Abstracts

We review Heisenberg homology of configurations in once bounded surfaces and extend the construction to the regular thickening of a finite graph with ribbon structure. The first part is based on joint work with Martin Palmer and Awais Shaukat.

In this talk we will introduce cochain level operations inducing the Steenrod power operations for topological spectra. These operations have been described for topological spaces by Steenrod himself and May, in which case they are compatible with a multiplication structure. The lack of this multiplication structure on the cohomology of a spectrum allows to impose axiomatically the vanishing of the negative power operations.

There exists a functorial universal finite-type invariant for tangles in thickened surfaces \Sigma\times[0,1]. It is obtained very closely to the recent construction by Habiro and Massuyeau (2017-20) of a similar invariant for bottom tangles in handlebodies. The values of these invariants are formal sums of colored chord diagrams. We show that, in view of constructions and results by Andersen, Mattes, and Reshetikhin (1996-98), this establishes a canonical deformation quantization of the Poisson algebra in the title. We will also look at the case of some skein algebras, and the case of Poisson algebras of regular functions on some moduli spaces {\cal M}(\Sigma,G) of flat G-connections on \Sigma.

Any Batalin–Vilkovisky algebra with a homotopy trivialization of the BV-operator gives rise to a hypercommutative algebra structure at the cochain level which, in general, contains more homotopical information than the hypercommutative algebra introduced by Barannikov and Kontsevich on cohomology. In this talk, I will explain how to use the purity of mixed Hodge structures to prove formality of certain hypercommutative algebras associated to Kähler and Calabi-Yau manifolds. This is joint work with Geoffroy Horel.

The main player in Gromov-Witten (GW) theory is the moduli space of stable maps to projective space \overline{\mathcal{M}}_{g,n}(\mathbb{P}^n,d). It parametrizes (nodal) curves of fixed genus g with n marked points and with a morphism to \mathbb{P}^n of degree d. One can obtain enumerative invariants, called GW invariants, using intersection theory on \overline{\mathcal{M}}_{g,n}(\mathbb{P}^n,d). However, for g\geq1 these invariants have contributions from lower genera.

In genus one, Vakil and Zinger defined reduced GW invariants, which count "honest" genus 1 maps, using a blow-up of \overline{\mathcal{M}}_{g,n}(\mathbb{P}^n,d). A similar approach was used by Hu, Li and Niu in genus two. We define reduced GW invariants in all genera using the notion of blow-up along a sheaf. This is joint work with Cristina Manolache, Etienne Mann and Renata Picciotto.

We present a greedy algorithm optimizing arrangements of lines with respect to a property and apply this algorithm to the case of simpliciality. An implementation produces a database with many surprising examples of moduli spaces of realizations of matroids of rank three.

Graphic subspace arrangements are associated to simple graphs. They generalize configuration spaces of points in Euclidean space, and have been studied in a few references (including by late Stefan Papadima). We use poset topology to compute the Poincare polynomials of the graph configuration spaces, or equivalently of the graphic arrangements. The answer is expressed in terms of acyclic orientations of the graph and of its bond poset. The stable homotopy type is deduced, and the result is expanded to treat more general families of configuration spaces of points in Euclidean space. This is joint work with Moez Bouzouita.

For g≥2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this talk, we will discuss a complete characterization of the infinite metacyclic subgroups of Mod(Sg) up to conjugacy. In particular, we will discuss equivalent conditions under which a pseudo-Anosov mapping class generates a metacyclic subgroup of Mod(Sg) with another mapping class. As an application, we establish the existence of infinite metacyclic subgroups of Mod(Sg) isomorphic to ℤ ⋊ ℤn, ℤn ⋊ ℤ, and ℤ ⋊ ℤ.

A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear dependency in vector spaces as well as many aspects of graph theory. Moreover, matroids form a cornerstone of tropical geometry and a deep link between algebraic geometry and combinatorics. After a gentle introduction to matroids, I will present parts of a new OSCAR module for matroids through several examples. I will focus on computing the moduli space of a matroid which is the space of all arrangements of hyperplanes with that matroid as their intersection lattice. If time permits I will discuss diverse applications of this module. I will not assume any prior knowledge of matroids. Based on joint work with Dan Corey and Benjamin Schröter.

I will present topological invariants counting graph configurations, following Witten, Kontsevich, Kuperberg, Thurston, Watanabe, and many other authors. I will develop two examples. First, I will show how to find the Casson invariant of a homology 3-sphere by counting graphs theta in such a 3-manifold. Next, I will outline how the ideas involved inspired Watanabe to produce exotic elements in the fundamental group of the group of diffeomorphisms of the 4-dimensional sphere.

Slides available here.

The Johnson-Morita theory provides an approach for the mapping class group of a surface by considering its actions on the successive nilpotent quotients of the fundamental group of the surface. In this talk, after an outline of the original theory, we will present an analogue of the Johnson-Morita theory for the handlebody group, i.e. the mapping class group of a handlebody. This is work in progress with Kazuo Habiro ; as we shall explain if time allows, our motivation is to recover the "tree reduction" of a certain functor on the category of bottom tangles in handlebodies that we introduced (a few years ago) using the Kontsevich integral.

The complement of an arrangement of hyperplanes in a complex vector space is a much studied interesting topological space. A fundamental problem is to decide when this space is aspherical. For special classes of arrangements, such as the braid arrangements or more generally so called supersolvable arrangements, this can be achieved by utilizing fibrations which connect complements of arrangements of different rank.

Another prominent space associated to an arrangement is its Milnor fiber -- the typical fiber of the evaluation map of the defining polynomial of the arrangement on its complement which is a smooth fibration by Milnor's famous result. This is a much more subtle topological invariant and it is still an open problem to understand its homology or even its first Betti number in conjunction with the combinatorial structure of the arrangement.

I will present a new combinatorial approach to study such fibrations for arrangements which can be defined over the reals via oriented matroids. This is partly joint work with Masahiko Yoshinaga (Osaka University).

Let M be a compact surface without boundary, and n\geq 2. We analyse the quotient group B_n(M)/\Gamma_2(P_n(M)) of the surface braid group B_{n}(M) by the commutator subgroup \Gamma_2(P_n(M)) of the pure braid group P_{n}(M). If M is different from the 2-sphere, we prove that B_n(M)/\Gamma_2(P_n(M)) \cong P_n(M)/\Gamma_2(P_n(M)) \rtimes_{\varphi} S_n, and that B_n(M)/\Gamma_2(P_n(M)) is a crystallographic group if and only if M is orientable. 

If M is orientable, we show a number of results regarding the structure of B_n(M)/\Gamma_2(P_n(M)). Finally, we construct a family of Bieberbach subgroups \widetilde{G}_{n,g} of B_n(M)/\Gamma_2(P_n(M)) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if {\mathcal X}_{n,g} is a flat manifold whose  fundamental group is \widetilde{G}_{n,g}, we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms. This is a joint work with Daciberg Lima Gonçalves, John Guaschi and Carolina de Mirana e Pereiro.

This is a continuation of the talk of Craig Westerland, reporting on joint work of the two of us with Bergström and Diaconu. I will explain that the stable homology of braid groups, with coefficients arising from the hyperelliptic representation, is related to a problem in arithmetic statistics: estimating the asymptotics for moments of families of quadratic L-functions. Our determination of the stable homology, combined with an improved stable range for these coefficients (proven in joint work with Miller--Patzt--Randal-Williams) establishes an asymptotic formula for all moments, for sufficiently large prime powers. This asymptotic formula had been conjectured by Conrey-Farmer-Keating-Rubinstein-Snaith.

Maximizing curves show up in a very classical problem in the theory of algebraic surfaces, namely how to construct surfaces with high Picard numbers. One of possible ways is to consider double covers of the complex projective plane branched along reduced plane curves with ADE singularities. This idea leads to the notion of maximizing curves. We show that maximizing curves are exactly free curves and we present several examples.

Based on a joint work with Alex Dimca.

We present a new approach to the Goodwillie-Sinha spectral sequence for the homology of the space of long knots in R^3 based on Fox-Neuwirth cells of configuration spaces that makes it more accessible to computations.

The twisted cohomology of mapping class groups of compact orientable surfaces (with one boundary) is very difficult to compute generally speaking. In this talk, I will describe the computation of the stable cohomology algebra of these mapping class groups with twisted coefficients given by the first homology of the unit tangent bundle of the surface. This type of computation is out of the scope of the traditional framework for homological stability. I will also present the computation of the stable cohomology algebras with twisted coefficients given by the exterior powers of the aforementioned representations. This represents a joint work with Nariya Kawazumi. I will finally present some of my recent progresses on the stable twisted cohomology computations given by other representations of the mapping class groups.

A famous theorem of Milnor says that every connected 3-manifold M admits a unique prime decomposition as a connected sum of prime manifolds.

I will explain joint work with Rachael Boyd and Corey Bregman in which describe the moduli space B Diff(M) as a colimit of the moduli spaces of prime pieces of M and their configuration spaces.

This allows us to prove a conjecture of Kontsevich according to which B Diff(M rel \partial M) is equivalent to a finite CW-complex whenever M  has non-empty boundary. We also give a full computation of the rational cohomology ring of B Diff( (S^1 x S^2) # (S^1 x S^2) ).

In this talk, I will explain how foams (some 2-dimensional CW complexes) allow to give a purely combinatorial definition of gl(N) link homologies, the case N = 2 being the celebrated Khovanov homology. This construction allows to see an sl(2) action on these homologies. We will also see how to extend the previous action functorially. Joint work with You Qi, Louis-Hadrien Robert and Joshua Sussan.

The braid group B_{2g+1} has a description in terms of the hyperelliptic mapping class group of a curve X of genus g.  This equips it with an action on V = H_1(X), and we may produce a wealth of new representations S^{\lambda}(V) by applying Schur functors to V.  The goal of this talk is to describe the stable (in g) group homology of these representations.  

Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space.  We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces.

This is joint work with Bergström, Diaconu, and Petersen.  Dan will give a talk later in the program on the application of these results to statistical questions about moments of quadratic L-functions in the function field setting.

Quantum representations of mapping class groups are closely related to three-dimensional topological field theory and play an important role in the representation theory of Hopf algebras and conformal field theory. The framework to discuss these representations is the notion of a modular functor. I will recall some classical constructions of modular functors and outline an approach to modular functors via cyclic and modular operads. Finally, the connection to factorization homology will be covered. (The talk is based on different joint works with Adrien Brochier and Lukas Müller.)

I will first give a review on the calculations of homology of mapping class groups for finite type surfaces. After that I will discuss what is the story in the case of infinite type surfaces. In particular, I will discuss how one can calculate the homology of mapping class groups for some well-known surfaces, including the disk minus Cantor set. This is based on joint works with Martin Palmer.