Session 1 -- Tu 15:30 - 16:30

• Patience Ablett (Warwick): Gotzmann's persistence theorem for smooth projective toric varieties
  The Hilbert scheme is a scheme which parameterises subschemes living inside a fixed ambient space. By understanding the geometry of the Hilbert scheme itself, we can learn about the geometry of the subschemes we parameterise. In the case that our fixed ambient space is projective space, this scheme is well understood using Gotzmann's regularity and persistence theorems. In this talk, we look at generalising Gotzmann's persistence result to the setting of any smooth projective toric variety.

• Leandro Meier (Jena): Bounding minimal log discrepancy of complexity one T-Varieties
  After shortly introducing the minimal log discrepancy and Shokurov's conjecture regarding its boundedness, I will present my result, confirming boundedness of the minimal log discrepancy at a closed point for varieties with a torus action of complexity one.

• Sean Monahan (TU Munich): Horospherical varieties with quotient singularities
  Horospherical varieties are a vast generalization of toric varieties where one replaces the acting torus by any reductive group. There is a combinatorial classification of horospherical varieties using so-called coloured fans, which generalizes the classification of toric varieties via fans. In this project, I give a combinatorial characterization of when horospherical varieties have quotient singularities, and I use this to answer a local-to-global question about quotient singularities posed by Fulton in this setting.

• Kaelyn S. Willingham (Minnesota): On the Pierce-Birkhoff Conjecture
  Garrett Birkhoff and Richard Pierce first introduced the f-ring construction in a 1956 paper on lattice-ordered rings, which have interesting algebraic & combinatorial features. The f-ring became a central component in a subsequent 1960s conjecture in real algebraic geometry, which states that any piecewise-polynomial function can be expressed as a maximum of (finite) minima of (finite) collections of polynomials. Now known as the Pierce-Birkhoff Conjecture, the problem remains open to this day despite many attempts at its resolution. In this short talk, I would like to introduce the conjecture and discuss its relationships with both tropical geometry and machine learning theory, with the hope of soliciting others to work on this problem with me.

• Elsa Maneval (EPFL): Non-coprime case of the Hausel-Thaddeus Conjecture and p-adic Integration
  I will introduce the moduli spaces of Higgs bundles that appear in the Hausel-Thaddeus mirror symmetry conjecture, with a focus on the distinction between the coprime and non-coprime cases. I will then discuss the p-adic integration method and how it extends to the non-coprime case.

• Giulia Iezzi (RWTH Aachen): Linear degenerations of Schubert varieties via quiver Grassmannians

• Matthew Dupraz (FU Berlin): Lattice counts and K-theory
  For complete toric varieties, we may compute the Euler characteristic of a line bundle by counting the number of lattice point in the polytope associated to the corresponding Cartier divisors. For incomplete toric varieties the Euler characteristic is not well defined, but for Bergman fans it's possible to define a closely related map that gives rise to a non-degenerate pairing on the K-theory. My aim is to define such a map using just the combinatorics of the fan of a larger class of toric varieties in order to obtain a combinatorial model for the K-theory of such toric varieties. In analogy to the approach of Dustin Ross, which relates the degree map on the Chow ring of a toric variety given by a tropical fan with the volume of an associated polyhedral complex, I would like to express this Euler characteristic in terms of some lattice count.

• Kyle Huang (BTU Cottbus): Generating Smooth 3-Polytopes
  Smooth lattice polytopes are an important class of lattice polytopes in combinatorial algebraic geometry, corresponding to projective embeddings of smooth toric varieties via complete linear series. For fixed integers d and n, there are only finitely many smooth lattice d-polytopes with <= n lattice points. We describe and implement a novel algorithm to classify smooth 3-polytopes, extending previous classifications to smooth 3-polytopes with <= 50 lattice points. This is ongoing joint work with Christian Haase.