TGiF-Tropical Geometry in Frankfurt
5th July 2019 - Second meeting in Summer Semester
13:15-14:15 Madeline Brandt (University of California at Berkeley)
Title: Matroids and their Dressians
Abstract: In this talk we will explore Dressians of matroids. Dressians have many lives: they parametrize tropical linear spaces, their points induce regular matroid subdivisions of the matroid polytope, they parametrize valuations of a given matroid, and they are a tropical prevariety formed from certain Plücker equations. We show that initial matroids correspond to cells in regular matroid subdivisions of matroid polytopes, and we characterize matroids that do not admit any proper matroid subdivisions. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. If time permits, we will also discuss an ongoing project extending these ideas to flag matroids.
14:45-15:45 Dhruv Ranganathan (University of Cambridge)
Tropical curves, stable maps, and singularities in genus one
In the early days of tropical geometry, Speyer identified an extremely subtle combinatorial condition that distinguished tropical elliptic space curves from arbitrary balanced genus one graphs. Just before this, Vakil and Zinger gave a very explicit desingularization of the moduli space of elliptic curves in projective space, with remarkable applications. Just after this, Smyth constructed new compactifications of moduli spaces of pointed elliptic curves, using worse-than-nodal singularities, as part of the Hasset-Keel program. A decade on, we understand these three results as part of a single story involving logarithmic structures and their tropicalizations. I will discuss this picture and how the unified framework extends all three results. This is joint work with Keli Santos-Parker and Jonathan Wise.
16:00-17:00 Yoav Len (Georgia Institute of Technology)
Algebraic and Tropical Prym varieties
My talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian vari- eties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concern- ing Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results con- cerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci. This is joint work with Martin Ulirsch.
7th June 2019 - First meeting in Summer Semester
Margarida Melo (Università degli studi Roma Tre):
Combinatorics and moduli of line bundles on stable curves.
The moduli space of line bundles on smooth curves of given genus, the so called universal Jacobian, has a number of different compactifications over the moduli space of stable curves. These compactificatons have very interesting combinatorial properties, which can be used to describe their geometry. In the talk I will explain different features and applications of these interesting objects, focusing on properties which have a natural tropical counterpart.
Farbod Shokrieh (University of Copenhagen):
Heights and moments of abelian varieties
We give a formula which, for a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron-Tate height of $(A,\lambda)$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a role.
(Based on joint works with Robin de Jong.)
Philipp Jell (Universität Regensburg):
The tropical Hodge conjecture for divisors
The Hodge conjecture is one of the big open questions in algebraic geometry. Mikhalkin and Zharkov formulated a tropical analogue of this conjecture. In joint work with Johannes Rau and Kristin Shaw, we established this conjecture for divisors. I will introduce the notions that are necessary to state the tropical Hodge conjecture and then sketch the proof and further directions of research.