Welcome to Combinatorics and Optimization

Title: Equivariant quasisymmetry

 Abstract: We introduce equivariant quasisymmetry, a version of quasisymmetry for polynomials in two sets of variables. Using this definition we define double fundamental polynomials and double forest polynomials, a quasisymmetric generalizations of the theory of double Schur and double Schubert polynomials, where the subset of noncrossing permutations play the role of $S_n$.This combinatorics is governed by the quasisymmetric flag variety, a new geometric construction which plays the role for equivariant quasisymmetry what the usual flag variety plays in the classical story.

In this talk, I will focus on the combinatorial aspect first, and with the remaining time discuss the geometrical implications.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title:Evolution of random graph orders and their dimensions

Abstract: A poset is a set X equipped with a partial order. In this talk I will briefly review the literature of different models on random orders. Then we discuss a particular model called the random graph order. The random graph order is classical model to generate a random causal set, which was introduced in physics to model and analyse the space-time universe. 

We will focus on an open problem proposed by Erdos, Kierstad and Trotter on the evolution of the dimensions of random graph orders. This problem has been studied by Albert and Frieze, and by Erdos, Kierstad and Trotter around 1990. Better bounds on the dimensions were obtained by Bollobas and Brightwell in 1997, for “non-sparse” random graph orders. We study the last piece of the puzzle, in the bipartite case, by investigating “a transition phase” that was predicted to occur in the sparse regime by Bollobas and Brightwel, and we prove a negative result to their prediction. We expect that a similar phenomenon would occur in the nonbipartite case.

This talk is based on a collaborated work with Arnav Kumar. 

Title: New perspectives on quasisymmetry via divided differences, flag varieties, etc.

 Abstract: In this talk we will introduce a new combinatorial approach to understanding quasisymmetric polynomials via a quasisymmetric analogue of "divided differences". The resulting combinatorial theory is paired with a geometric theory that closely follows the classical development of Schubert calculus and Schubert varieties. Next week Nantel Bergeron will continue with the torus-equivariant story.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,