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Abstracts for A Tropical Day at the MPIM

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Combinatorial Nullstellensatz, fewnomials and Shub-Smale's conjecture: tropical versions

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Speaker: 
Dima Grigoriev
Affiliation: 
CNRS Lille/Bonn
Date: 
Mon, 2019-12-02 09:30 - 10:30
Location: 
MPIM Seminar Room
Parent event: 
A Tropical Day at the MPIM

Combinatorial Nullstellensatz (due to N. Alon) provides conditions in terms of the support of a polynomial when it can't vanish on a subset of an integer grid. We prove its tropical version. Moreover, we establish a sharp bound on the number of points in a grid at which a tropical polynomial can vanish (for classical polynomials it is called Schwartz-Zippel lemma). We estimate the size of universal sets at which no tropical fewnomial (with a fixed number of monomials) can vanish. This relates to Erdos problem from convex combinatorial geometry.

Tropical convex hulls of infinite sets

Posted in
Speaker: 
Sara Lamboglia
Affiliation: 
University of Frankfurt
Date: 
Mon, 2019-12-02 11:00 - 12:00
Location: 
MPIM Seminar Room
Parent event: 
A Tropical Day at the MPIM

In this talk  I will present some recent results on the interplay between tropical and classical convexity.
In particular,  I will focus on the tropical convex hull of convex sets and polyhedral complexes and
their explicit computation. This will lead to a lower bound on the degree of tropical curves. This is
joint work with Cvetelina Hill and Faye Pasley Simon.

Non-Archimedean uniformization and tropicalization: Teichmueller space and M_g

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Speaker: 
Martin Ulirsch
Affiliation: 
Goethe-Universität Frankfurt
Date: 
Mon, 2019-12-02 13:30 - 14:30
Location: 
MPIM Seminar Room
Parent event: 
A Tropical Day at the MPIM

The theory of uniformization for maximally degenerate curves over non-Archimedean curves (by Mumford) and for abelian varieties (by Raynaud) are one of the big achievements of modern arithmetic algebraic geometry. In recent years it has become clear that this story also has a tropical aspect: In fact, one may think of the construction as a two-step process: first construct a tropical uniformization, then use the combinatorial data of this tropical uniformization to build the non-Archimedean uniformization. In this talk, I will illustrate this principle in the case of curves.

Using Methods of Tropical Geometry for Model Reduction and Model Analysis of Biological and Chemical Networks

Posted in
Speaker: 
Andreas Weber
Affiliation: 
University of Bonn
Date: 
Mon, 2019-12-02 15:00 - 16:00
Location: 
MPIM Seminar Room
Parent event: 
A Tropical Day at the MPIM

Model reduction of bio-chemical networks relies on the knowledge of slow
and fast variables. We provide a geometric method, which is
based on the Newton polytopes,  to identify slow variables of a
bio-chemical network with polynomial rate functions.
The gist of the method is the notion of tropical equilibration that
provides approximate descriptions of slow invariant manifolds.
Compared to extant numerical algorithms such as the intrinsic low
dimensional manifold method, our approach is symbolic

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