# Topology Seminar

This is the home page of the seminar. Usual meeting time is Tuesdays at 11am, usually followed by a lunch with the speaker.

## Fall 2021 Schedule

Unfortunately we only had one talk this semester, but it was a wonderful in-person treat.

Date Speaker Title Abstract
Nov 9 Rok Gregoric Chromatic homotopy theory via spectral algebraic geometry A key aspect of chromatic homotopy theory is that structural properties of the stable homotopy category are reflected in the algebro-geometric properties of the moduli stack of formal groups. In this talk, we will discuss how to make that connection precise in the context of non-connective spectral algebraic geometry, using the stack of oriented formal groups.

## Spring 2021 Schedule

One more semester of virtual seminar; Zoom links will be sent to the topology mailing list (uiuc-topology google group); if you are interested, join the list.

A virtual social lunch meeting will follow the seminar talk. Bring your own food or drink or just yourself.

Date Speaker Title Abstract
Jan 26 XiaoLin Danny Shi Models of Lubin-Tate spectra via Real bordism theory In this talk, we will present Real-oriented models of Lubin-Tate theories at p=2 and arbitrary heights. For these models, we give explicit formulas for the action of certain finite subgroups of the Morava stabilizer groups on the coefficient rings. This is an input necessary for future computations. The construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory. As a consequence, we will describe how we can use these models to prove periodicity theorems for Lubin-Tate theories and set up an inductive approach to prove differentials in their slice spectral sequences. This talk is based on several joint projects with Agnès Beaudry, Jeremy Hahn, Mike Hill, Guchuan Li, Lennart Meier, Guozhen Wang, Zhouli Xu, and Mingcong Zeng.
Feb 2 Jeremy Hahn Redshift in algebraic K-theory I will describe work, joint with Dylan Wilson, about the redshifting properties of algebraic K-theory. I will focus on concrete examples, sketching proofs at the prime 2 that K(ko) has chromatic height 2 and K(tmf) has chromatic height 3.
Feb 16 Charles RezkUnderstanding accessible infinity-categories Lurie introuded the very important notion of "accessible infinity-category", a generalization of the more classical notion of "accessible category". These are (infinity-)categories which are produced from two pieces of data: a small (infinity-)category and a "regular cardinal". The goal of this talk is to give an introduction to some of the ideas surrounding these, and to put them in a broader context.
Feb 23 William BalderramaThe Borel $$C_2$$-equivariant $$K(1)$$-local sphere I'll talk about the structure of the Borel $$C_2$$-equivariant $$K(1)$$-local sphere. This captures $$Im~J$$-type phenomena in $$C_2$$-equivariant and $$\mathbb{R}$$-motivic stable stems, and gives a concise approach to understanding the $$K(1)$$-localizations of stunted projective spaces.
Mar 9 Tomer Schlank Cyclotomic Galois extensions in the chromatic homotopy. The chromatic approach to stable homotopy theory is "divide and conquer". That is, questions about spectra are studied through various localizations that isolate pure height phenomena and then are put back together. For each height n, there are two main candidates for pure height localization. The first is the generally more accessible K(n)-localization and the second is the closely related T(n)-localization. It is an open problem whether the two families of localizations coincide. One of the main reasons that the K(n)-local category is more amenable to computations is the existence of well understood Galois extensions of the K(n)-local sphere. In the talk, I will present a generalization, based on ambidexterity, of the classical theory of cyclotomic extensions, suitable for producing non-trivial Galois extensions in the T(n)-local and K(n)-local context. This construction gives a new family of Galois extensions of the T(n)-local sphere and allows to lift the well known maximal abelian extension of the K(n)-local sphere to the T(n)-local world. I will then describe some applications, including the study of the T(n)-local Picard group, a chromatic version of the Kummer theory, and interaction with algebraic K-theory. This is a joint project with Shachar Carmeli and Lior Yanovski.
Apr 6 Brian Shin A multiplicative theory of motivic infinite loop spaces From a spectrum $$E$$ one can extract its infinite loop space $$\Omega^\infty E = X$$. The space $$X$$ comes with a rich structure. For example, since $$X$$ is a loop space, we know $$\pi_0 X$$ comes with a group structure. Better yet, since $$X$$ is a double loop space, we know $$\pi_0 X$$ is in fact an abelian group. How much structure does this space $$X$$ possess? In 1974 Segal gave the following answer to this question: the structure of an infinite loop space is exactly the structure of a grouplike $$E_\infty$$ monoid. In fact, this identification respects multiplicative structures. In this talk, I'd like to discuss the analogue of this story in the setting of motivic homotopy theory. In particular we'll see that the motivic story, while similar to the classical one, has a couple interesting twists.
Apr 20 Ningchuan Zhang Exotic $$K(h)$$-local Picard groups when $$2p-1=h^2$$ and the Vanishing Conjecture The study of Picard groups in chromatic homotopy theory was initiated by Hopkins-Mahowald-Sadofsky. By analyzing the homotopy fixed point spectral sequence for the $$K(h)$$-local sphere, they showed that the exotic $$K(h)$$-local Picard group $$\kappa_h$$ is zero when $$(p-1)\nmid h$$ and $$2p-1>h^2$$. In this joint work in progress with Dominic Culver, we study $$\kappa_h$$ when $$p-1=h^2$$ and show that its vanishing is implied by a special case of Hopkins' Chromatic Vanishing Conjecture. Goerss-Henn-Mahowald-Rezk defined an algebraic detection map for $$\kappa_h$$, which is injective in this case. We will use the Gross-Hopkins duality reduce the target of the detection map to some Greek letter element computations. The vanishing of $$\kappa_h$$ is then implied by some bounds on the divisibility of those Greek letter elements. At height 3 and prime 5, the Miller-Ravenel-Wilson computation implies that exotic elements in $$\kappa_3$$ are not detected by a type 2 complex. The full vanishing of $$\kappa_3$$ requires a bound on the $$v_1$$-divisibility of $$\gamma$$-family elements. Using the same duality argument, we can also reduce the mod-$$p$$ Homological Vanishing Conjecture to some bounds on the divisibility of Greek letter elements. By comparing the bounds in both cases, we conclude that the Vanishing Conjecture implies $$\kappa_h=0$$ when $$2p-1=h^2$$.
Apr 27 Agnès Beaudry Equivariant quotients and localizations of norms of $$BP_{\mathbb{R}}$$ Quotients, localizations and completions of $$BP$$ play a central role in chromatic homotopy theory. For example, the Johnson-Wilson spectra $$E(h)$$ obtained by a quotient and localization of $$BP$$ are key players in the chromatic story at height $$h$$. However, working only with $$E(h)$$, the equivariance inherent to the chromatic story coming from the Morava stablizer group is obscured. A first step is to instead consider $$E_{\mathbb{R}}(h)$$, the Real Johnson-Wilson $$C_2$$-spectrum. However, for many heights $$h$$ there are bigger subgroups of the Morava stabilizer group lurking and $$E_{\mathbb{R}}(h)$$ does not capture their action. Indeed, restricting to finite cyclic 2-groups and for $$h=2^{n-1}m$$, the stabilizer group contains a subgroup isomorphic to $$C_{2^n}$$. In this talk, I will explain how one can instead consider quotients of norms of $$N_{C_2}^{C_{2^n}}BP_{\mathbb{R}}$$ to construct height $$h$$, $$C_{2^n}$$-equivariant analogues of $$E(h)$$.
May 4 Elizabeth Tatum Splitting $$BP\langle 2 \rangle \wedge BP \langle 2 \rangle$$ At Odd Primes In the 1980's, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $$bo \wedge bo$$ and $$l \wedge l$$. In this talk, we will construct an analogous splitting for the spectrum $$BP\langle 2 \rangle \wedge BP \langle 2 \rangle$$ at odd primes.

## Fall 2020 Schedule

This semester, the seminar is virtual. Zoom links will be sent to the topology mailing list (uiuc-topology google group); if you are interested, join the list.

A virtual social lunch meeting will follow the seminar talk. Bring your own food or drink or just yourself.

Date Speaker Title Abstract
Sep 15 Dan Berwick-Evans How do field theories detect the torsion in topological modular forms? Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms ($$TMF$$) for which cocycles are 2-dimensional (supersymmetric, Euclidean) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in $$TMF$$. However, the abundant torsion in $$TMF$$ has always been mysterious from the field theory point of view. In this talk, we will describe a map from Stolz and Teichner's category of field theories to a cohomology theory that approximates $$TMF$$. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target $$S^3$$ determines a cocycle representative of the generator of $$\pi_3(TMF)=\mathbb{Z}/24.$$
Oct 20 Richard Wong The Picard Group of the Stable Module Category for Quaternion Groups One problem of interest in modular representation theory of finite groups is in computing the group of endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by Carlson-Thevenaz using the theory of support varieties. However, I provide new, homotopical proofs of their results for the quaternion group of order 8, and for generalized quaternion groups, using the descent ideas and techniques of Mathew and Mathew-Stojanoska. Notably, these computations provide conceptual insight into the classical work of Carlson-Thevenaz.
Oct 27 Niny Arcila Maya Decomposition of Topological Azumaya Algebras Let $$\mathcal{A}$$ be a topological Azumaya algebra of degree $$mn$$ over a CW complex $$X$$. We give conditions for the positive integers $$m$$ and $$n$$, and the space $$X$$ so that $$\mathcal{A}$$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $$m$$ and $$n$$. Then we prove that if $$m<n$$ and the dimension of $$X$$ is higher than $$2m+1$$, $$\mathcal{A}$$ has no such decomposition.
Nov 3 Christopher Lloyd Calculating the nth Morava K-theory of the real Grassmannians using $$C_4$$-equivariance In this talk we will demonstrate how letting the cyclic group of order four act on the real Grassmannians can show the Atiyah-Hirzebruch spectral sequence calculating their 2-local nth Morava K-theory collapses. This uses chromatic fixed point theory coming from the classification of the equivariant Balmer spectrum of the cyclic groups. This work is joint with Nicholas Kuhn.
Nov 10 Eva Belmont $$\mathbb{R}$$-motivic homotopy theory and the Mahowald invariant The Mahowald invariant is a highly nontrivial map (with indeterminacy) from the homotopy groups of spheres to itself, with deep connections to chromatic homotopy theory. In this talk I will discuss a variant of the Mahowald invariant that can be computed using knowledge of the $$\mathbb{R}$$-motivic stable homotopy groups of spheres, and discuss its comparison to the classical Mahowald invariant. This is joint work with Dan Isaksen.
Nov 17 Morgan Opie Vector bundles on projective spaces Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on $$\mathbb{C}P^5$$. In particular, I will describe a classification of such bundles which involves a surprising connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on $$\mathbb{C}P^5$$ from simple ones; and future directions related to this project.

 Department of Mathematics 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350    Fax: (217) 333-9576     Email: math@illinois.edu