Abstract: Wilson Loop Diagrams are the Feynman diagrams of SYM N=4 theory. Rather than studying the amplitude as a whole, in this talk, I present the appearance of positive Grassmannians in these calculations, and show how they simplify the combinatorics of the integrals. I also present an interesting class of Wilson loop diagrams with curious integrals.

Abstract: In his seminar work, Baxter established that the spectrum of the 6 (and 8)-vertex model can be described in terms of polynomials and of the Baxter's QT-relation. Frenkel-Reshetikhin conjectured that the there is an analog form for the spectrum of more general quantum integrable systems.

Our recent proof (with E. Frenkel) of this conjecture for arbitrary untwisted affine types is based on the study of the "prefundamental representations". We had previous constructed these prefundamental representations with M. Jimbo as asymptotical limits of the Kirillov-Reshetikhin modules over the quantum affine algebra. One of the crucial point for our proof of the conjecture is to establish generalized Baxter's as relations in the Grothendieck ring of a category O containing these prefundamental representations.

In a joint work  with B. Leclerc, we use this category O and such asymptotical representations to obtain new monoidal categorification of (infinite rank) cluster algebras. In particular, the celebrated Baxter relations for the 6-vertex model get interpreted as Fomin-Zelevinsky mutation relations.

This project is supported by the European Research Council under the European Union's Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.

Abstract: There is a natural torus action on the quantum grassmannian (a deformation of the homogeneous coordinate ring of the grassmannian). Remarkably, the invariant prime ideals under this action are parameterised by exactly the same combinatorial objects as the cells in the totally nonnegative grassmannian -- Cauchon-Le diagrams. The talk will survey known and conjectured results on this invariant prime spectrum and the corresponding results for the algebra of quantum matrices. This is joint work (or work in progress) at various times with: Karel Casteels, Ken Goodearl, Ann Kelly, Stephane Launois, Laurent Rigal, and Ewan Russell.

Abstract: Joint work with J.Scott.

By a result of J. Scott, the homogeneous coordinate ring of the Grassmannian Gr(k,n) can be realised as a cluster algebra.

We introduce a twist map on Gr(k,n). We show that it is related to a twist of Berenstein-Fomin-Zelevinsky and can be implemented by a maximal green sequence, up to frozen variables. The Pluecker coordinates of the Grassmannian are all cluster variables. We give Laurent expansions for twists of Pluecker coordinates as scaled dimer partition functions (matching polynomials) on weighted versions the plabic (planar bicoloured) graphs arising in the cluster structure.

Abstract: The twistor action for N=4 super Yang-Mills can be used to define a diagram formalism for Wilson-loops that are dual to amplitudes and other correlators. We show how these diagrams naturally relate to grassmannians and tile  a corralahedron by analogy with the original amplituhedron for the Wilson loop.  We use the framework to find invariant forms for low lying stress-energy correlators. This covers some joint work with Susama Agarwala, Burkhard Eden and Paul Heslop.

Abstract: Frieze patterns of numbers are combinatorial objects introduced by Coxeter in the early 70's. The recent revival of friezes is due to connections with the theory of cluster algebras. After a short and elementary introduction to Coxeter's friezes and their generalizations, I will explain how the spaces of friezes are identified with the moduli spaces of points in projective spaces. In addition, these two spaces are also isomorphic to certain spaces of periodic linear difference equations. This “triality” allows us to combine analytic, geometric and combinatorial approaches to study the spaces, and convert information from one to another. We will illustrate this principle with the Gale transform. [joint work with Ovsienko, Schwartz, Tabachnikov]

Abstract: I will describe a supersymmetric analog of the Coxeter frieze patterns, related with linear difference operators generalizing the classical Hill's operators. (This part of the talk is a joint work with S. Morier-Genoud and S. Tabachnikov.) Using superfriezes as a starting example, I will then present an attempt to develop the notion of cluster superalgebra.

Abstract: Among the many recent developments in the study of scattering amplitudes in maximally supersymmetric Yang-Mills theory is its dual formulation in terms of an integral over the Grassmannian Gr(k,n) -- the space of k-planes in n dimensions. This formulation is tied to the concept of on-shell diagrams, bipartite graphs which relate field theory with combinatorics and graph theory.

The planar limit (SU(N) gauge group with N large) is well understood and the Grassmannian description simplifies to a much simpler subset of the whole space - the Positive Grassmannian.

In this talk I will discuss which of the above features are still present when moving beyond the planar limit and what new aspects arise.

Abstract: In my first lecture, I'll give an introduction to the totally non-negative Grassmannian, including the combinatorics of its cell decomposition.  In my second lecture I'll give an introduction to the KP equation, and explain how points in the real Grassmannian give rise to soliton solutions to the KP equation.  The remaining lectures will describe our joint work with Kodama, which shows how the rich combinatorial structure of the non-negative Grassmannian can be used to obtain very precise information about the corresponding solutions of the KP equation.