The totally nonnegative part of the Grassmannian Gr(k,n) is the set of k-dimensional subspaces of Rⁿ whose Plücker coordinates are all nonnegative. The amplituhedron is the image in Gr(k,k+m) of the totally nonnegative part of Gr(k,n), through a (k+m) x n matrix with positive maximal minors. It was introduced in 2013 by Arkani-Hamed and Trnka in their study of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. Taking an orthogonal point of view, we give a description of the amplituhedron in terms of sign variation. We then use this perspective to study the case m=1, giving a cell decomposition of the m=1 amplituhedron and showing that we can identify it with the complex of bounded faces of a cyclic hyperplane arrangement. It follows that the m=1 amplituhedron is homeomorphic to a ball. This is joint work with Lauren Williams.

In this talk, I will discuss the combinatorics of the affine Grassmannian, the affine flag variety and the central degeneration, with an emphasis on some convex polytopes involved. The affine Grassmannian and affine flag variety are the infinite-dimensional analogs of the ordinary Grassmannian and flag variety. The central degeneration is the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves. Our convex polytopes are the moment polytopes of some schemes, and turn out to be special cases of the alcoved polytopes studied by Lam and Postnikov.

I will present two structures of combinatorial interest:

The first structure is a partial sorting algorithm on permutations of size n studied by Gwen McKinley in her undergraduate thesis. Take the first entry in the one-line notation of the permutation and move it into the place of its value. This gives rise to a forest structure, which we call fixed point forest, with derangements as leaves. Despite its simple description it exhibits a rich structure. In joint work with Toby Johnson and Erik Slivken (arXiv:1605.09777) we analyze the local structure of the tree at a random permutation in the limit as n goes to infinity.

The second is a Markov chain on semaphore codes that appeared in joint work with Pedro Silva and John Rhodes (arXiv:1509.03383). A semaphore code is a suffix code with a right action. We compute the stationary distribution and hitting time to reset explicitly.

In the early 70's, Vapnik and Chervonenkis defined the combinatorial condition now called finite VC-dimension and showed that a family of finite VC-dimension of subsets of a measure space satisfies a uniform law of large numbers. Families of finite VC-dimension have since been studied in combinatorics and in model theory (amongst others). The last 10 years have seen some interaction in both directions between those two areas. I will define a strengthening of this condition called distality which was introduced in model theory to capture combinatorial properties of real geometry. I will explain how this turns out to be a useful notion to generalize the so-called strong Erdős-Hajnal property for graphs with semi-algebraic edge relation, proved by Alon, Pach, Pinchasi, Radoicic and Sharir. I will try to show the advantages and drawbacks of the model-theoretic approach. (This is based on works of Chernikov-Starchenko and myself.)

The positive Grassmannian is a nice geometrical object with a rich combinatorial structure. It was linked with many areas of mathematics and physics, including cluster algebras, statistical physics, and scattering amplitudes. We'll discuss combinatorics of the positive Grassmannian from the point of view of convex geometry. We'll talk about a joint work with Miriam Farber about arrangements of equal minors on the positive Grassmannian. In one case, they are related to plabic graphs, weakly separated sets, etc., and, in another case, they are related to alcoved polytopes, triangulations of hypersimplices, etc. We'll also talk about a joint project with Thomas Lam on polypositroids and membranes.

Generalized permutahedra are a beautiful family of polytopes which are known to have a rich combinatorial structure. We explore the Hopf algebraic structure of this family, and use it to unify old results, prove new results, and answer open questions about families of interest such as graphs, matroids, posets, trees, set partitions, hypergraphs, and simplicial complexes. As a corollary, we obtain a new polyhedral approach for computing the multiplicative and compositional inverses of a power series.

The talk will be based on joint work with Marcelo Aguiar, and will assume no previous knowledge of Hopf algebras or generalized permutahedra.

(Please note the unusual time and place of this talk.)

Permutation tableaux, a subset of Postnikov's Le-diagrams, are 0-1 fillings of Ferrers diagrams with a 1 in each column, and a 1 on every square that has a 1 above it and to its left. They are in one-to-one correspondence with permutations whose weak excedances correspond to rows, and they were shown by Corteel and Williams to have a strong connection to the steady state distribution of the Partially Asymmetric Exclusion Process, a one-dimensional model in statistical mechanics.

Earlier, in 2004, Ehrenborg and van Willigenburg introduced another set of fillings of Ferrers diagrams, which are in one-to-one correspondence with certain bipartite graphs. These EW-tableaux have a unique fixed (but arbitrary) row of all 1s (corresponding to a unique sink in an acyclic orientation of the corresponding graph), and no occurrence of X-patterns, that is, no four corners of a rectangle in a tableau are allowed to have 0s in one pair of diagonally opposite corners and 1s in the other two.

Ehrenborg and van Willigenburg showed that these tableaux are in one-to-one correspondence with permutations, where rows correspond to strict excedances, but their proof was not bijective. We present a bijective proof, which also associates fixed points in a permutation to certain columns of the corresponding tableaux. This bijection also applies to a variation of these tableaux, where excedances manifest themselves in a slightly different way. The bijection leads to (poorly understood) bijections between these three kinds of tableaux.

This is joint work (in progress) with Thomas Selig.

The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank d has the homotopy type of a sphere of dimension d-1. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space. The talk will focus on the connection of this problem with polytopes and tilings of polytopes; no knowledge of matroids or oriented matroids is required.

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. Based on joint work with Laura Escobar.

This talk is about joint work with Ilke Canakci. I will present a combinatorial realization of continued fractions in terms of perfect matchings of the so-called snake graphs, which are planar graphs that have first appeared in expansion formulas for the cluster variables in cluster algebras from triangulated marked surfaces. This realization of continued fractions gives rise to applications to cluster algebras, as well as elementary number theory.