ORGANIZERS:
The moduli space of genus g tropical curves with n marked points is a fascinating topological space, with a combinatorial flavor and deep algebro-geometric meaning. In the algebraic world, forgetting the n marked points gives a fibration whose fibers are configuration spaces of a surface, and Serre's spectral sequence lets one compute the cohomology "in principle". In joint work with Bibby, Chan and Yun, we construct a surprising tropical analog of this spectral sequence, manifesting as a graph complex and featuring the cohomology of compactified configuration spaces on graphs.
I will talk about the study of Alexander modules of algebraic varieties using Gabber and Loeser's Mellin transform. The main strength of this approach is that it allows the application of the full machinery of the theory of perverse sheaves, and even mixed Hodge modules. We obtain new results about the structure of Alexander modules, especially about their torsion part and, in the multivariable case, their artinian submodules. It also yields a mixed Hodge structure on the maximal artinian submodules of the Alexander modules. This is based on joint work with Eva Elduque, Laurentiu Maxim and Botong Wang.
The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss implications for homogeneous polynomials and other special cases.
An important task in algebraic geometry and Hodge theory is to control the monodromy of families of subvarieties in a given variety. In their recent work on the Shafarevich conjecture, Lawrence and Sawin have shown that any non-isotrivial family of smooth hypersurfaces in an abelian variety has big monodromy when twisted by a generic local system of rank one. I will explain how to go beyond hypersurfaces: The same big monodromy theorem holds for every family of subvarieties of dimension at most half the dimension of the abelian variety. The proof uses a combination of geometric arguments, representation theory and perverse sheaves; this is joint work with Ariyan Javanpeykar, Christian Lehn and Marco Maculan.
I will describe investigations of the entire "population" of Morse functions on a smooth compact manifold. I will concentrate on two questions. How does a typical/average Morse function look like? Are there many atypical Morse functions?
We briefly recall the definition of hyperplane arrangements, toric arrangements and their generalization called abelian arrangements. The cohomology ring of the complement is known from a result by Orlik and Solomon (1980) in the hyperplane case. The toric case is due to De Concini and Procesi (2005) and to Callegaro, D’Adderio, Delucchi, Migliorini, and I (2020). In this talk, we present a new and unified presentation of the cohomology ring of all abelian (non-compact) arrangements. This is a work in progress with Evienia Bazzocchi e Maddalena Pismataro.
A long-standing open question about mapping class groups of surfaces is whether they are linear, i.e. act faithfully on finite-dimensional vector spaces. In genus zero, for the braid groups, the answer is yes, as proven by Bigelow and Krammer using one of the family of Lawrence representations of the braid groups. Motivated by this, I will describe an analogue of the family of Lawrence representations for higher-genus surfaces -- depending on a chosen representation V of the discrete Heisenberg group. A subtlety is that these mapping class group representations are in general twisted, essentially as a consequence of the non-commutativity of the discrete Heisenberg group. However, I will explain how to untwist them for particular choices of V (and for any V if we restrict to the Torelli group). This all represents joint work with Christian Blanchet and Awais Shaukat. The appearance of the discrete Heisenberg group in the construction arises from the study of the lower central series of (partitioned) surface braid groups: I will also outline recent joint work with Jacques Darné and Arthur Soulié that answers the stopping question for these lower central series.
Jones and Alexander polynomials are two important knot invariants and our aim is to see them both from a unified model constructed in a configuration space. More precisely, we present a common topological perspective which sees both invariants, based on configurations on ovals and arcs in the punctured disc. The model is constructed from a graded intersection between two explicit Lagrangians in a configuration space. It is a polynomial in two variables, recovering the Jones and Alexander polynomials through specialisations of coefficients. Then, we prove that the intersection before specialisation is (up to a quotient) an invariant which globalises these two invariants, given by an explicit interpolation between the Jones polynomial and Alexander polynomial. We also show how to obtain the quantum generalisation, coloured Jones and coloured Alexander polynomials, from a graded intersection between two Lagrangians in a symmetric power of a surface.
In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive "generic" corank 1 projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In this paper we completely settle the case of dimensionality type 2, by studying Zariski equisingular families of surfaces singularities, not necessarily isolated, in the three-dimensional space.
In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod cup-one products and compatible binomial operations. Given such an \(R\)-dga, \((A,d)\), defined over the ring \(R=\mathbb{Z}\) or \(\mathbb{Z}_p\) (for \(p\) a prime) and with \(H^1(A)\) a finitely generated, free \(R\)-module, we show that \(A\) admits a functorially defined \(1\)-minimal model, unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, \(G(A)\), which only depends on the \(1\)-quasi-isomorphism type of \(A\). These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic (torsion-free) cohomology that share the same rational \(1\)-minimal model, yet whose integral \(1\)-minimal models are not isomorphic. This is joint work with Richard Porter.