May 13, 2025 (starting time: 13:35 pm): Guendalina Palmirotta (Paderborn University): Resonance chains on funneled tori
Abstract: Already in many scattering systems such as the n-disk systems and the symmetric 3-funneled hyperbolic surfaces, resonance chains have been rigorously studied. Their existence is related to the analyticity of the Selberg zeta function and they can be described explicitly by a polynomial. In a recent paper, Li, Matheus, Pan, and Tao introduced a different method to explicitly compute the polynomial derived from the intermediate zeta function, enabling the description of resonance chains not only on symmetric 3-funneled hyperbolic surfaces, but also for, funneled tori. In this talk, I will show you the numerics that I obtained for the funneled tori and try to explain the mathematical concepts behind them. This is an ongoing project with Tobias Weich.
May 20, 2025: Sean Monahan (TU Munich): Horospherical varieties and stacks
Abstract: To begin, I will introduce horospherical varieties and their known combinatorial theory. Horospherical varieties generalize toric varieties and their correspondence with polyhedral fans: instead of a torus action, we can use any reductive group, and instead of fans, we use a generalization called "coloured" fans. After this, I will highlight some of my research on extending this combinatorial theory to the world of algebraic stacks, namely a correspondence between horospherical stacks and "stacky" coloured fans.
May 27, 2025: Marvin Hahn (Trinity College Dublin): Tropical geometry of b-Hurwitz numbers
Abstract: The Goulden-Jackson b-conjecture is a remarkable open problem in algebraic combinatorics. It predicts an enumerative meaning for the coefficients of the expansion of a certain expression of Jack symmetric functions. Major progress was made in recent work of Chapuy and Dołęga, which led to the introduction of b-Hurwitz numbers. These invariants are generalisations of classical Hurwitz numbers obtained by including non-orientable surfaces. They are polynomials in a parameter b, which measures the "non-orientability" of the coverings involved. In this talk, we develop a tropical theory of b-Hurwitz numbers and the recently introduced monotone b-Hurwitz numbers. As part of this development, we resolve an open question posed by Chapuy–Dołęga, as well as an open question posed by Bonzom–Chapuy–Dołęga. This talk is based on joint work in progress with Raphaël Fesler, Maksim Karev and Hannah Markwig.
June 10, 2025: Pedro Souza (Goethe University Frankfurt): On the topology of the moduli space of tropical Z/pZ-covers
Abstract: We study the topology of the moduli space of (unramified) Z/pZ-covers of tropical curves of genus g≥2 where p is a prime number. By recent work of Chan-Galatius-Payne, the (reduced) homology of this tropical moduli space computes (with a degree-shift) the top-weight (rational) cohomology of the corresponding algebraic moduli space. We prove contractibility of certain subcomplexes of the tropical moduli space and use this result to show that it is simply connected and to fully determine its homotopy type for g=2 and all p.
June 17, 2025: Terry (Dekun) Song (University of Cambridge): Dual complex of genus one mapping spaces
Abstract: The dual complex of a smooth variety encodes the combinatorial structure that underlies all its possible normal crossings compactifications. We prove that the dual complexes of genus zero and genus one mapping spaces are contractible (in degrees > 0 and > 1 respectively) via an explicit deformation retraction. In genus one, the key geometric input comes from the Vakil - Zinger space and its tropical interpretation due to Ranganathan - Santos-Parker - Wise. Joint work with Siddarth Kannan (MIT). Time permitting, I will discuss ongoing work on understanding the full cohomology of the Vakil - Zinger space.
June 24, 2025: Siao Chi Mok (University of Cambridge): Logarithmic Fulton—MacPherson configuration spaces
Abstract: The Fulton—MacPherson configuration space is a well-known compactification of the ordered configuration space of a projective variety. We describe a construction of its logarithmic analogue: it is a compactification of the configuration space of points on a projective variety X away from a simple normal crossings divisor D, and is constructed via logarithmic and tropical geometry. Moreover, given a semistable degeneration of X, we construct a logarithmically smooth degeneration of the Fulton—MacPherson space of X. Both constructions parametrise point configurations on certain target degenerations, arising from both logarithmic geometry and the original Fulton–MacPherson construction. The degeneration satisfies a “degeneration formula” – each irreducible component of its special fibre can be described as a proper birational modification of a product of logarithmic Fulton–MacPherson configuration spaces. Time permitting, we explore some potential applications to enumerative geometry.
July 1, 2025: Soham Karwa (Duke University): Non-archimedean periods for log Calabi-Yau surfaces
Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman. We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will see how non-archimedean periods recover the complex analytic periods for log Calabi-Yau surfaces, proving the first instance of a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.
July 15, 2025: Giulia Iezzi (RWTH Aachen): Quiver Grassmannians for the Bott-Samelson resolution of type A Schubert varieties
Abstract: Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. In this talk, we construct a special quiver with relations and consider two classes of quiver Grassmannians for this quiver. For an appropriate choice of dimension vector for this quiver, we provide an isomorphism between the corresponding quiver Grassmannians and certain Bott-Samelson resolutions of type A Schubert varieties. Furthermore, for smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety.