Title: Edge-coloured graph enumeration, the Ising model on random graphs, and Lee-Yang zeros Abstract: Certain exponential integrals serve as generating functions of labelled edge-coloured graphs. Based on this, we derive asymptotics for the number of edge-coloured regular graphs with arbitrary weights assigned to different vertex structures. The asymptotic behaviour is governed by critical…

Title: The variety of orthogonal frames Abstract: Orthogonal frames are tuples of pairwise orthogonal vectors in a quadratic vector space. We study the variety of all orthogonal frames and its defining ideal, and discuss their algebraic, geometric and topological properties. We discuss applications to Lovász-Saks-Schrijver ideals of graphs. Joint work with Alessio Sammartano.

Title: Polytopes from heaps Abstract: The Birkhoff polytope is the convex hull of all permutation matrices in a fixed symmetric group. We study certain subpolytopes of the Birkhoff polytope whose vertex sets coincide with prefixes of a set of commutation-related reduced words. The vertices are indexed by order ideals (downwards-closed sets) of a certain poset called a heap. The set of order ideals…

Title: Logarithmic coherent sheaves Abstract: As a variety undergoes a simple normal crossing degeneration, its coherent sheaves and their moduli spaces degenerate unpredictably. Since Gieseker, one geometric idea has proved indispensable in constructing these degenerations: study sheaves on expansions of the degenerate variety. This talk will introduce logarithmic coherent sheaves, which arrange…

Title: The tropical Donagi theorem Abstract: The Torelli morphism assigning to a smooth projective curve its Jacobian is well known to be injective. Generalizing to Prym varieties, the situation becomes much richer. Indeed, the Prym-Torelli morphism assigning to an étale double cover of smooth curves its Prym variety is never injective. Donagi introduced the tetragonal construction and gained…

Title: The algebraic geometry of Ornstein--Uhlenbeck processes in equilibrium Abstract: The goal of this talk is to explain a few recent results about a class of statistical models. While these results answer statistical questions, their proofs are carried out entirely in terms of algebraic geometry. I will explain the applied motivation behind all this and show how to carry out such a translation…

Title: Continuity of the tropical Prym-Torelli map Abstract: Given an étale double cover of smooth curves one can associate a principally polarised abelian variety called the Prym variety. In tropical geometry, the naive tropical definition of Prym variety does not behave well under degeneration. Recently, Röhle and Zakharov suggested a slightly different definition of the Prym, called the…

Title: Hamiltonicity in acyclic orientation graphs Abstract: Given a graph G, we are interested in enumerating all possible acyclic orientations of G by use of a Gray code, i.e. an enumeration where the change between two subsequent elements is small in some sense. In the case of graph orientation, this small change is an edge flip. This can be modelled through the corresponding graphic hyperplane…

Title: Positive del Pezzo Geometry Abstract: A positive geometry is, roughly speaking, a complex projective variety with a distinguished semialgebraic set in its real points, considered as the nonnegative part. Important to a positive geometry is a differential form, called the canonical form, that is compatible with the combinatorics of the boundary of the nonnegative part. Positive geometries…

Title: A universal decomposition of orbifold Gromov-Witten invariants of root stacks Abstract: For a pair (X|D) of a smooth projective variety X relative normal crossings divisor D one can study maps from curves with fixed tangencies along D via two theories; Logarithmic Gromov--Witten theory (LogGW) of (X|D) and Orbifold Gromov--Witten theory (OrbGW) of the root stack X_{D,r}. Each theory has its…

Title: The polytope of all $q$-rank functions Abstract: A $q$-rank function is a real-valued function defined on the subspace lattice of $\mathbb{F}_q^n$ that is non-negative, upper bounded by the dimension function, non-decreasing, and satisfies the submodularity law. Each such function corresponds to the rank function of a $q$-polymatroid. Intuitively, we can view these objects as $q$-analogues…

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Title: Minuscule Coxeter Dressians Abstract: Matroids combinatorially abstract independence systems and have deep connections throughout mathematics. One characterization of matroids is given by (0,1)-polytopes with edge directions in the type A root system. In this case, edges capture "symmetric exchange" between bases. Regular matroidal subdivisions of polytopes are well studied. In particular,…

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…

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…