No abstract available.
This is a continuation of the previous talks. We will be working through the paper of Karp and Williams, to understand the Amplituhedron as the image of a matrix map from the totally positive Grassmannian. This talk will work through the basic definitions of the amplituhedron, and then walk through some examples when k=1. It will follow closely to section 1 and section 3 of the Karp and Williams paper.
Amplituhedra were introduced by Arkani-Hamed and Trnka as part of a program to provide an alternative to the classical Feynman diagram approach to scattering amplitudes, via a surprising link with Grassmannian geometry. I will attempt to provide some physics context, and then give the original definition of the amplituhedron (as a certain image of a totally non-negative Grassmannian) and a new proposed definition. This talk is based on joint work with Nima Arkani-Hamed and Jaroslav Trnka.
This talk is the third and final part of our working through the paper by Karp and Williams. We will discuss (finally) the poset structure that you can put on the elements of the m=1 amplituhedron and how they can be made into a hyperplane arrangement.
As part of or program on noncommutative laurent phenomenon, we
introduce and study noncommutative Catalan 'numbers' as Laurent
polynomials in infinitely many free variables and related theory of
noncommutative binomial coefficients. We also study their (commutative
and noncommutative) specializations, relate them with Garsia-Haiman
(q,t)-versions, and establish total positivity of the corresponding
Hankel matrices. Joint work with Arkady Berenstein (Univ. of Oregon).
In 1980, Michael Somos invented integer sequences that have later been popularized and generalized (among others) by David Gale. A certain mystery around this class of sequences is probably due to their relation with a wealth of different topics, such as: elliptic curves, continued fractions, and more recently with cluster algebras and integrable systems.
I will describe a way to extend Somos-4 and Somos-5 and more general Gale-Robinson sequences, and construct a great number of new integer sequences that also look quite mysterious. The construction is based on the notion of 'cluster superalgebra' (which can be used as a machine to produce integer sequences).
Most of the talk will be accessible to non-experts in any of the above mentioned subjects.
We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k, n), whenever k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k, n). Then we apply our results to the Grassmannians of 'finite mutation type'. We prove the n = 9 case of a conjecture of Fomin-Pylyavskyy describing the cluster combinatorics for Gr(3, n), in terms of Kuperberg’s basis of non-elliptic webs, and prove a similar result for the Grassmannian Gr(4,8).
Department of Mathematics
Michigan State University
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College of Natural Science