Title: Connecting tropical intersection theory with polytope algebra in types A and B
Abstract: 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.
Title: From Gentle algebras to S-matrix in Quantum Field Theory
Ayush Kumar Tewari
Title: Forbidden patterns in tropical planar curves
Abstract: 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).
Title: Quantum geometry of matroids
Abstract: Gromov-Witten theory probes the geometry of manifolds by studying the space of algebraic curves in that manifold. In the last decade, it has emerged that matroids, which are combinatorial abstractions of a certain class of manifolds, possess many of the structures that one typically attaches to manifolds. This is often true even when there is no direct connection to geometry. I will discuss a new direction in this interplay between matroids and geometry, by defining and examining the Gromov-Witten theory of matroids. The talk is based on joint work with Jeremy Usatine (Brown).
Title: Update on tropical schemes
Abstract: A tropical subscheme of a tropical toric variety is given by an ideal in a semiring of tropical polynomials (the tropicalization of the Cox ring) that satisfies certain matroidal conditions. These ideals generalize the tropicalization of ideals of subschemes of toric varieties, and share many of their properties. I will introduce these concepts, and give an update on the status of this program: The varieties of tropical ideals satisfy the balancing condition, and a version of the strong Nullstellensatz. This is joint work with Felipe Rincon.
Title: Algebraic and Convex Geometry of Sums of Squares on Varieties
Abstract: 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.
Title: Bitangents to plane quartics - tropical, real and arithmetic count
Abstract: A plane quartic defined over the complex numbers has 28 bitangent lines. A real quartic may have 4, 8, 16 or 28 bitangent lines. Tropical quartics have 7 bitangent classes. We study their lifting behavious over various fields, and also discuss arithmetic counts which can be viewed as a way to simultaneously count over any field.
Joint work in progress with Sam Payne and Kris Shaw.
Title: Large deviations for random hives and the spectrum of the sum of two random matrices
Abstract: 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.
Title: Tropical Quantum Field Theory, Mirror Polyvector Fields and Multiplicities of Tropical Curves
Abstract: We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably, these structures include a tropical quantum field theory and an L∞-structure. The latter is an instance of Getzler's gravity algebra, and the l2-bracket is a restriction of the Schouten-Nijenhuis bracket. I will explain its relationship to string topology. The work is joint with Travis Mandel.
Jacob P. Matherne
Title: Singular Hodge theory for combinatorial geometries
Abstract: Here are two problems about hyperplane arrangements (or, more generally, matroids).
Problem 1: If you take a collection of planes in R^3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the "Top-Heavy Conjecture", that Dowling and Wilson conjectured in 1974.
Problem 2: Given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it. These polynomials should have nonnegative coefficients.
Both of these problems were formulated for all matroids, and in the case of hyperplane arrangements they are controlled by the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. For arbitrary matroids, no such variety exists; nonetheless, I will discuss a solution to both problems for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.
Title: A continuous associahedron of type A
Abstract: Generalized associahedra appear throughout mathematics; in particular, they encode much of the combinatorics of cluster algebras of finite type. In this talk I will give a construction of a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. This continuous associahedron is convex and is related to the theory of cluster algebras. It manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. We will also see that the associahedron shares important properties with the generalized associahedron of type A.
Title: Vector bundles on Riemann surfaces and metric graphs
Abstract: 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.
Title: Optimization and Tropical Combinatorics
Abstract: 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.
Title: Circular Fence Posets and associated polytopes with unexpected symmetry
Abstract: Circular fence posets are a natural class of posets that arise in enumerative combinatorics, cluster algebras and quiver representations. In recent work joint with Ezgi Kantarci Oguz, we showed that the order ideals of circular fence posets admit a surprising and non-trivial symmetry.
In this talk, I will talk about a natural class of polytopes (that we for obscure reasons have named "Hector-Louis polytopes") associated to circular fence posets, which naturally lead to infinite families of pairs of polytopes which are non-isomorphic but which have the same Ehrhart polynomial.
A famous example of such a phenomenon is Stanley’s theorem which says that two natural (non-isomorphic) polytopes associated to a poset, the so called Order and Chain polytopes have the same Ehrhart polynomial. Stanley proved this by exhibiting unimodular triangulations of both polytopes and constructing a piecewise unimodular linear map between the two.
In our case however, the equality of the Ehrhart polynomial seems to be of a different nature. Our proof proceeds by first coming up with a combinatorial reinterpretation of this fact and proving this using linear algebraic techniques.
Subsequent to describing this, I will discuss the connection between these polytopes and the Alcoved polytopes of Postnikov and Lam and conjecture some possible extensions.
This is joint work with Ezgi Kantarci Oguz at Galatasaray University and Cem Yalim Ozer, a graduate student at Bogazici University.
Title: Vertex gluings and Demazure products
Abstract: Finite graphs and metric graphs provide useful combinatorial analogs of algebraic curves. Since the work of Cools-Draisma-Payne Robeva, it has been known that chains of loops are particularly useful graphs in this regard. I will describe a new perspective on these chains of loops, by describing a version of Brill-Noether theory for curves or graphs with two marked points. Divisors on twice-marked graphs are associated with permutations; the extent to which a divisor is special is measured by the inversions of the permutation. This twice-marked Brill-Noether theory is well-suited to inductive arguments; when two twice-marked graphs are glued together, permutations are combined by an operation called the Demazure product. I will describe how this framework provides a short proof of some of the results of Cools-Draisma-Payne Robeva, as well as more recent results on tropical Hurwitz-Brill-Noether theory, and may provide a route to identifying new classes of Brill-Noether-General graphs.
Title: Newton Polytopes and parameter estimation in reaction networks.
Abstract: 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.
Title: Shrinking dynamics on tropical series
Abstract: We discuss tropical series, certain dynamics on them, and several open questions. Roughly speaking, a tropical series is a tropical polynomial with infinite number of monomials. Motivated by a sandpile model, we introduce so-called shrinking operator S_p on tropical series (p is any point, f is a series), such that S_pf passes through p. Numerical evidence suggests that when points p_i are chosen uniformly randomly in a unit square, S_p_1S_p_2…S_p_n 0 converges to a square grid (open question). Also, S_p lifts to an operator on usual series, but only in characterisic two.
Title: Geometry of tropical varieties with a view toward applications
Abstract: These lectures provide an introduction to tropical geometry and the interactions between combinatorics and algebraic geometry, with focus on combinatorial aspects of Hodge theory.
Title: The tropical limit of string theory and Feynman integrals
Abstract: Besides its goal as a theory of quantum gravity, from the quantum field theory (QFT) standpoint, string theory is a remarkable mathematical tool that allows to compute certain quantities in QFT (that is, roughly, particle’s physics) more efficiently than using standard QFT approach.
In this talk, I will describe how scattering amplitudes in QFT, which are quantities that characterise the interactions between particles, are recovered using the so called tropical limit of string theory scattering amplitudes. The idea, which has been known since the onset of string theory, is that when strings become point-like, they reproduce particle’s dynamics. In https://arxiv.org/abs/1309.3551, I have shown how that limit is actually described by a tropical limit, which allowed me to derive some practical results on scattering amplitudes at higher order and reformulate the sum over Feynman diagrams as an integral over the tropical moduli space of weighted graphs of https://arxiv.org/abs/0907.3324.
I will try to convey the essential ideas without assuming knowledge in quantum field theory or string theory, and conclude with open questions and perspectives.
Title: Complex dynamics: degenerations, and irreducibility problems
Abstract: 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.
Title: Dimers and Beauville integrable systems.
Abstract: 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.
Tony Yue YU
Title: Generalizing GKZ secondary fan using Berkovich geometry
Abstract: 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.