# 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 and we are returning to face to face talks this semester in Peter Hall 107 (the occasional seminar *may be held online)*. You can sign up for our mailing list by emailing either Marcy or Diarmuid.

## Next Talk

**♦ **Monday October 3rd (2**:**15pm-3:15pm): Yossi Bokor (ANU/Sydney)

**Title: Geometric and Topological Data Analysis: Investigating and summarising the shape of data**

**Abstract:**The shape of data can provide many insights, and is useful in both reconstructing and classifying objects. In this talk, we look at some examples of reconstruction and classification problems.

**Semester 2, 2022**

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**♦ **Monday August 1st (2:15pm – 3.15pm): **Agnese Barbensi**

**Title: The 3D shape of spatial curves**

**Abstract: **The characterisation and classification of spatial curves’ 3D structure is a problem of rapidly increasing importance, with applications in the most varied scientific disciplines, ranging from biology to finance and ecology. Motivated by concrete examples arising in nature, we propose a topological model to analyse the structure of spatial curves that combines persistent homology, hypergraph theory and network science.

**♦ **Monday August 8th (2:15pm – 3.15pm): **Daniele Celoria**

**Title: Überhomology of simplicial complexes**

**Abstract: **We introduce a natural filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories, which generalise simplicial homology and are closely related to discrete Morse matchings on X. We show that, by placing these homologies in a poset, we obtain a triply graded homology theory which we call überhomology. This latter homology is not a homotopy invariant, but nonetheless encodes both combinatorial and topological information on X. Time permitting we’ll talk about a recent collaboration with Caputi and Collari, relating a specialisation of the überhomology to connected dominating sets in graphs.

**♦ **Monday August 15th (2:15pm – 3.15pm): **Diarmuid Crowley**

**Title: What is a manifold structure?**

**Abstract: **By a “manifold structure”, I mean additional structure on a smooth manifold M, which can be specified via typically topological, data on the tangent TM of M (or sometimes the normal bundle of an embedding of M into Euclidean space). Key examples include *orientations, spin structures, almost complex structures and almost contact structures.*

In this talk I will review some different perspectives on manifold structures and discuss how to move between (stable) tangential and normal structures. I will also take a glimpse at spaces of manifold structures on M and their moduli spaces; specifically I will look at how two important groups, the diffeomorphisms of M and the gauge group of TM, act on spaces of manifold structures.

**♦ **Monday August 22nd (2:15pm – 3.15pm): **Marcy Robertson**

**Title: What is a Morse-Smale category?**

**Abstract: **The idea of Morse theory is to study smooth manifolds by studying differentiable functions on them. In this talk, I’ll introduce the Morse-Smale complex: a chain complex constructed from a Morse function and show that one can recover the homology and, sometimes, homotopy type of the original space from this construction. This talk is aimed at graduate students and non-experts and will focus on the construction of examples.

**♦ **Monday August 29th (2:15pm – 3:15pm): **Tamara Hogan**

**Title: **A knot-theoretic approach to comparing the Grothendieck-Tiechmüller and Kashiwara-Vergne symmetry groups

**Abstract: **The relationship between the Grothendieck-Teichmuller groups and Kashiwara-Vergne (KV) groups has been the subject of a growing body of research in recent years. These two sets of groups are symmetries of objects called Drinfel’d associators and Kashiwara-Vergne (KV) solutions respectively. Work of Alekseev-Torossian and Alekseev-Enriquez-Torossian has established that every Drinfel’d associator can be used to construct a unique KV-solution and that there are unique maps of the symmetry groups which corroborate this construction. Work of Bar-Natan and Bar-Natan Dancso then puts a topological spin on this problem by showing that what are called ‘homomorphic expansions’ of parenthesized braids are equivalent to Drinfel’d associators, and similarly, homomorphic expansions of ‘welded foams’ (types of surfaces embedded in R^4) are equivalent to KV-solutions. In this talk, I’ll give a brief overview of these two perspectives. Then, I’ll explain some joint work with Dancso and Robertson where we constructed a map between the graded symmetry groups of these expansions.

**♦ **Monday September 5th (2:15pm – 3.15pm): **Daniele Celoria**

**Title: An introduction to Heegaard Floer homology**

**Abstract: **In this first talk I will sketch the definition of Heegaard Floer homology (HF), a suite of remarkable 3-manifold invariants introduced 20 years ago by Ozsváth and Szabó. I will then outline the main properties of HF, and discuss some of the main related results.

**♦ **Monday September 12th (2:15pm – 3.15pm): **Daniele Celoria**

**Title: ****An introduction to knot Floer homology**

**Abstract: **In this talk I will define a relative version of HF, called knot Floer homology (HFK). These groups are invariants of knots in 3-manifolds, and categorify the classical Alexander polynomial. I will discuss how knot Floer homology can be used to study the knot concordance group.

**♦ **Monday September 19th (**5:00pm-6:00pm**): Adrien Brochier (Université de Paris)

**Title: Virtual tangles and deformation quantization**

**Abstract: **The Shum-Reshetikhin-Turaev theorem states that the category of framed oriented tangles is universal among ribbon categories. This is a cornerstone of the deep relation between low-dimensional topology and deformation quantization (and the category of representation of quantum groups in particular). In this talk, I’ll explain a similar interpretation of an appropriate category of framed oriented virtual tangles. This result explains how these topological objects are related to quantum groups themselves (as opposed to their category of representations), and more generally to the Etingof-Kazdhan quantization of quasi-triangular Poisson groups.

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**Semester 1, 2022**

**♦ **Monday February 28th (1:00pm – 3.30pm): **Victor Turchin** **(Kansas State)**

**Title: The rational homotopy type of embedding spaces**

**Abstract: **I will talk about my joint work with Benoit Fresse and Thomas Willwacher. Using embedding calculus and methods of the rational homotopy theory we construct L-infinity algebras of diagrams that encode the rational type of connected components of embedding spaces in R^n. This type depends on the component. Different known invariants of embeddings seem to be related to the rational homotopy theory invariants that we discovered. In the talk I will consider as the main example the spaces of string and spherical links. I will also be happy to discuss after the main talk the case of embeddings of 4-folds in R^7 and of 3-folds in R^6.

**♦ **Monday March 7th (2:15 pm): **Edmund Heng (ANU)**

**Title: Surfaces, Triangulated Categories and Dynamics**

**Abstract: **Recent developments in the theory of Bridgeland’s stability conditions have established astounding analogues of dynamics and Teichmuller theory in triangulated categories. In this talk, I will introduce the study of dynamical systems in triangulated categories. In particular, I will introduce the notion of categorical entropy, which aims to measure the complexity of endofunctors of triangulated categories. If time allows, I will briefly explain a categorical Nielsen-Thurston classification for the rank two Artin groups, coming from a notion of HN-automaton that serves as a train-track automaton in the categorical world.

**♦ **Monday March 14th (2:00 pm): **Ivo Vekemans** **(ANU)**

**Title: Coherent G-Commutative Monoids: A Bicategorical Story
**

**Abstract: **Fixing a finite group G, G-Mackey and G-Tambara functors arise in the context of G-actions, encoding operations which behave like induction, restriction, conjugation, and tensor induction. In equivariant homotopy theory Mackey and Tambara functors replace abelian groups and commutative rings respectively. Tambara functors have been characterised as those Mackey functors which are coherent G-commutative monoids, however the definitions involved are ad hoc constructions. By developing a relative theory we unify important constructions previously thought distinct, provide more general definitions, and show that Mackey functors can be characterised as coefficient systems which are coherent G-commutative monoids. Further we can extend these characterisations to incomplete Mackey functors and bi-incomplete Tambara functors.

**♦ **Monday March 21st (12:00pm): **Alexander Kupers (Toronto)**

**Title: Embedding calculus in codimension zero**

**Abstract: **In this talk I will give an introduction to a recent approach to the classification of smooth manifolds and the homotopy theory of their diffeomorphism groups. This approach is based on embedding calculus, which is a tool that provides approximations to spaces of embeddings by restricting them to open balls. Classically it is used in codimension at least 3, where it is known the approximation map is an equivalence, but one can apply it in lower codimensions as well. Of particular interest is codimension 0, when embeddings between closed manifolds are diffeomorphisms. I will explain some results on the strengths and weaknesses of the resulting approach to the study of smooth manifolds, as well as connections to operads and some open problems. This is joint work with Ben Knudsen and with Manuel Krannich.

**♦ **Monday March 28th (2:00 pm): **Morgan Opie** **(UCLA)**

**Title: Chromatic invariants of vector bundles on projective spaces
**

**Abstract: **In this talk, I will discuss my ongoing work on complex rank 3 topological vector bundles on CP^5. I will describe a classification of such bundles using twisted, topological modular form-valued invariants, and the subtleties involved in actually computing this invariant. As time allows, I will outline future chromatic directions suggested by this result and by prior work of Atiyah and Rees.

**♦ **Monday April 4th (11:00am) : **Robert Burklund (MIT)**

**Title: Classification of manifolds and the adams spectral sequence
**

**Abstract: **A classical problem in differential topology is the following: Classify all simply-connected, closed, smooth (2n)-manifolds whose only non-trivial homology groups are H_0, H_n and H_{2n}. In this talk I will survey the history of the high dimensional side of this question and how its resolution requires a surprisingly deep understanding of the Adams spectral sequence computing the stable homotopy groups of spheres. I will then discuss how the situation changes as we relax our topological restrictions on the manifold (for example allowing H_{n-e}, H_{n-e+1}, … H_{n+e} to be non-trivial for a small number e). This talk represents joint work with Jeremy Hahn and Andy Senger.

**♦ **Monday April 11th (2:00pm) : **Hiro Lee Tanaka** **(Texas State)**

**Title: Morse theory on a point: Broken lines and associativity
**

**Abstract: **I’ll introduce a stack of Morse trajectories on a point. It turns out this stack classifies associative algebras in a large class of categories, and this is a first step toward constructing stable homotopy enrichments of invariants that people in mirror symmetry care about (Lagrangian Floer theory and, more generally, Fukaya categories). I’ll begin with a basic review of Morse theory and give some feel for what this stack is doing. This is joint work with Jacob Lurie.

**♦ **Monday May 2nd (6:15pm) : **Oscar Randal-Williams (Cambridge)**

**Title: On topological Pontrjagin classes
**

**Abstract: **Classical results of Sullivan and Kirby–Siebenmann may be used to see

that the map from the space BO (classifying stable vector bundles) to

the space BTop (classifying stable bundles of Euclidean spaces) is a

rational homotopy equivalence. Therefore the familiar Pontrjagin

classes of vector bundles must arise from more mysterious invariants

of bundles of Euclidean spaces. For bundles of d-dimensional Euclidean

spaces, one may ask whether the identities among Pontrjagin classes

familiar from d-dimensional vector bundles continue to hold: to

everyone’s great surprise, Weiss has shown that they do not. I will

explain an elaboration of Weiss’ results, using unrelated methods.

This is joint work with S. Galatius.

**♦ **Monday May 9th (2:00pm) : Short talks *face to face in Peter Hall 107*

2:00 – 2:30 **Kurt Stoeckl**

**Title: Quadratic presentations of Operads governing Operadic structures**

**Abstract: **There exist coloured operads whose algebras are other operadic structures such as modular operads, wheeled properads and props. In “Massey Products for Graph Homology”, Ben Ward gives a quadratic presentation of a groupoid coloured operad whose algebras are modular operads and shows this operad is Koszul. In this talk we’ll informally discuss what operads governing operadic structures look like, how we can get nice presentations of these operads using ideas of Ward, and some consequences.

2:30 – 3:00 **Alex Clark**

**Title: Tensor-triangular geometry and the classification t-structures**

**Abstract: **Associated to any essentially small tensor-triangulated category (tt-category) is a particular (appropriately universal) locally ringed space, called the Balmer spectrum of the category. Examples of essentially small tt-categories include derived categories of perfect complexes, stable module categories, and the stable homotopy category of finite CW-spectra. In this talk I will give an outline of the general setup of tensor-triangular geometry, and explain some of my work towards a classification of t-structures on tt-categories.

**♦ **Monday May 16th (2:00pm) : Short talks *face to face in Peter Hall 107*

2:00 – 2:30 **Olivia Borghi**

**Title: G-monoidal infinity catgeories
**

**Abstract: **Given a monoidal category C we can require the the commutativity of its monoidal product to be governed by certain sequences of groups. The simplest examples are symmetric monoidal categories where those groups are the symmetric groups, and braided monoidal categories which are governed by the braid groups. Other, lesser known, examples include ribbon braided monoidal categories and coboundary categories which are governed by the ribbon braid and cactus groups respectively. I will define these G-monoidal categories as algebras over a construction called a parenthesized action operad. I will then proceed to define infinite dimensional analogs of these categories. This definition is based on Jacob Lurie’s definition for braided monoidal infinity categories as coCartesian fibrations over the 2-dimensional little cubes infinity operad.

2:30 – 3:00 **Kai Machida**

**Title: ****Lambda-schemes and resolutions of singularities**

**Abstract: **F1-geometry is a hypothetical geometry over a base deeper than SpecZ. The origins of this theory can be attributed to Jacques Tits, who in the 1950’s postulated the existence of such a geometry that would explain analogues between projective geometries and finite sets. Since the 1990’s there have been many approaches to defining appropriate categories for F1-geometry, one of which is Borger’s theory of Lambda-schemes. We will discuss details of Lambda-schemes and how for specific examples we can apply a resolution theorem of Bierstone and Milman for toric varieties to give Lambda-equivariant resolutions.

**♦ **Monday May 23rd (5:15pm) : **Alexander Berglund (Stockholm)**

**Title: Algebraic models for classifying spaces of fibrations**

**Abstract: **For a simply connected finite CW-complex X, we construct an algebraic model for the rational homotopy type of Baut(X), the classifying space of fibrations with fiber X. The model reduces the computation of the rational cohomology ring of Baut(X) to the computation of cohomology of arithmetic groups and dg Lie algebras. In special cases, this reduces further to calculations with modular forms and invariant theory. We also show that the representations of the homotopy mapping class group of X in the higher rational homotopy groups of Baut(X) are algebraic in a suitable sense, extending a classical result of Sullivan and Wilkerson. Our results moreover improve and generalize certain earlier results of Madsen and myself on Baut(M) for highly connected manifolds M. This is joint work with Tomas Zeman.

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**Semester 2, 2021**

**♦ **Monday July 26th (2.15pm – 3.15pm)**: Paul Lessard (Macquarie)**

**Title: Spectra as Locally Finite Pointed Z-groupoids**

**Abstract: **For nearly 40 years the notion of __ω-groupoid has guided the development of higher category theory, a synthesis of homotopy theory and category theory. More recently that same notion, that of ____ω-groupoid, has in part given rise to the synthesis of logic and homotopy theory called homotopy type theory. While the quest for a maximally algebraic definition of the notion has waxed and waned in popularity over the years, the degree to which the notion of ____ω-groupoid has reorganised these branches of mathematics is hard to overstate.__

Homotopy theory however naturally splits into a dimension dependent and a dimension invariant sort – unstable and stable homotopy theory respectively. What sort of algebraic structure might one hope could anchor a similar constellation for stable homotopy theory? In this talk I’ll explain how Kan’s model for spectra from 1963 can be reimagined as an answer to this question.

In this talk we’ll:

- provide a fully 2-categorical treatment generalising Kan’s model; and
- we’ll show that spectra are locally finite pointed Z-groupoids in a precise sense, by way of:
- A rumination on Cisinski model category theory; and
- a new limit reflection result regarding the
__2____-category of categories with arities.__

♦ Monday August 2nd (2.15pm – 3.15pm): **Marcy Robertson**

**Title: What is a symmetric spectrum?**

**Abstract:** This is an expository talk on spectra and symmetric spectra.

♦ Monday August 9th (2.15pm – 3.15pm): **Diarmuid Crowley**

**Title: What is a Postnikov Tower?**

**Abstract:** This is an expository talk on Postnikov towers are related ideas.

♦ Monday August 16th (2.15pm – 3.15pm): **Lance Gurney**

**Title: Ring stacks and cohomology: why I’m interested in derived algebraic geometry**

♦ Monday August 23rd (3.30pm – 4.30pm): **David Baraglia**** (Adelaide)**

**Title: Tautological Classes of Definite 4-manifolds**

**Abstract: **Tautological classes are characteristic classes of manifold bundles. They have been extensively studied for bundles of surfaces, where they were first introduced by Mumford in the setting of moduli spaces of curves. In higher dimensions there are not many examples of manifolds for which the tautological ring, the ring generated by tautological classes, is known. We will use gauge theory to study tautological classes of 4-manifolds with positive definite intersection form. Amongst other things, this allows us to compute the tautological ring for CP^2 and the connected sum of CP^2 with itself.

♦ Monday August 30th: **No seminar**

♦ Monday September 6th: **No seminar**

♦ Monday September 13th (5.15pm – 6.15pm): **Fabian Hebestreit**** (Bonn)**

**Title: Symplectic groups and Poincaré categories**

**Abstract: **The primary goal of the talk will be to explain recent joint work M. Land and T. Nikolaus, in which we compute much of the so-called stable part of the cohomology of symplectic and orthogonal groups over the integers, in particular at the previously mysterious prime 2.

Our approach is via the group completion theorem, which relates the stable cohomology of these arithmetic groups over general rings to that of Grothendieck-Witt spaces (a.k.a. hermitian K-theory). The latter have long been understood relative to usual algebraic K-theory, if 2 is assumed a unit in the input ring. The secondary goal of the talk is to indicate how Lurie’s set-up of Poincaré categories can be combined with ideas from cobordism theory to remove this assumption; these results are further joint with Calmès, Dotto, Harpaz, Moi, Nardin and Steimle.

♦ Monday September 27th (5.15pm – 6.15pm): **Luciana Bonatto**** (Oxford)**

**Title: Decoupling monoids of configurations on surfaces **

**Abstract:**The monoid of oriented surfaces with one boundary component has featured prominently in the works of Miller and Tillmann, and has been essential to understanding the Mumford conjecture. On another direction, Segal’s monoid of configurations in euclidean space originated a branch of scanning results. In this talk, we are going to discuss a combination of these and look at the monoid of configurations on oriented surfaces. More than being a model for the monoid of punctured surfaces, this allows us to look at configurations with labels and even with collision rules. We will show that the group completion of this monoid does not detect that the points in the configurations are constrained to the surface: it simply sees surfaces and particles in the infinite euclidean space. In other words, the particles get decoupled and this group completion splits as a product of the well known spaces originated from the surface and Segal’s monoids.

♦ Monday October 4th (2.15pm – 3.15pm): **Zsuzsanna Dancso (University of Sydney)**

**Title: Duflo’s Theorem for Arbitrary Lie Algebras: a Topological Proof**

**Abstract: **The Duflo map is an algebra isomorphism between the centre of the universal enveloping algebra of a Lie algebra, and the invariant part of its symmetric algebra. The heart of Duflo’s Theorem is the multiplicativity of the Duflo map. In this talk we present a topological construction of the Duflo map and a topological proof of its multiplicativity, using “expansions” (aka universal finite type invariants).

♦ Monday October 18th (2.15pm – 3.15pm): **Diarmuid Crowley**

**Title: What is a transfer map?**

**Abstract: **Transfer maps are stable “wrong way maps” from the base space to the total space of certain fibre bundles. These surprising maps were studied in detail be Becker and Gottlieb and have a variety of unexpected applications in topology.

This talk will review the definition of the transfer map and discuss some applications to, for example, a proof of the Adam’s Conjecture (Becker and Gottlieb), the cohomology of Lie groups (Feshbach and others) and, if time permits, elements of the stable stems represented by Lie groups (Knapp).

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**Semester 1, 2021**

♦ 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.

♦ 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.

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**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.