Title: Tropical trigonal curves Abstract: The moduli space of algebraic curves of genus g admits a natural stratification by gonality, defined as the smallest positive integer d for which a curve admits a degree d morphism to the projective line, or equivalently, a linear series of degree d and dimension one. After motivating the study of the tropical analogue of the moduli space of trigonal…

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…