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…
