ICTS

Tropical curves in $\RR^2$ correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs that cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most six. We also define a special family of lattice polytopes namely panoptigons and enumerate all possible panoptigons and show how they can be used to find a forbidden pattern in tropical planar curves.

This talk is based on work in Tewari(2022) and joint work with Michael Joswig (2020) and Ralph Morrison (2020).

In biochemical reactions, the interactions between species are represented by directed graphs and their dynamics over time is modelled with ordinary differential equations. Under certain assumptions (mass-action) the ODEs are parametric polynomials, the problem of understanding steady states of these systems is equivalent to understanding the solutions of the parametric polynomials. I will lay down the basic groundwork for this formalism and discuss some of the methods that are used to explore the parameter space which ensures multiple positive steady states of the dynamics. The main focus will be on the family of phosphorylation networks.

This is based on joint works with Elisenda Feliu, Timo de Wolff, Oguzhan Yürük.

The intersection theory of tropical linear spaces, under the name of the matroid Chow ring, has recently been in the spotlight for its involvement in matroid Hodge theory.  Berget, Eur, Spink and Tseng have developed tools to easily pass between tropical intersection (Chow) and polytope algebra (K-theory) computations in this setting, including introducing some tautological vector bundles for matroids.  I'll present this as well as ongoing joint work of mine with Eur, Larson and Spink which extends this to delta-matroids, their Coxeter type BC counterpart.

We study large deviations of discrete concave functions on a triangular lattice, termed hives by Knutson and Tao, and relate this to the “surface tension” of continuum versions of the same. We also prove a large deviations Principle for sums of random Hermitian matrices, through a relation with hives.

This is joint work with Scott Sheffield.

We explore several classical problems in combinatorial optimization and how they can benefit from being viewed through the lens of tropical geometry.  The includes parameterized versions of the shortest-path problem, applications to linear programming and phylogenetics.

A polynomial with real coefficients is called nonnegative if it takes only nonnegative values. For example, any sum of squares of polynomials is obviously nonnegative. The study of the relationship between nonnegative polynomials and sums of squares is a classical area in real algebraic geometry. The lectures will be about the convex cones of nonnegative polynomials and sums of squares on a variety. Convex-geometric considerations will lead to new insights in algebraic geometry. The main questions we will consider are: when are all nonnegative polynomials sums of squares, and the number of squares needed to write a sum of squares. I will also introduce applications in matrix completion and optimization.

Gelfand-Kapranov-Zelevinski introduced the notion of secondary fan in the study of the Newton polytopes of discriminants and resultants. It also controls the geometric invariant theory for toric varieties. We propose a generalization of the GKZ secondary fan to general Fano varieties using ideas from Berkovich geometry and Mori theory. Furthermore, inspired by mirror symmetry, we propose a synthetic construction of a universal family of Kollár-Shepherd-Barron-Alexeev stable pairs over the toric variety associated to the generalized secondary fan. This generalizes the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. We gave a detailed construction and proved the stability in the case of del Pezzo surfaces. This is joint work with Hacking and Keel.

We explore several classical problems in combinatorial optimization and how they can benefit from being viewed through the lens of tropical geometry.  The includes parameterized versions of the shortest-path problem, applications to linear programming and phylogenetics.

A polynomial with real coefficients is called nonnegative if it takes only nonnegative values. For example, any sum of squares of polynomials is obviously nonnegative. The study of the relationship between nonnegative polynomials and sums of squares is a classical area in real algebraic geometry. The lectures will be about the convex cones of nonnegative polynomials and sums of squares on a variety. Convex-geometric considerations will lead to new insights in algebraic geometry. The main questions we will consider are: when are all nonnegative polynomials sums of squares, and the number of squares needed to write a sum of squares. I will also introduce applications in matrix completion and optimization.

Associated to a convex integral polygon $N$ in the plane are two integrable systems:
(1) The cluster integrable system of Goncharov and Kenyon constructed from the planar dimer model.
(2) The Beauville integrable system, associated with the toric surface of $N$.
We show that these integrable systems are isomorphic. Based on joint work with Alexander Goncharov and Rick Kenyon, and Giovanni Inchiostro.

We explore several classical problems in combinatorial optimization and how they can benefit from being viewed through the lens of tropical geometry.  The includes parameterized versions of the shortest-path problem, applications to linear programming and phylogenetics.

A polynomial with real coefficients is called nonnegative if it takes only nonnegative values. For example, any sum of squares of polynomials is obviously nonnegative. The study of the relationship between nonnegative polynomials and sums of squares is a classical area in real algebraic geometry. The lectures will be about the convex cones of nonnegative polynomials and sums of squares on a variety. Convex-geometric considerations will lead to new insights in algebraic geometry. The main questions we will consider are: when are all nonnegative polynomials sums of squares, and the number of squares needed to write a sum of squares. I will also introduce applications in matrix completion and optimization.

The geometry of various moduli spaces of bundles on a compact Riemann surface is a pillar of modern geometry, since they are at the sweet spot of being at same time very intricate but also accessible to a manifold of different techniques. Recently the perspective that compact metric graphs are, in a way, a natural combinatorial, or ""tropical"", analogue of compact Riemann surfaces has gained significant traction. This is, in particular, due to its numerous applications in the context of enumerative geometry, the cohomology of moduli spaces, and Brill-Noether theory.

A tropical analogue of line bundles on metric graphs is, by now, well-understood and reflects the various compactifications of the Jacobian over semistable degenerations of compact Riemann surfaces. The goal of this talk is to propose an up-to-now still missing analogue of vector bundles of higher rank on metric graphs. After defining these objects I will, in particular, talk about a tropical analogue of the Weil-Riemann-Roch-Theorem and of the Narasimhan-Seshadri correspondence. Then I will outline a tropicalization procedure that lets us connect this a priori only combinatorial theory with the classical story. As it turns out, this will work best in the case of the Tate curve.

Given time, I might indulge in some speculations concerning a new approach to degenerations of vector bundles using methods from logarithmic geometry that incorporates and expands on both the algebro-geometric and the tropical story.

The non-speculative aspects of this talk are based on joint work with Margarida Melo, Sam Molcho, and Filippo Viviani as well as with Andreas Gross and Dmitry Zakharov.

Per_n is an affine algebraic curve, defined over Q, parametrizing (up to change of coordinates) degree-2 self-morphisms of P^1 with an n-periodic ramification point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-2 self-morphisms of C with an n-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is Per_n is irreducible over C? (2) Is G_n is irreducible over Q? 

We show that if G_n is irreducible over Q, then Per_n is irreducible over C. In order to do this, we find a Q-rational smooth point of a projective completion of Per_n. This Q-rational smooth point represents a special degeneration of degree-2 morphisms, and as such admits a tropical interpretation.