homology, and we construct the Real Witt vectors of rings with anti-involution.

Title: The RO(C_2)-graded homology of C_2-equivariant Eilenberg-Maclane spaces

Abstract: This talk describes work in progress computing the H\underline{\mathbb{F}}_2 homology of the C_2 - equivariant Eilenberg-Maclane spaces associated to the constant Mackey functor \underline{\mathbb{F}}_2. We extend a Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the RO(C_2)-graded setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which is quite complicated, lacking an explicit E^2 page and having arbitrarily long equivariant degree shifting differentials. We avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson along with norm and restriction maps to complete the computation.

Title: Towards Splitting $BP<2> ⋀ BP<2>$ at Odd Primes

Abstract: In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo ⋀ bo$ and $l ⋀ l$. These splittings helped make it feasible to do computations using the $bo$- and $l$-based Adams spectral sequences. I will discuss progress towards an analogous splitting for $BP<2> ⋀ BP<2>$ at odd primes.

Title: The cohomology of C_2-surfaces with constant integral coefficients

Abstract: Let C_2 denote the cyclic group of order 2. In this talk,

we’ll explore some recent computations done in RO(C_2)-graded cohomology

with constant integral coefficients for C_2-surfaces. We’ll also explore

some interesting patterns in these computations, and discuss how these

might generalize to C_2-manifolds of higher dimension.

  • February 8: No seminar

  • February 15: Catherine Ray, Northwestern

Title: Galois Theory in Homotopy Theory

Abstract: We construct ramified families of curves to explicitly model the Lubin-Tate action, the action of a formal group law on its deformation space, for a maximal finite subgroup G. We will see that as a G-representation, this deformation space is a quotient of a regular representation of a finite cyclic group! This allows us to compute the E2 page of the homotopy fixed point spectral sequence of the K(h, p) - local homotopy groups of spheres for height h=p^{k-1}(p-1), for all such h and p simultaneously.

  • February 22: Guchuan Li, Michigan

Title: Vanishing results in Chromatic homotopy theory at prime 2

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 fixed points of Lubin--Tate theories $E_h$. These fixed points are 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 $N(h,G)$.

This vanishing result has proven to be computationally powerful, as demonstrated by Hill--Shi--Wang--Xu’s recent computation of $E_4^{hC_4}$. Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem. As an application, we extend Kitchloo--Wilson’s $E_n^{hC_2}$-orientation results to all $E_n^{hG}$-orientations at the prime 2. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.

Title: The Homology of BP_R<n>

Abstract: The truncated Brown-Peterson spectra admit actions by the cyclic group of order 2 via complex conjugation. Their fixed point spectra are higher height analogues of real K-theory. We describe how to use Tate square methods along with the slice spectral sequence to compute their mod 2 homology. This is joint work in progress with Mike Hill and Doug Ravenel.

Title: The chromatic Nullstellensatz

Abstract: 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 joint work with Robert Burklund and Tomer Schlank on a chromatic version of Hilbert’s Nullstellensatz in which Lubin-Tate theories play the role of algebraically closed fields. I will then sample some applications of our results to chromatic support, redshift, and orientation theory for E-infty rings.

Student Talks

  • January 11: Jordan Benson

  • January 18: Scotty Tilton

  • January 25: Scotty Tilton

  • February 1: Cheng Li

  • February 8: Maxwell Johnson

  • February 15: Arseniy Kryazhev

  • February 22: Shangjie Zhang

  • March 1: Shangjie Zhang

  • March 8: Maxwell Johnson

Titles and Abstracts -- Fall 2021

Research Talks

Title: The RO(G) graded cohomology of G-equivariant classifying spaces

Abstract: The cohomology of classifying spaces is an important classical topic in algebraic topology.

However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces.

The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic.

In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum.

The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

Title: A comparison between C_2-equivariant and classical squaring operations

Abstract: For any C_2-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from the RO(C_2)-graded cohomology to the classical cohomology. In this talk, I will compare the RO(C_2)-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.

Title: Calculations in nonabelian equivariant cohomology

Abstract: Calculating the coefficients of equivariant generalized cohomology theories has been a fundamental question for equivariant homotopy theory. In this talk, I will talk about some calculations when the group is nonabelian. Examples include RO(G)-graded Eilenberg-MacLane cohomology of a point with constant coefficient when G is a dihedral group of order 2p or the quaternion group Q_8, and coefficient ring of \Sigma_3-equivariant complex cobordism. I will discuss techniques in such computations: isotropy separation, cellular structures and dualities. This is joint work with Po Hu and Igor Kriz.

Title: Deformation Theory and Supersymmetric Quantum Mechanics

Abstract: There is a deep relationship between deformation theory for symplectic manifolds and quantizing field theories. In this talk, I'll discuss this story for symplectic supermanifolds and supersymmetric mechanics. We will approach these questions using modern descent techniques that work more generally for factorization algebras associated to higher-dimensional field theories. Relations to manifold invariants such as the L-genus will also be discussed. No physics knowledge is required.

Title: Equivariant Steenrod Operations

Abstract: Classical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory. It led to the theory of characteristic classes, which is key to some of the most celebrated applications of homotopy theory to geometry. The G-equivariant Steenrod algebra is not known beyond the group of order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and generalize it to construct G-equivariant Steenrod operations. Time permitting, I will discuss potential applications to equivariant geometry.

Student Talks

  • November 2: Scotty Tilton

Title: Introduction to Spectra I

  • November 9: Arseniy Kryazhev

Title: Introduction to Spectra II

  • November 16: Shangjie Zhang

Title: Models of Spectra

  • November 23: Maxwell Johnson

Title: The Adams spectral sequence

  • November 30: Jordan Benson

Title: The May spectral sequence