Title: Positive tropicalization and its applications
Abstract: Tropical geometry builds a bridge between algebraic and polyhedral geometry by transforming an algebraic variety into a polyhedral object that preserves key properties of the original variety. Recently, there has been increasing interest in the tropicalization of the positive part of algebraic varieties, and more generally, in the…
Title: Geometry of Severi varieties on toric surfaces
Abstract: In this talk I will give an overview of a series of works together with Xiang He and Ilya Tyomkin on the geometry of Severi varieties of toric surfaces. The main focus will be on the question of their irreducibility, the so called Severi problem, and its applications to the irreducibility of other moduli spaces of curves.
The…
Title: Hirzebruch-Riemann-Roch in Combinatorial K-Theory
Abstract: The Hirzebruch-Riemann-Roch Theorem provides an intricate connection between the K-theory and the intersection theory of an algebraic variety. What data is needed to describe this connection explicity? For smooth projective toric varieties, it turns out that all the necessary information is contained in the Ehrhart polynomial. I…
Title: Quantum mirrors, tropical curves and curve counting on surfaces
Abstract: Certain counts of rational curves in (log Calabi–Yau) surfaces can be read off from the geometry of the Gross–Hacking–Keel mirror to the surface. I will discuss a generalisation of this to higher-genus curves, whose counts can be extracted from the geometry of a deformation quantisation of this mirror via a…
Title: 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…
Title: Non-archimedean periods for log Calabi-Yau surfaces
Abstract: 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.…
Title: 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…
Title: 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…
Title: 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…
Title: 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…
Title: 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…
Title: 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…
