Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS167, part 3: Computational tropical geometry
Time:
Friday, 12/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F013
53 seats, 74m^2

Presentations
3:00pm - 5:00pm

Computational tropical geometry

Chair(s): Kalina Mincheva (Yale University), Yue Ren (Max Planck Institute for Mathematics in the Sciences, Germany)

This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Tropicalized quartics and curves of genus 3

Marvin Hahn1, Hannah Markwig2, Yue Ren3, Ilya Tyomkin4
1Goethe Universität Frankfurt, 2Eberhard Karls Universität Tübingen, 3Max Planck Institute for Mathematics in the Sciences, Germany, 4Ben Gurion University

Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus 3 can be realized as a tropicalization of a quartic in the euclidean space. We focus on the interior of the maximal cones in the moduli space and study all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropical plane and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that any realizably hyperelliptic curve cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems, and recent advances in the realizability of sections of the tropical canonical divisor.

 

Tropical Jucys Covers and refined quasimodularity

Marvin Hahn1, Felix Leid2, Danilo Lewanski3, Jan-Willem van Ittersum4
1Goethe Universität Frankfurt, 2Universität des Saarlandes, 3Max Planck Institute for Mathematics, 4universiteit utrecht

Hurwitz numbers count genus $g$, degree $d$ covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d'enfant count. We study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields a new interpretation in terms of tropical covers involving local multiplicities given by Gromov-Witten invariants. We further discuss applications of this new interpretation with regards to quasimodularity results and wall-crossing formulae.

 

Tropical lines on tropical surfaces

Michael Joswig1, Marta Panizzut1, Bernd Sturmfels2, Magnus Dehli Vigeland3
1Technische Universität Berlin, 2Max Planck Institute for Mathematics in the Sciences, UC Berkeley, 3University of Oslo

In 1849, Arthur Cayley and George Salmon proved that every smooth cubic surface in P3 contains exactly 27 lines. Since the early development of tropical geometry, two natural problems were to understand whether the same statement holds for smooth tropical cubic surfacss and to classify combinatorial positions of their tropical lines. The answer to the first turned out to be false, as smooth tropical surfaces might contain families of tropical lines. Moreover, classifying positions of tropical lines reveals some computational challenges due to the massive number of combinatorial types of smooth tropical cubic surfaces.

In this talk we will tell this tropical story. We will introduce motifs of tropical lines on tropical surfaces and study their configurations. Throughout the talk we will highlight the computational aspects.

 

Polyhedral tropical geometry of higher rank

Marcel Celaya, Josephine Yu
Georgia Tech

In a recent paper, Jell, Scheiderer, and Yu define a notion of real tropicalization for semialgebraic sets. In this talk I will discuss what happens when the semialgebraic set is a linear subspace. In this setting, the real tropicalization is a polyhedral fan that is best understood using the theory of oriented matroids. The main focus of this talk will be on understanding the topological and combinatorial properties of this fan.