Speaker: Valentin Bonzom

Title: Deformations of Hurwitz numbers via algebraic combinatorics

Abstract: Hurwitz numbers count factorisations in the symmetric groups and as such can be studied using algebraic combinatorics. Following the works of Goulden and Jackson, we will explain how algebraic combinatorics unravels beautiful properties and conjectures. We will explain how the KP hierarchy arises and can be used to extract powerful recursion formulas. Moreover, the theory of symmetric functions has suggested deformations of classical Hurwitz numbers by Jack and Macdonald polynomials, where positivity conjectures still hold. We will discuss the recent results in the field, including functional equations that characterize the generating functions of those deformed Hurwitz numbers, and integrability via the BKP hierarchy in some cases of non-oriented Hurwitz numbers such as the monotone ones. This is based on works by various combinations of Ben Dali, Chapuy, Dolega and myself.

Speaker: Lucia Caporaso

Title: Algebraic hyperbolicity and hypertangency of curves on surfaces.

Abstract: The talk will report on joint work with Amos Turchet on a standard conjecture of diophantine geometry usually attributed to Vojta. The conjecture predicts that in a surface of log general type there are only finitely many  rational curves. For the projective plane, this is implies that   the complement of a general  curve of degree at least 4 contains finitely many copies of ℂ∗ (where ℂ∗ is the set of nonzero complex numbers).  We prove some cases of this conjecture for the plane and for other surfaces.

Speaker: Maciej Dołęga

Title: b-Hurwitz numbers from W-constraints and refined topological recursion

Abstract: b-Hurwitz numbers are a one-parameter deformation of classical Hurwitz numbers, interpolating between them and their real (non-orientable) analogue. We explain how to encode combinatorially, using lattice paths, a system of partial differential operators that uniquely determines these numbers, derived from a representation of W-algebras. This result implies a remarkable property of undeformed Hurwitz numbers: they can be computed using topological recursion. We demonstrate that in the deformed case, a large class of Hurwitz numbers can be computed by the deformed version of topological recursion known as refined topological recursion. This talk is based on joint work with N. Chidambaram and K. Osuga.

Speaker: Elise Goujard

Title: Hurwitz numbers and flat surfaces

Abstract: I'll present some counting problems on moduli space of flat surfaces related to Hurwitz theory. After motivating these problems via some questions on billiards dynamics, I will explain how the count of integer points in these moduli spaces is related to Hurwitz theory. Then I'll present some results about polynomiality of certain 2-orbifold Hurwitz numbers appearing in this context, summarizing an old joint work with M. Möller and a new joint work with E. Duryev and I. Yakovlev.

Speaker: Xiang He

Title: The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces

Abstract: In 1969, Fulton introduced classical Hurwitz spaces in the algebro-geometric setting and established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than the degree of the coverings. In this talk, I will give a prove of the irreducibility over any algebraically closed field. The main ideal is to construct a dominant map from a certain Severi variety on P1\times P1 to the Hurwitz space, and show that the Severi variety is irreducible. I will also establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces. This is joint work with Karl Christ and Ilya Tyomkin.

Speaker: Ilia Itenberg

Title: Real enumerative invariants relative to the toric boundary

Abstract: The talk is devoted to real enumerative invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate signed enumeration of real algebraic curves of genus 1 and 2. It turns out that two different rules of signs in the enumeration lead to the same collection of invariants. The proof of this surprising fact uses the tropical counterparts of the invariants under consideration. This is a joint work with Eugenii Shustin.

Speaker: Maksim Karev

Title: CJT-refined action of the algebra of symmetric functions on the Fock space

Abstract: We propose a representation-theoretical framework, that conjecturally underlies the notion of  b-Hurwitz numbers. The talk is based on a joint work in progress with R.Fesler, M. A. Hahn and H. Markwig.

Speaker: Reinier Kramer

Title: Topological recursion for leaky Hurwitz numbers

Abstract: Hurwitz numbers, which count covers of Riemann surfaces with given ramification profiles, have in the last view decades been shown to have strong relations to many different areas of mathematics, ranging from Gromov-Witten theory to integrable hierarchies and topological recursion. Recently, Cavalieri-Markwig-Ranganathan introduced leaky Hurwitz numbers. These no longer count maps to the Riemann sphere, i.e. meromorphic functions, but rather meromorphic sections of pluricanonical bundles. These numbers up to now lack some of the interpretations given above, but are a natural extension from the integrability perspective. In my talk I will introduce these numbers, and report on progress on including them in the topological recursion framework. In particular, I will highlight how the spectral curve, i.e. the base step of the recursion, arises from a Hamiltonian flow along the cut-and-join operator. This is joint work in progress with Marvin Anas Hahn.

Speaker: Danilo Lewanski

Title: Leaky Hurwitz numbers

Abstract: Leaky Hurwitz numbers are yet another variation of classical Hurwitz numbers. However, this variation is fundamentally different because instead of changing the ramification type, or the finite group whose representation theory define the numbers, this variation a priori breaks their geometric definition, and yet several Hurwitz numbers-like results apply.

Contributed talks

Speaker: Marco Fava

Title: On compactified Jacobians of reduced curves. 

Abstract: Several solutions to the problem of compactifying the generalised Jacobian of (families of) singular curves by means of torsion-free rank 1 sheaves have been produced since the 1950's. However, in general not all the possible compactified Jacobians arise from these classical constructions. We develop the notion of V-compactified Jacobians for families of reduced curves, and show that they give a complete classification of smoothable compactified Jacobians of nodal curves. In particular, we classify compactified universal Jacobians (i.e. over the moduli stacks of pointed stable curves). This talk is based on a joint work with Nicola Pagani and Filippo Viviani.

Speaker: Matthias Hippold

Title: A Logarithmic Hurwitz Space with Wild Ramification

Abstract: Hurwitz Spaces parametrizing covers of curves where the characteristic of the base field divides the degree of the map have received only little attention in current research as wild ramification is not well enough understood yet. However, new results from Berkovich analytic and logarithmic geometry provide explicit conditions for covers of prime degree p over a field of characteristic p to be liftable to covers over a DVR of mixed characteristic. Making use of these insights in the basic case of degree p covers of curves of genus 0, we will introduce a logarithmic Hurwitz space parametrizing lifting data that consist of a finite cover of a logarithmic smooth curves of genus 0 together with a piecewise linear function on the tropicalization of the source curve and a differential form that recovers missing geometrical information at components where the cover is given by the geometric Frobenius. Furthermore, we will give an outline of a proof showing that this Hurwitz space is logarithmically smooth over the DVR and discuss an example of degree 3 covers of curves of genus 0 by curves of genus 2 in characteristic 3 where all interesting types of deformations appear.

Speaker: David Klompenhouwer

Title: Spin refinement of moduli spaces of meromorphic differentials and the BKP hierarchy 

Abstract: In the moduli space of curves, the strata of curves carrying meromorphic differentials with even orders at zeros and poles can be decomposed into even and odd components, based on the parity of spin structures on the curve. This so-called spin refinement of strata of meromorphic differentials with vanishing residues form a partial cohomological field theory (CohFT). We apply the DR hierarchy construction of Buryak to this partial CohFT and show that the resulting system of evolutionary PDEs coincides with the BKP hierarchy, after a reduction. This result relies on some intersection-theoretic computations that are made possible by recent work by Costantini-Sauvaget-Schmitt and the recently proved DR/DZ correspondence. This is based on joint work with Stijn Velstra.

Speaker: Amy Li

Title: Computing the Chow ring of Hurwitz spaces with marked ramification

Abstract: The Picard Rank Conjecture for the Hurwitz space of simply branched admissible covers states that A^1 is generated by the codimension-1 boundary strata. This was proved for covers of degree 3,4, and 5 by Deopurkar and Patel (2015). Little is known about the higher codimension Chow groups. In this talk, I will present a result showing that A^2 of the Hurwitz space of degree 3 simply branched admissible covers is generated by the codimension-2 boundary strata. The proof follows a method by Canning and Larson (2022) which realises the open Hurwitz space inside a vector bundle over a particular moduli stack of vector bundles. We do this for open Hurwitz spaces with marked ramification, thereby showing that such Hurwitz spaces have trivial Chow ring, and obtaining the result by excision. This is joint work with E. Clader, Z. Hu, H. Larson, and R. Lopez.