Learning Seminar in Algebraic Combinatorics @ University of Michigan

We will look at a mathematical formulation of the theory of scattering amplitudes from physics. We will see how combinatorial objects like the Associahedron naturally come up in this setting. Building on this connection, we will see how these amplitudes are related to the totally positive tropical Grassmannian and related cluster algebras.

Tropical geometry has found a newfound role in quantum field theory, in studying Feynman integrals, scattering amplitudes, and asymptotic behaviour. I'll give a gentle introduction to these applications, and as a bonus topic introduce a link between Bergman fans and the so-called 'wavefunction of the universe'.

This is a follow-up on Jayden's talk. Tropical Grassmannians are polyhedral fans and we introduce how to construct a polyhedral complex that captures their combinatorics. In the case k=2 (Grassmannian of lines), we establish an isomorphism between this polyhedral complex and the simplicial complex of phylogenetic trees. We also establish a one-to-one correspondence between the tropical Grassmannian and a subset of the Dressian given by tropical linear spaces of "realizable" valuated matroids.

Last week we introduced cluster algebra with principal coefficients, and F-polynomials. We will pick up from the definition of g-vectors, and talk about several different works on how to construct g-vector fans. In Jahn-Löwe-Stump, the authors constructed the g-vector fan by common refinement of normal fans of Newton polytopes of F-polynomials. In Holhweg-Lange-Thomas, they constructed the generalized associahedron (whose normal fan is the g-vector fan) by deleting inequalities of certain facets. In Bazier-Matte et al., the authors constructed the generalized associahedron by extending the construction of the kinematic associahedra in ABHY to all simply laced Dynkin quivers. In this talk, we will try to give a brief survey of each construction, illustrating them with a type A example.

Cluster Algebra is a commutative ring with a special finite set of elements (clusters) that can produce all its generators in an algorithmic way. Cluster complex is an abstract simplicial complex encoding how different clusters are related to each other. Finding polyhedral realizations of these complexes has been a problem of long-standing interest. In a previous talk, Mia sketched an example from Speyer-Williams illustrating that the fan associated to Trop^+(Gr2,n) is isomorphic to the normal fan of an associahedron. More generally, Speyer-Williams conjectured that if A is a cluster algebra of finite type, then the fan of Trop^+(SpecA) modulo its lineality space should be isomorphic to the fan which is the cone over the cluster complex. In the acyclic case, Jahn-Löwe-Stump proved that Trop^+(SpecA)/L is linearly isomorphic to the g-vector fan, which is combinatorially isomorphic to the cluster complex. To unwrap their statement, I will give an introductory talk on Cluster Algebra, Generalized Associahedron (polyhedral realization of cluster complex), and g-vector fans.

In a previous lecture, we learnt about tropical linear spaces in terms of valuated matroids, or tropical Plucker vectors. In this talk, I'll give a quick overview of some other ways to think about tropical linear spaces: matroid decompositions of the hypersimplex, Chow quotients of the Grassmannian, and membranes. I'll focus on the last of these. A result of Keel and Tevelev states that tropical linear spaces can be identified with membranes, certain subcomplexes of a (Bruhat-Tits) affine building.

Building off of last week's talk, we will discuss the tropical totally positive Grassmannian, Trop+(Gr(k,n)). First introduced by Speyer and Williams in 2003, the tropical totally positive Grassmannian came out of work studying the totally positive part of the tropicalization of an affine variety. After briefly looking at those results, we'll focus our attention on the fan structure of the tropical totally positive Grassmannian. If there's time, we'll show that the fan associated to Trop+(Gr(2,n)) is combinatorially the fan dual to the type A_{n-3} associahedra.

Matroids are hard. Valuated matroids generalize matroids. Tropical linear spaces are equivalent to valuated matroids. Therefore, tropical linear spaces are hard, so are their parameter spaces: the Dressians, the tropical analog of the Grassmannians. Inside the Dressian there is a nicer subset, called the tropical Grassmannian, that parametrizes the tropicalized linear spaces. I will familiarize the audience with these concepts with a lot of examples.

We introduce the notion of a balanced weighted polyhedral complex, and prove the structure theorem for tropical varieties. This says that the tropicalization of an irreducible d-dimensional variety is a balanced weighted rational polyhedral complex which is pure of dimension d and connected through codimension 1.

In this talk, we introduce the tropical hypersurface associated to a multivariate Laurent polynomial, as well as more general tropical varieties arising from classical subvarieties of the torus. We will discuss Kapranov's theorem and present the generalized Fundamental Theorem of Tropical Algebraic Geometry, both of which unite classical algebro-geometric objects to their tropical counterparts.

In this talk I will introduce Gröbner basis in a field with valuation and the construction of Gröbner complexes, which will be the ambient space for tropical varieties. Relevant background on valuations and polyhedral geometry will also be covered in the talk.

I will give an introduction to the subject and go over some basics of tropical geometry, following chapter one of Maclagan-Sturmfels' book "Introduction to Tropical Geometry".