3:00pm - 5:00pmComputational 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)
Massively parallel methods with applications in tropical geometry
Dominik Bendle1, Kathrin Bringmann2, Arne Buchholz3, Janko Boehm1, Christoph Goldner4, Hannah Markwig4, Mirko Rahn5, Yue Ren6, Benjamin Schroeter7
1Technische Universität Kaiserslautern, 2Universität Köln, 3Universität des Saarlandes, 4Eberhard Karls Universität Tübingen, 5Fraunhofer ITWM, 6Max Planck Institute for Mathematics in the Sciences, Germany, 7Binghamton University
In this talk, I will discuss the use of massively parallel methods in the context of tropical geometry. I will first address the technical framework, which combines the computer algebra system Singular with the workflow management system GPI-Space. I will then focus on the computation of tropicalizations, and algorithms to determine generating series for Gromov-Witten invariants via Feynman integrals.
Tropical Grassmannians Gr_p(3,8) and the Dressian Dr(3,8)
Dominik Bendle1, Janko Boehm1, Yue Ren2, Benjamin Schroeter3
1Technische Universität Kaiserslautern, 2Max Planck Institute for Mathematics in the Sciences, Germany, 3Binghamton University
The pointwise valuation of an algebraic variety is a polyhedral complex, the tropical variety, which carries information about the algebraic set. A class of prominent examples are tropical Grassmannians Gr_p(d,n) over fields of characteristic p which are set theoretically included in tropical prevarieties, called Dressians.
These fans have close connections to many mathematical areas, e.g., matroid theory, mathematical biology, cluster algebras and physics. More precisely, the Dressian Dr(d,n) parametrizes d-dimensional tropical linear spaces in n-dimensional space also known as valuated matroids, while the Grassmannian Gr_p(d,n) contains those that are realizable. Moreover, these moduli spaces agree for d=2, and additionally they parametrize both all phylogenetic trees, and tropical curves of genus zero.
I will introduce these fans with their natural fan structure, inherit from Gröbner bases and regular subdivisions. Moreover, I report about new theoretical results additional to Janko Böhm's presentation of massiv parallelized computations. My focus will be on the relationship between Gr_0(3,8) and Dr(3,8).
Computing unit groups of curves
Justin Chen1, Sameera Vemulapalli2, Leon Zhang1
1UC Berkeley, 2Princeton University
The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus.
Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves.
A numerical algorithm for tropical membership
Taylor Brysiewicz
Texas A&M University
In 2012, Hauenstein and Sottile proposed a numerical oracle for the Newton polytopes of a hypersurface. Drawing from ideas of Hept and Theobald, we describe how this algorithm may be used to numerically verify membership in tropical hypersurfaces.
