# Topology Seminar

The Topology Seminar is a research seminar organised by Diarmuid Crowley, Guillaume Laplante-Anfossi and Marcy Robertson. We generally meet from 2.15pm – 3.15pm on Mondays in Peter Hall 162 (the occasional seminar *may be held at another time and/or with an online speaker)*. You can sign up for our mailing list by emailing either Marcy or Diarmuid.

## Next Talk

**♦ **Monday May 27th (2:15pm – 3.15pm): **Nir Gadish (Michigan)**

**Title: Infinitesimal calculations in fundamental groups**

**Abstract:** How can you tell if some power of a word in a group is an n-fold commutator? Thinking of a topological space as a presentation of its fundamental group, Ozbek, Sinha, Walter and I clarify the way in which rational cochains measure fundamental groups, resulting in the first algorithm resolving the above question about commutators in general presented groups.

The talk will introduce the universal “Lie dual” of a group and construct it as the derived indecomposables of a commutative cochain model of a topological space. Furthermore, these indecomposables are concretely modeled by the Harrison complex, which we will review. Intended geometric applications include formulas for Milnor invariants of links and general 3-manifolds in terms of intersections between link components and Seifert surfaces.

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## Semester 1, 2024

**♦ **Monday March 4th (2:15pm – 3.15pm): **Diarmuid Crowley**

**Title: What is a (degree-d) normal map?**

**Abstract: **If we embed a compact connected smooth n-manifold M in R^\infty, it’s stable normal bundled is well-defined, up to contractible choice, and this fact forms the basis for the study of manifolds via “normal maps”; i.e. bundle maps from the stable normal bundle of M to some stable vector bundle over an auxiliary space B.

If B = X itself is a connected n-dimensional Poincare complex and orientations are chosen, then we can assign an integer, the degree, to a normal map f : M \to X.

In this talk, I will recall the role of bordism of degree one normal maps in classical surgery theory and also describe recent joint work with Cs. Nagy, where we use the theory degree-d normal maps to get new results on the smooth classification of complete intersections.

**♦ **Monday March 11th (2:15pm – 3.15pm): **Gufang Zhao**

**Title: Cousins of relative Donaldson-Thomas theory in dimension 4**

**Abstract: **The goal of this talk is to give a few examples of moduli spaces originated from relative Donaldson-Thomas theory in dimension 4. Attempts in finding numerical invariants via these moduli spaces lead to a question of functoriality of the cohomology or K-theory of these moduli spaces. Invariants arising from the functoriality in examples will be given. The original parts of the talk are based on an on-going project joint with Cao, and partially with Zhou.

**♦ **Monday March 18th (2:15pm – 3.15pm): **Scott Mullane**

**Title: Volumes of moduli spaces of cone surfaces, stability conditions and wall-crossing**

**Abstract: **We consider Weil-Petersson volumes of the moduli spaces of conical hyperbolic surfaces. The moduli spaces are parametrised by their cone angles which naturally live inside Hassett’s space of stability conditions on pointed curves. The space of stability conditions decomposes into chambers separated by walls. Assigning to each chamber a polynomial corresponding to the Weil-Petersson volume of a moduli space of conical hyperbolic surfaces, we use algebraic geometry to compute wall-crossing polynomials relating the polynomials to each other, and to the volume of the maximal chamber, given by Mirzakhani’s polynomial. Time permitting, I will discuss possible future directions.

**♦ **Monday March 25th (2:15pm – 3.15pm): **Thomas Geisser (Rikkyo University)**

**Title: Brauer groups and Neron-Severi groups of surfaces over finite fields**

**Abstract: **For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behaviour of the zeta-function at 1 to other invariants of the surface. We give a refinement which equates invariants related to the Brauer group to invariants of the Neron-Severi group. If time permits we give some applications for abelian surfaces. For example, the largest Brauer group over a field of order q=p^{2r} has order 16q, and the largest Brauer group of a supersingular abelian surface over a prime field has order 36.

**♦ **Monday April 8th (2:15pm – 3.15pm): **Rhuaidi Burke (UQ)**

**Title: Developments in computational 4-manifold topology**

**Abstract: **Dimension 4 is the first dimension in which exotic smooth manifold pairs appear — manifolds which are topologically the same but for which there is no smooth deformation of one into the other. On the other hand, smooth and piecewise-linear manifolds (manifolds which can be described discretely) do coincide in dimension 4. Despite this, there has been comparatively little work done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this talk, I will present some developments in this direction: a new software implementation of an algorithm to produce triangulations of 4-manifolds from handlebody diagrams, as well as a new heuristic for simplifying these triangulations. Using these new software tools, we present small triangulations of exotic 4-manifolds, and related objects. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations, and time permitting I will discuss recent work towards a structure and decomposition theory for such triangulations.

**♦ **Monday April 15th (2:15pm – 3.15pm): **Marcy Robertson**

**Title: Tangles, cyclic ribbons and duality**

**Abstract:** In this talk we will present an operadic interpretation of a classic result of

Kassel and Turaev about invariants of tangles. As a consequence, we will

describe new results about Galois actions on ribbon categories. The talk includes joint

work with Chandan Singh.

**♦ **Monday April 22nd (2:15pm – 3.15pm): **Johanna Knapp**

**Title: Calabi-Yau manifolds and physics**

**Abstract:** I will give a fairly general overview on Calabi-Yaus and their significance in string theory. This includes: definition, how Calabi-Yaus arise in string theory, topological classification, constructions and example of the quintic threefold, and a sketch of how the mathematics of Calabi-Yaus and their moduli spaces can determine physical properties.

**♦ **Monday April 29th (2:15pm – 3.15pm): **Kyle Broder**** (UQ)**

**Title: Invariant metrics in complex analysis and a conjecture of Kobayashi and Lang**

**Abstract:** A compact complex manifold X is declared Kobayashi hyperbolic if every holomorphic map from the complex plane into X is constant. Kobayashi hyperbolic manifolds have maintained a central role in our understanding of the landscape of complex manifolds since their introduction in 1967. A striking feature of complex geometry is the capacity to encode this highly transcendental notion of hyperbolicity in the coarse geometric language of distance functions that are invariant under the automorphism group and decrease under holomorphic maps. A long-standing conjecture of Kobayashi (1970) and Lang (1986) predicts that a compact Kobayashi hyperbolic Kähler manifold admits a Kähler—Einstein metric of negative Ricci curvature. We will present the most general evidence for the Kobayashi—Lang conjecture: A compact Kähler manifold with a pluriclosed metric of negative holomorphic curvature admits a Kähler—Einstein metric of negative Ricci curvature. This result is a joint work with James Stanfield and comes from the first general improvement on the Schwarz lemma for holomorphic maps between Hermitian manifolds since 1978.

**♦ **Monday May 6th (2:15pm – 3.15pm): **Kurt Stoeckl**

**Title: Graphical and Segal Infinity Props**

**Abstract:** Simplicial sets are of fundamental importance in homotopy theory, as they provide a means to weaken the associativity of categorical composition. For instance, on the category of simplicial sets, the Joyal model structure provides a model of infinity categories, and the Kan-Quillen model structure provides a model of homotopy spaces. A prop is a type of symmetric monoidal category with two types of composition, a categorical composition, and a monoidal composition.

In this talk, we will introduce a graphical set model for props, which allows us to weaken both compositions up to coherent homotopy. We will outline a Quillen equivalence to a new definition of a Segal infinity prop, and time permitting, relate these models to existing structures in literature. This talk is based on joint work with Philip Hackney and Marcy Robertson.

**♦ **Monday May 20th (2:15pm – 3.15pm): **Guillaume Laplante-Anfossi**

**Title: Kashiwara—Vergne operads**

**Abstract: **Gluing genus zero surfaces along boundaries endows their mapping class groups with the structure of an operad. A deep theorem of Boavida de Brito, Horel and Robertson from 2017 identifies the homotopy automorphisms of this operad with the Grothendieck—Teichmüller group, a mysterious profinite group containing the absolute Galois group of the rational numbers.

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## Semester 2, 2023

**♦ **Monday July 31st (2:15pm – 3.15pm): **Diarmuid Crowley**

**Title: What is a mapping class group?**

**Abstract: **generally conceived, a mapping class group is a group of equivalences classes of automorphisms of some mathematical object of interest. A paradigm example from differential topology is the group of isotopy classes of diffeomorphisms of a smooth manifold M; i.e. the mapping class group of M is \pi_0 Diff(M), for appropriate topologies on Diff(M), the space of diffeomorphisms of M.

The computation of mapping class groups is a fascinating and much studied problem. In this talk I will give a selective overview of the study of the mapping class groups of some familiar classes of manifolds.

As time permits, I will also discuss structure preserving diffeomorphisms; e.g. what is a “spin diffeomorphism” of a spin manifold, and how mapping class groups appear in the study of the higher homotopy groups of Diff(M) and more generally the topology of the classifying space BDiff(M).

**♦ **Monday August 7th (2:15pm – 3.15pm): **Daniele Celoria**

**Title: What is the concordance group?**

**Abstract: **The concordance group is a fascinating object in low-dimensional topology, combining 1,2,3 and 4-dimensional manifolds. After giving the basic definitions and examples, I will give some applications and talk about a number of related topics.

**♦ **Monday August 14th (2:15pm – 3.15pm): **Francesco Lin** **(Columbia)**

**Title: Homology cobordism and the geometry of hyperbolic three-manifolds?**

**Abstract:** The three-dimensional homology cobordism group is a fundamental object of study in low-dimensional topology. A major challenge in the study of its structure is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, after introducing the main protagonists, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying some natural geometric constraints.

**♦ **Monday August 21st (2:15pm – 3.15pm): **Marcy Robertson**

**Title: What is an infinity category?**

**Abstract:** An infinity category is a variation on a classical category in which many of the categorical properties are only defined up to coherent homotopy. In this talk I’ll explain some basics of infinity categories, with an emphasis on their historical use in topology.

**♦ **Monday September 4th (2:15pm – 3.15pm): **Guillaume Laplante-Anfossi**

**Title: Convex polytopes and higher categories**

**Abstract:** An omega-category is a category where, in addition to (1-)morphisms between objects, there are (n+1)-morphisms between n-morphisms, for any positive integer n. Such categories are required in the study of non-abelian cohomology and topological quantum field theory. In 1991, M. Kapranov and V. Voevodsky suggested that convex geometry could be the source of loop-free omega-categories, where there is no infinite sequence of composable morphisms. More precisely, they conjectured that a polytope together with a generic basis forms a loop-free omega-category, where the n-morphisms are given by the n-dimensional faces of the polytope.

I will explain why this conjecture of Kapranov and Voevodsky is true up to dimension 3, and fails for polytopes of dimension 4 and higher. The situation is as bad as it can be: there are polytopes for which any choice of basis induces a loop, and the canonical basis induces loops on high dimensional random simplices with high probability. Further, omega-category structures on polytopes are universal in the sense of Mnëv, i.e. they can behave « as badly as any semi-algebraic set », obeying what R. Vakil called Murphy’s law in algebraic geometry. This is joint work with A. Padrol and A. Medina-Mardones.

**♦ **Monday September 18th (2:15pm – 3.15pm): **Tamara Hogan**

**Title: A knot theoretic interpretation of the Goldman-Turaev Lie bialgebra **

**Abstract: **Arising in the early study of string topology, the Goldman-Turaev Lie bialgebra is a well-studied structure on the space of free immersed loops on a surface of genus g with n+1 punctures. The bracket and co-bracket in this structure involve smoothings of intersections between pairs of loops or between a loop and itself. More recently, in a series of papers, Alekseev, Kawazumi, Kuno and Naef showed that expansions (or finite-type invariants) of these structures are equivalent to solutions of the Kashiwara-Vergne (KV) equations. A similar correspondence between expansions of welded tangles and solutions to the KV equations is known due to Bar-Natan and Dancso.

Motivated by trying to understand the connection between these two different spaces and their expansions, in this talk I will describe the construction of a lift of the Goldman-Turaev structure in genus zero to the space of tangles in a handlebody. This is a report on forthcoming joint work with Dror Bar-Natan, Zsuzsanna Dancso, Jessica Liu and Nancy Scherich.

**♦ **Monday October 2nd (4:15pm – 5.15pm) (Zoom talk): **Cristina Anghel ****(Geneva)**

**Title: Globalising the Jones and Alexander Polynomials (and their quantum generalisations) via configurations in the punctured disc**

**Abstract:** The Jones and Alexander polynomials are two important knot invariants and we obtain both of them from a *common topological perspective*, based on configurations on ovals and arcs in the punctured disc. The model is a graded intersection (in two variables) between *two explicit Lagrangians in a configuration space*. We consider links as closures of braids which, in turn, as mapping classes, act on the punctured disc. Then, we prove that the intersection before specialisation is (up to a quadratic quotient) an invariant which globalises the two invariants, given by an explicit interpolation between the Jones and Alexander polynomials.

Secondly, we discuss *a unified topological model* for their quantum generalisations, the *coloured Jones and Alexander polynomials*, from a graded intersection in a *symmetric power of a surface*. If time permits, we present topological models for the corresponding 3-manifold quantum invariants.

**♦ **Monday October 9th (2:15pm – 3.15pm): **Marcy Robertson**

**Title: Towers of Kashiwara-Vergne Solutions**

**Abstract:** The motivating idea behind “Grothendieck-Teichmüller theory” is to study Galois actions on geometric spaces via the endomorphisms they induce on towers of fundamental groups of certain well-understood moduli spaces. The idea behind this talk is to import the fundamental aspects of this theory into the study of actions by the so-called “Kashiwara-Vergne groups” on free loop spaces.

**♦ **Monday October 16th (2:15pm – 3.15pm): **Chris Rogers (Nevada)**

**Title: Formal deformation problems and the unicity of homotopy transfer**

**Abstract:** Here’s a common scenario in homological algebra: Suppose we are given a cochain complex A, an algebra B of some particular type P (e.g., P = associative, commutative, . . . ) and, finally, a quasi-isomorphism of complexes A –> B. Then a solution to the “homotopy transfer problem” is a pair consisting of a homotopy P-algebra structure on A, and a lift of the quasi-isomorphism to an equivalence of homotopy P-algebras A ~ B.

In this talk, I’ll describe an explicit relationship between the homotopy theory of dg-Lie algebras and formal deformation problems in characteristic zero. As an application, I will give an explicit construction of the moduli space (as an 1-groupoid) of solutions to the above homotopy transfer problem. I will show that: (1) the space of solutions is non-empty, and (2) it is contractible. The rst statement implies the well known fact that a homotopy equivalent transferred structure always exists, and the second implies that this structure is unique, up to homotopy, in the strongest possible sense (perhaps less well-known). The talk is based on the paper 10.1016/j.jpaa.2023.107403.

**♦ **Monday October 23rd (12 noon) (Zoom talk): **Nir Gadish (Michigan)**

**Title: Letter-braiding invariants of words in groups (slides)**

**Abstract: **How can one tell if a group element is a k-fold commutator? A computable invariant of words in groups that does not vanish on k-fold commutators will help.

For free groups this is achieved by Fox calculus, whose geometric applications include Milnor invariants of links, and there are generalizations for braid groups and RAAGs, but beyond that little is known. We introduce a complete and computable collection of such invariants for any group that detect its dimension series, using the algebraic Bar construction. Consequences of this theory include nonvanishing of mod p Massey products for finite groups, and a dual version of the Johnson homomorphism defined for automorphisms of arbitrary groups.

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## Semester 1, 2023

**♦ **Friday February 10th (3:15pm – 4.15pm): **Rinat Kashaev (Université de Genève)**

**Title: ****Generalized 3d TQFT’s from local fields**

**Abstract: **Let F be a local field. Based on a particular quantum dilogarithm associated to F, one can construct at least three different generalised distribution valued 3d TQFT’s two of which are of Turaev-Viro type. The associated 3-manifold invariants are expected to be enumerative invariants counting with specific weights representations of \pi_1 into PL_2F. This is the ongoing work in collaboration with Stavros Garoufalidis.

**♦ **Monday March 6th (2:15pm – 3.15pm): **Jonathan Bowden (Regensburg)**

**Title: Exotic contact structures on standard spheres**

**Abstract: **In the 1989 Eliashberg showed that the 3-dimensional sphere has a unique tight contact structure. In contrast there are many families of exotic contact structures on odd-dimensional spheres in higher dimension, appearing for example as links of singularities of complex hypersurfaces. These examples all appear as boundaries of complex analytic Stein manifolds – they are Stein fillable. There are various other notions of fillability, which have only recently been shown to be distinct (in all dimensions). In general constructing such examples requires topology in order to verify tightness and the problem of determining whether contact structures on spheres with exotic fillability properties exist was unsolved.

In this talk I will put these problems into context and will report on recent progress on these questions in joint work with F. Gironella, A. Moreno and Z. Zhou.

**♦ **Monday March 13th (2:15pm – 3.15pm): **Guillaume Laplante-Anfossi**

**Title: Diagonals of polytopes and higher structures**

**Abstract: **The set-theoretic diagonal of a polytope has the crippling defect of not being cellular: its image is not a union of cells. One is thus looking for a cellular approximation to the diagonal. Finding such an approximation in the case of the simplices and the cubes is of fundamental importance in algebraic topology: it allows one to define the cup product in cohomology. I will present a general method, coming from the theory of fiber polytopes of Billera and Sturmfels, which permits one to solve this problem for any family of polytopes. I will then explain how this machinery, applied to new families of polytopes, gives us the tools to define higher algebraic objects such as the tensor product of homotopy operads or a functorial tensor product of A-infinity categories.

**♦ **Monday March 20th (2:15pm – 3.15pm): **June Park**

**Title: The space of morphisms & moduli stacks of fibrations**

**Abstract: **The defining property of fine moduli stacks (of curves or abelian varieties) is that they have ‘universal families’ which translate the study of a family of (curves) to the study of morphisms to moduli stacks. I will explain this idea using Hom_n(C, Mbar_{g,m}), the moduli space of curves on the moduli stack of stable genus-g curves with m-markings. Once we have the Hom-space, we will compute some arithmetic invariants using topological methods.

**♦ **Monday April 3rd (2:15pm – 3.15pm): **Paul Norbury**

**Title: What is Teichmüller space?**

**Abstract: **Teichmüller space parametrises geometric structures on a surface together with a marking which is essentially a memory of the topological structure of the surface. The geometric structure is realised in different ways, in particular via a conformal structure or a hyperbolic metric on the surface. In this lecture I will give an introduction to Teichmüller spaces and describe some simple examples. I will also introduce natural functions on Teichmüller space which give rise to different sets of convenient coordinates, notably Fenchel-Nielsen coordinates and Penner coordinates.

**♦ **Monday April 17th (4:15pm – 5.15pm, PH 107 & on line talk): **Thorsten Hertl (Freiburg)**

**Title: Line Bundle Twists for Unitary Bordism are Ghosts**

**Abstract: **Classical theorems of Conner-Floyd and Hopkins-Hovey say that complex K-theory is completely determined by unitary bordism and spin^c bordism respectively. The isomorphisms appearing in these theorems are induced by the maps that send a bordism class to its orientation-class in complex K-theory. Despite this geometric description, the proofs that they are indeed isomorphism are rather abstract and homotpy-theoretical. Motivated by theoretical physics, Baum, Joachim, Khorami and Schick extend Hopkins and Hovey’s result in a forthcoming paper to twisted spin^c bordism and twisted K-theory. Here, the twists are given by (representatives of) elements in third integral cohomology. Since every almost complex structure induces a spin^c structure and since the classical Conner-Floyd orientation factors through the Hopkins-Hovey orientation, one may wonder whether there is a twisted unitary bordism theory and a twisted Conner-Floyd orientation that extends the result of Baum, Joachim, Khorami and Schick ‘to the left’. In this talk, I answer this question in the negative.

**♦ **Monday April 24th (2:15pm – 3.15pm): **Jayden Hammet**

**Title: Ruled complex bordism**

**Abstract: **The calculation of bordism groups is a classical problem in algebraic topology; indeed, much of the work done on such calculations was carried out in the ’50s and ’60s. However, recent work by Bowden, Crowley, and Stipsicz identifies a bordism obstruction which detects when an almost contact manifold is Stein fillable. In this setting certain bordism groups arise which, while having been investigated previously by Stong and Höhn, have not been fully calculated.

In this talk I will introduce these ruled complex bordism groups, and describe their computation as well as the analysis of the corresponding Thom spectra. I will also describe some applications of these computations to the Stein fillability of almost contact manifolds.

**♦ **Monday May 1st (2:15pm – 3.15pm): **Dionne Ibarra** **(Monash)**

**Title: Jones-Wenzl Idempotents in the Temperley-Lieb algebra-module over the Möbius Band**

**Abstract: **In this talk we will give an introduction to Jones-Wenzl idempotents of the Temperley-Lieb algebra and how they are used to define SU(2)-quantum knot and 3-manifold invariants. We will then focus our discussion on calculations of the trace of Jones-Wenzl idempotents in the Kauffman bracket skein module of the Twisted I-bundles of the Möbius band and their properties in the Temperley-Lieb algebra-module over the Möbius Band.

**♦ **Monday May 8th (2:15pm – 3.15pm): **Lukas Anagnostou**

**Title: Volumes of moduli spaces of hyperbolic surfaces with cone points**

**Abstract: **The moduli space of hyperbolic surfaces of genus g with n cusps has a natural compactification corresponding to the Deligne-Mumford compactification of the moduli space of algebraic curves. It can be shown that the same is true if one replaces cusps with geodesic boundary components. A famous result by Mirzakhani expresses volumes of these moduli spaces as polynomials of even degree in boundary lengths, with coefficients given by intersection numbers of the Deligne-Mumford compactificatied moduli space. By letting boundary geodesics take on purely imaginary values, one obtains crude formulas for volumes of hyperbolic surfaces with cone points of arbitrary cone angle. Such formulas have been proven successful for particular cases of small cone angles, however, for larger cone angles these formulas can lead to a variety of contradictions. In this talk, we discuss when these volume formulas hold. Based on work of Hassett, I will also discuss how one obtains meaningful volumes outside of the regions where Mirzakhani’s results hold.

**♦ **Monday May 15th (2:15pm – 3.15pm): **Jonathan Spreer** **(Sydney)**

**Title: The complexity of 3-manifolds obtained by Dehn filling**

**Abstract: **In this talk, I will demonstrate how to use normal surface theory and layered triangulations to construct upper and lower complexity bounds for closed 3-manifolds obtained as even Dehn fillings of a 3-manifold M with single torus boundary component.

In the first part of the talk, I will go over the procedure of normalising surfaces embedded in triangulations of 3-manifolds.

In the second part of the talk, I will show that there exist infinite families of even Dehn fillings of M for which we can determine the complexity of the filled manifolds up to an additive constant. This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of M, or the number of crossings of a knot diagram. I will also compute the gap for explicit families of fillings of knot complements in the 3-sphere. The practicability of this approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot P(−2,3,7), and the trefoil.

**♦ **Monday May 22nd (2:15pm – 4.15pm): **Victor Turchin** **(Kansas State)**

**Title: Smoothing theory, delooping of disc embeddings and diffeomorphism spaces**

**Abstract:** Smoothing theory provides delooping of the groups of relative to the boundary disc diffeomorphisms Diff_\del(D^m) = \Omega^{m+1} Top_m/O_m, m\neq 4; Diff_\del(D^m) = \Omega^{m+1} PL_m/O_m, any m.

The result was established in the 70s and is due to contributions of several people: Cerf, Morlet, Burghelea, Lashof, Kirby, Siebenmann, Rourke, etc. Less well known is a similar statement for the spaces of relative to the boundary (framed) disc embedding spaces D^m -> D^n. This latter result implicitly appeared in a work of Lashof from the 70s and was stated explicitly by Sakai nine years ago. The range stated by Sakai is n>4, n-m>2. After careful reading of the literature with help of Sander Kupers, we found out that the delooping in question holds for any codimension n-m and any n (except n=4 in the topological version of the delooping). Moreover, we proved that the delooping in question is compatible with the Budney E_{m+1}-action. In case of framed embedding spaces Emb_\del^{fr}(D^m,D^n), we showed that by means of smoothing theory delooping, the Budney E_{m+1}-action and the Hatcher O_{m+1}-action on Emb_\del^{fr}(D^m,D^n) can be combined into a framed little discs operad E_{m+1}^{fr} action.

This is part of joint work with Paolo Salvatore.

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## Semester 2, 2022

**♦ **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.

**♦ **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.

**♦ **Monday October 10th (2:15pm – 3.15pm): **Christian Haesemeyer**

**Title: What is K-Theory?**

**Abstract: **In this talk I will give a brief introduction to algebraic K-theory. This invariant is both a way to organise (stable) linear algebra over rings, and to capture geometric and topological information. Using a few concrete examples I will try to illuminate this connection.

**♦ **Monday December 12th (2:30pm – 3.10pm): **Adrian Hendrawan**

**Title: On the Weyl invariants of the projective unitary groups**

**Abstract: **Let i : BT \to BG be the classifying map for the inclusion of a maximal torus T into a connected Lie group G. A classical strategy for studying the cohomology groups H^*(BG) is to study the image of the restriction map

i^*: H^*(BG) \to H^*(BT). The ring H^*(BT)

is a polynomial algebra which admits an action by the Weyl group W of G, and the image of i^* is contained in the Weyl-invariant subalgebra H^*(BT)^W \subset H^*(BT).

In the case of the unitary group U_n, H^*(BU_n; Z) = Z[c_1, …, c_n] is a polynomial algebra generated by the Chern classes c_i, and the map i^* is an isomorphism. For the projective unitary group, G = PU_n = U_n/U_1, the ring contains torsion and H^*(BPU_n; Z), remains unknown in general for n > 3. In this thesis we construct a canonical extension map

e^* : H^*(BPT_n; Z)^W → H^*(BPU_n; Z).

and show that it is a ring map, which is a right inverse to the restriction map i^*. Thus we exhibit that the Weyl-invariant algebra as a canonical subalgebra of H^*(BPU_n; Z).

**♦ **Monday December 12th (3:10pm – 4.10pm): **Csaba Nagy**** (Glasgow)**

**Title: Homotopy equivalence and simple homotopy equivalence of manifolds**

**Abstract: **Abstract: A homotopy equivalence between finite CW-complexes is called simple if it is homotopic to a composition of elementary collapses and expansions. Lens spaces provide famous examples of manifolds that are homotopy equivalent but not simple homotopy equivalent to each other, in all ≥ 3 odd dimensions. However, no even-dimensional examples are known in the literature.

We construct even-dimensional manifolds that are homotopy equivalent (in fact h-cobordant) but not simple homotopy equivalent to each other. More generally, we give a purely algebraic characterisation of groups G with the property that there exists a pair of manifolds with fundamental group G that are h-cobordant but not simple homotopy equivalent. We also consider a second type of example: manifolds that are homotopy equivalent, but not equivalent under the equivalence relation generated by simple homotopy equivalence and h-cobordism. For the latter equivalence relation we construct a complete invariant, which is defined on all n-manifolds with no [n/2]-cells for n > 6.

This is joint work with Johnny Nicholson and Mark Powell.

——————————————————————————————

**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.

——————————————————————————————

**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).

——————————————————————————————

**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.

——————————————————————————————

**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.