Tropical geometry studies a piecewise linear, combinatorial shadow of degenerations of algebraic varieties. In many cases, usual algebro-geometric objects such as divisors or line bundles on curves have tropical analogues that are closely tied to their classical counterparts. In this talk, I will present an elementary theory of tropical principal bundles on metric graphs, generalizing the case of tropical line bundles to bundles with arbitrary reductive structure group. Our approach is based on tropical matrix groups arising from the root datum of the corresponding reductive group. Building on Fratila’s description of the moduli space of semistable principal bundles on an elliptic curve, we describe a tropicalization procedure for semistable principal bundles on a Tate curve. This is joint work with Andreas Gross, Martin Ulirsch and Dmitry Zakharov.
Hilbert's 16th problem asks for the topological classification of real
plane algebraic curves. Viro's patchworking theorem provides a purely
combinatorial route to this: a triangulation together with a sign
distribution on lattice points determines the topology of the resulting
curve.
We present an efficient algorithm for extracting this topology, enabling
exhaustive enumeration over billions of sign distributions and
triangulations. Using this, we show that all 121 real schemes of degree
seven are realizable as patchworked curves, settling a question of
Itenberg and Viro from 1996. In degree eight, we prove that patchworking
is no longer sufficient.
This based on joint work with Zoe Geiselmann, Michael Joswig, Lars
Kastner, Konrad Mundinger, Sebastian Pokutta, Christoph Spiegel and Max
Zimmer.
The Demazure product on integer permutations is an associative operation that can be understood as a greedy multiplication in Bruhat order. In this talk, I will describe some surprising uses of this combinatorial construction for degeneration arguments in Brill--Noether theory. Roughly speaking, the Demazure product provides a mild generalization of the compatibility conditions for limit linear series. I will describe how this point of view unifies some arguments of classical Brill--Noether theory with the newer Hurwitz--Brill--Noether theory, and discuss some open problems.
A locally symmetric variety is a non-compact complex algebraic variety obtained as the quotient of a Hermitian symmetric domain by the action of an arithmetic group. I will start by reviewing the theory of toroidal compactifications of these varieties, originally due to Ash-Mumford-Rapoport-Tai. Building on this construction, we define the tropicalization of a locally symmetric variety to be a combinatorial (polyhedral) object encoding the boundary strata of a toroidal compactification of the variety. I will discuss applications of this theory to the cohomology of moduli spaces and arithmetic groups, with an emphasis on the case of moduli of abelian varieties and general linear groups. Based on joint work with Assaf, Brandt, Bruce, and Chan.
The Gale transform is an intriguing map between spaces of point configurations in projective spaces. A point configuration is called self-dual if its Gale transform is equivalent to the configuration itself. In this talk, I report recent work in progress with Alessio Caminata and Luca Schaffler on algebraic/geometric properties of the space of self-dual configurations. We provide set-theoretic defining equations and show that its normalization is almost Gorenstein.
