# Topology Seminar

The Topology Seminar is a research seminar featuring a mixture of guest and local speakers organised by Diarmuid Crowley and Marcy Robertson. The seminar has returned to its usual meeting time 2.15pm – 3.15pm on Mondays. Due to the ongoing pandemic the seminar is *currently running online*. You can sign up for our mailing list by emailing either Marcy or Diarmuid.

**Next Talk:**

♦ Monday May 24th (**5:15-6:15**): Markus Land **(Copenhagen)**

**Title: **On the K-theory of pushouts and pullbacks

**Abstract: **I will report on joint work with Tamme. First, I will give a brief review of (a slight generalisation) of our previous theorem on the K-theory of pullbacks and will discuss some of its applications such as Suslin’s famous excision theorem and the fact that invariants which we call truncating have many pleasant properties like excision, nilinvariance, and cdh-descent. I the remaining time, I will then indicate how, under certain assumptions, we can obtain a formula for the circle-dot ring appearing in our previous theorem as a pushout of E_1-ring spectra, and explain some examples to illustrate that the resulting pushouts can often be calculated explicitly, allowing for new calculations of K-theories to be performed.

**Semester 1, 2021**

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♦ Monday March 8th: **Diarmuid Crowley**

**Title: **The Topological Period-Index Problem and Surgery

**Abstract:** The Topological Period-Index Problem is about finding the minimal rank of a topological Azumaya Algebra representing a given element of the Brauer group TH^3(X) of a space X. It is related to similar period-index problems in algebraic geometry and its study is organised the Topological Period-Index Conjecture (TPIC).

While the TPIC holds for almost complex 6-manifolds, it is open for almost complex 8-manifolds. In this talk I will present a conjectural counter example to the TPIC for almost complex 8-manifolds, which is constructed using a new definition of “success” in surgery.

This is part of joint work with Mark Grant, Xing Gu and Christian Haesemeyer and also Csaba Nagy.

♦ Monday March 15th: **Marcy Robertson**

**Title: **The game of Lego Teichmüller

**Abstract:** In his famous text Esquisse d’un Programme, Grothendieck proposed that by studying the action of the absolute Galois group on fundamental groups of moduli spaces of curves we could obtain information about the elements of the group that we could not obtain by studying number fields directly. In this talk we describe an operad built from profinite mapping class groups of surfaces and explain how the absolute Galois group acts on this object.

The talk will be introductory and covers joint work in progress with Pedro Boavida, Luci Bonatto and Geoffroy Horel.

♦ Monday March 22th **(11 am)**: **Urs Fuchs (Monash)**

**Title: **Pseudoholomorphic curves in symplectic geometry

**Abstract:** Pseudoholomorphic curves were introduced by Gromov as a fundamental tool for the study of symplectic manifolds. I will review some applications of pseudoholomorphic curves in symplectic geometry, which often crucially rely (among other things) on a version of Gromov’s compactness result for pseudoholomorphic curves. Then I will discuss some basic facts at the core of Gromov’s compactness result, with the goal of conveying why and in what settings such a result can be expected.

♦ Monday March 29th: **Ben Williams (University of British Columbia)**

**Title: **Classifying spaces for Algebras with Generating r-Tuples

**Abstract: **Suppose A is a finite-dimensional complex algebra and G is its group of automorphisms, then G acts freely on the space E(r) of r-tuples of elements in A that generate A as an algebra. The quotient E(r) -> E(r)/G serves as a good approximation to the universal principal G-bundle EG–>BG, improving as r increases. The space E(r)/G is a classifying space for bundles of algebras, locally isomorphic to A, equipped with r-tuples of generating sections. I will exploit this to show that for any A having reductive automorphism group, and any natural number n, there exists some ring R/C and a twisted form B of A over R that cannot be generated by fewer than n elements. If time permits, I will explain refinements of the result in the specific cases when A is a matrix algebra (so that the forms B are Azumaya algebras) or A= C^s with termwise operations (so that the forms B are étale algebras). Different parts of this talk represent joint work with Uriya First & Zinovy Reichstein, Sebastian Gant, and Abhishek Kumar Shukla.

♦ Monday April 12th (**12:00pm**): **Inbar Klang (Columbia University)**

**Title: **Isovariant fixed point theory

**Abstract: **If X and Y are spaces with an action of a group G, an isovariant map between them is an equivariant map that preserves isotropy groups. In this talk, I will discuss joint work with Sarah Yeakel, in which we study the homotopy theory of isovariant maps, and use this to provide complete obstructions to eliminating fixed points of isovariant maps between manifolds.

♦ Monday April 19th: **Aravind Asok (University of Southern California)**

**Title: **Vector bundles on algebraic varieties

**Abstract: **Suppose X is your favorite (smooth) complex algebraic variety. When does a complex vector bundle on the complex manifold defined by X admit an algebraic structure? This is a long-studied question with close links to various classical questions in the cohomology of algebraic varieties. I will explain how when X is affine, this problem can be reformulated in entirely (motivic) homotopy theoretic terms, and discuss an approach to analyzing this question for arbitrary X. In particular, I will discuss some results obtained jointly with Jean Fasel and Mike Hopkins about algebraic vector bundles on low-dimensional smooth projective varieties and projective spaces (not necessarily low dimensional). I will not assume any familiarity with motivic homotopy theory.

♦ Monday April 26th (**12:00**): **Liam Watson (University of British Columbia) **

**Title: **Symmetry and mutation

**Abstract: **Mutation is a relatively simple process for altering a knot in a non-trivial way, but it turns out to be quite tricky to see the difference between mutant pairs—a surprisingly wide range of knot invariants are unable to distinguish mutants. In the first part of the talk, I will give some background on the symmetry group associated with a knot, and show that this group is sometimes able to see mutation. In the second part of the talk, I will outline some work with Andrew Lobb, in which we appeal to a symmetry—when present—in order to define a refinement of Khovanov homology that is able to separate mutants.

♦ Monday May 3rd: No seminar

♦ Monday May 10th: **Csaba Nagy**

**Title:** \Theta-groups and classifying manifolds

**Abstract:** We define a family of abelian groups Theta_n(K) for every n>5 and simply-connected CW-complex K of dimension less that n/2. These groups generalise the groups of homotopy spheres Theta_n=Theta_n(*) introduced by Kervaire and Milnor, as well as the groups of high-dimensional links studied by Haefliger. They have a number of equivalent definitions which can be used to prove their functionality, find methods for calculating them and show how they can play a role in the classification of a large class of high-dimensional simply-connected manifolds.

♦ Monday May 17th (**12:00**): **David Gepner** **(Johns Hopkins University)**

**Title: **Integral representation theorems for DQ-modules

**Abstract: **We identify the type of ℂ[[ℏ]]-linear structure inherent in the ∞-categories which arise in the theory of Deformation Quantization modules. Using this structure, we show that the ∞-category of quasicoherent cohomologically complete DQ-modules is a deformation of the ∞-category of quasicoherent sheaves. We also obtain integral representation results for DQ-modules similar to the ones of Toën and Ben-Zvi-Nadler-Francis, stating that suitably linear functors between ∞-categories of DQ-modules are integral transforms.

**Semester 2, 2020**

♦ Monday December 7th: **Ethan Armitage **(GTS)

♦ Monday November 30th: **Jayden Hammet **(GTS)

♦ Monday November 23rd, *12 noon*: **Jeremy Hahn (MIT)**

**Title: Higher algebra as a tool for smoothing manifolds**

**Abstract: **Suppose M is a simply connected, closed topological manifold. Given a smoothing of M away from a disk, we may ask what the obstructions are to producing a smooth structure on all of M. After giving a general introduction to the question, I will explain work, joint with Robert Burklund and Andrew Senger, that proves the obstructions vanish in certain cases. Specifically, if M is (2n+1)-dimensional, n>300, and H_*(M;Z)=0 for 0<*<0.9n, then the obstructions vanish.

Our proof makes contact with a classical, but still largely mysterious, vanishing region in pictures of the homotopy groups of spheres.

♦ November 3rd: **Ethan Armitage** (GTS)

♦ October 27: **Martin Frankland (University of Regina)**

**Title: On good morphisms of exact triangles**

**Abstract: **The Adams spectral sequence is available in any triangulated category, a general setup that has found various applications. When studying the triangulated Adams spectral sequence, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution. Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman’s work on good morphisms of exact triangles.

♦ October 20: **Alexander Campbell (Johns Hopkins/Macquarie University)**

**Title: A model-independent construction of the Gray monoidal structure for (∞,2)-categories**

**Abstract:** The category of 2-categories admits a (non-symmetric, biclosed) monoidal structure due to John Gray, first constructed by him in the early 1970s. The tensor product of a pair of 2-categories is called their Gray tensor product; the internal homs are the 2-categories of 2-functors, (op)lax natural transformations, and modifications between them. These structures are fundamental to the “lax” aspects of 2-category theory, i.e. those aspects which are not captured by viewing 2-category theory simply as an instance of (homotopy coherent) enriched category theory.

There has been much recent interest in extending the Gray monoidal structure from 2-categories to (∞,2)-categories. Such an extension has been proposed for only a few of the many models for (∞,2)-categories, and much remains to be proven about these constructions. Furthermore, the treatment of (∞,2)-categories by Gaitsgory and Rozenblyum in the appendix to their book on derived algebraic geometry rests on a number of unproven assertions about (yet another approach to) the Gray monoidal structure for (∞,2)-categories.

In this talk I will describe joint work with Yuki Maehara in which we give a model-independent (i.e. a purely ∞-categorical) construction of the Gray monoidal structure on the ∞-category of (∞,2)-categories. Our construction is a generalisation to the ∞-categorical setting of a construction of the Gray monoidal structure for 2-categories due to Ross Street, which uses the techniques from Brian Day’s PhD thesis for extending a monoidal structure along a dense functor. The proof of our construction uses, among other things, the results from Yuki’s recent PhD thesis on the Gray tensor product for 2-quasi-categories. I will also describe a few of the open problems concerning the Gray monoidal structure for (∞,2)-categories, and how our results can be used to simplify (though not yet solve) one of these problems.

♦ October 13: **Yaping Yang**

**Title: Frobenii on Morava E-theoretical quantum groups**

**Abstract:** Let G be a simple, simply connected algebraic group over an algebraically closed field of characteristic p > 0. Lusztig in 2015 introduced a family of new characters, E^n, labelled by a prime number p and a positive integer n so that the limit characters E^{\infty} are related to the character formulas of the irreducible modular representations of G. This opens a door to the existence of new quantum groups whose irreducible representations have characters given by E^n. In my talk, I will construct a family of new quantum groups labelled by p and n using the Morava E-theories and Nakajima quiver varieties. I will explain the quantum Frobenius homomorphisms among these quantum groups. The main ingredient is the transchromatic character maps of Hopkins, Kuhn, Ravenal, and Stapleton. I will also give evidence of a Steinberg tensor product formula using these Frobenii.

♦ September 29: **Ethan Armitage** (GTS)

♦ September 22: **Marcy Robertson**

**Title: A higher dimensional notion of planar algebra **

**Abstract: **V. Jones introduced the notion of a planar algebra as the axiomatization of the standard invariant of a finite index subfactor — These are algebras whose multiplication is parameterized by planar tangles (circles and intervals embedded into a disc with holes). Since this original incarnation, planar algebras have arisen in almost any context in which one has a category with a good notion of duals. Indeed, there is a known classification of planar algebras as pivotal categories with a self-symmetric generator. A higher dimensional version of planar algebras was introduced by Bar-Natan and Dancso to provide a convenient formalism for describing surfaces embedded in 4-space. In this talk we describe a classification of these higher dimensional planar algebras as a type of rigid tensor category called a wheeled prop. Time permitting, we will describe applications to the quantisation of Lie bialgebras.

♦ September 15: **Poster Session**

**Plan:** We ran a Zoom poster session, with one breakout room for each poster presenter who shared their screen with their posters. Other attendees were co-hosts and could wander between the poster breakout rooms and discuss the posters with the presenter. Special thanks to our *poster presenters:*

**Zhongtian Chen**: “Hitchin fibers and stable pairs”**Alex Clark**: “The Period-Index Problem in the Language of Classifying Maps”**Songqi Ha**n: “The Parametrised H-Cobordism Theorem for Smooth Manifolds with Tangential Structures”

♦ September 8: **Vigleik Angeltveit **(Australian National University)

**Title: The Picard group indexed slice spectral sequence**

**Abstract:** Equivariant stable homotopy groups are usually graded on the real representation ring. I will explain how we can grade them on the Picard group of G-spectra instead, and how this decreases the number of indices we have to keep track of dramatically. I will give some sample Picard group computations. Finally, I will explain how this makes the slice spectral sequence manageable for G=C_p, and give some sample slice spectral sequence computations. I will not assume knowledge of equivariant stable homotopy theory, Picard groups, or the slice spectral sequence.

♦ September 1: **Ethan Armitage** (GTS)

♦ August 25: **Nora Ganter**

**Title: ****Categorical Traces and Hochschild (Co-)homology**

**Abstract: **I will describe two notions of categorical traces that have been used in categorical representation theory. I will describe their interactions with each other and consider in some detail the example of Hochschild cohomology and homology. This circle of ideas builds on work of Simon Willerton and is the basis for joint work in progress with Willerton and Baranovksy.

♦ August 18: **Jayden Hammet **(GTS)

♦ August 11: **No seminar** for the MATRIX conference on Operads.

**Semester 1, 2020 **

Due to the disruption caused by the pandemic, this semester was slightly unusual in that we intertwined the Graduate Topology Seminar with the Topology Seminar.

June 15: **Daniel Mathews** (Monash University)

**Title: A-polynomials, Ptolemy varieties and Dehn filling**

**Abstract:** The A-polynomial is a 2-variable knot polynomial which encodes topological and hyperbolic geometric information about a knot complement. In recent times it has been shown that the A-polynomial can be calculated from Ptolemy equations. Historically reaching back to antiquity, Ptolemy equations arise all across mathematics, often alongside cluster algebras.

In recent work with Howie and Purcell, we showed how to compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann-Zagier matrix to change basis, eventually arriving at a set of Ptolemy equations. This work refines methods of Dimofte, and the result is similar to certain varieties studied by Zickert and others. Applying this method to families of manifolds obtained by Dehn filling, we find relations between their A-polynomials and the cluster algebra of the cusp torus

June 9: **Marcy Robertson**

**Title: A new proof of the cyclic Deligne conjecture **

**Abstract: **Last week we talked about the motivation behind Deligne’s conjecture that the Hochschild Homology of an associative algebra acts like a double loop space (i.e. is a Gerstenhaber algebra). This week we will discuss the cyclic variant on Deligne’s conjecture, accounting for the S^1 action on Hochschild homology, and give a new proof that arises from an (infinity, cyclic) operad of ”normalised cacti.” The goal of this talk will again be mostly expository, motivating how something like a Gerstenhaber algebra structure can arise from operadic compositions of tree-like configurations of circles in the plane.

In the background of this talk are results obtained jointly with (various subsets of): Luci Bonatto, Safia Chettih, Philip Hackney, Abi Linton, Sophie Raynor, Nathalie Wahl, and Donald Yau.

May 25: **Marcy Robertson**

**Title: The beginners guide to the Deligne conjecture.**

**Abstract:** Deligne’s conjecture says that the Hochschild cohomology of an associative (or A-infinity) algebra, HH(A;A), “behaves” like a 2-fold loop space. More explicitly, it is a classical result that the Hochschild cohomology of an associative algebra is a Gerstenhaber algebra and since it was well known that Gerstenhaber algebras arise on the (co)homology of (chain-level) E_2-algebras (algebras that behave like a 2-fold loop space), Deligne conjectured that one could see that the Gerstenhaber algebra structure on HH(A;A) lifted from an E_2-algebra structure Hochschild cochains. The purpose of this talk is to give a gentle introduction to the words above and tell you why this seemingly random observation ties into quantised deformation theory, the existence of knot invariants and BV algebras.

May 4: **Diarmuid Crowley**

**Title: Projective Chern Classes II**

**Abstract:** In the first talk, I reported on joint work with Xing Gu showing that the integral invariant map is onto for the Lie groups G = PU_n.

In this follow-up talk, I will explore the following topics:

1) Weyl-chambers, cellular decompositions of BT and the assumption that Z^*(BT)^W \to H^*(BT)^W is onto;

2) Topological Azumaya algebras and their invariants;

3) Motivation for computing H^*(BPU(n)) coming from the topological period-index problem;

4) A question of Kari Vilonen about the invariant map H^*(BG) \to H^*(BT)^W.

April 27: **Diarmuid Crowley**

**Title: Projective Chern Classes**

**Abstract:** Let G be a connected Lie group with classifying space BG. The computation of H^*(BG), the integral cohomology of BG, is a classical problem in topology. An important technique is to look at the image of the invariant map I_G : H^*(BG) \to H^*(BT)^W, with target the cohomology of a maximal torus T \subset G, invariant under the action of the Weyl group. The map I is a rational isomorphism, as proven by Chern-Weil theory, however the integral situation is more subtle. In this talk I will report on joint work with Xing Gu, where we prove that I_G is onto for G = PU_n, the projective unitary group.