Speaker: Donald Richards (Penn State)
Title: Hyperdeterminantal total positivity
Abstract: This talk will cover some recent joint research with K. W. Johnson.
Let m be a positive integer, and let K be real-valued function defined on 2m-dimensional Euclidean space. We define the concept of hyperdeterminantal total positivity for the kernel K, thereby generalizing the classical concept of total positivity. We construct examples of hyperdeterminantal totally positive kernels, extending in particular the fundamental classical example, K(x,y) = exp(xy), x,y ∈ ℝ, of a totally positive kernel. By applying a hyperdeterminantal Binet-Cauchy argument, a generalization of Karlin's Basic Composition Formula is derived and then applied to construct numerous additional examples of hyperdeterminantal totally positive kernels. Further generalizations of the concept of hyperdeterminantal total positivity by means of the theory of finite reflection groups are described, and some open problems are posed.
Speaker: Alexander Glazman (University of Innsbruck)
Title: Random-cluster model on $\mathbb{Z}^2$ at the transition point
Abstract: The random-cluster model is defined on subgraphs of $\mathbb{Z}^2$ and has two parameters: cluster-weight $q>0$ and edge-probability $0<p<1$. It is classical that, for each $q\geq 1$, the model undergoes a percolation phase transition when $p=p_c(q)$. Beffara and Duminil-Copin in 2010 computed $p_c(q)$, and later works established the type of the phase transition: it is continuous when $1 \leq q \leq 4$ and discontinuous when $q>4$. The former is characterised by Russo-Seymour-Welsh estimates, while the latter asserts non-uniqueness of the infinite-volume DLR/Gibbs measure.
In this talk we revisit both parts of this diagram. When $1 \leq q \leq 4$, we give a new proof of continuity that does not use parafermionic observable, nor Bethe Ansatz. When $q>4$, we establish invariance principle under Dobrushin boundary conditions: the interface converges to the Brownian bridge. Both arguments rely on the Baxter-Kelland-Wu correspondence that relates the random-cluster model to a certain height function (six-vertex model). Remarkably, we obtain also some result when $q<1$, though only at the self-dual point.
Joint works with Moritz Dober, Piet Lammers and Sebastien Ott.
Speaker: Evita Nestoridi (Stony Brook)
Title: Shuffling via transpositions
Abstract: In their seminal work, Diaconis and Shahshahani proved that shuffling a deck of $n$ cards sufficiently well via random transpositions takes $1/2 n log n$ steps. Their argument was algebraic and relied on the combinatorics of the symmetric group. In this talk, I will focus on a generalization of random transpositions and I will discuss the underlying combinatorics for understanding their mixing behavior and indeed proving cutoff. The talk will be based on joint work with S. Arfaee.
Speaker: Bogdan Ion (Pittsburgh)
Title: The braid skein algebra of the Heisenberg manifold
Abstract: We explicitly describe the structure of the braid skein algebras of the 3-dimensional Heisenberg manifold (a circle fibration over the 2-dimensional torus), in various flavors (general configurations, special configurations, under gauge equivalence), identifying them with the double affine Hecke algebras of type A (or rather, type GL or type PSL). I will also discuss some consequences of such considerations: 1) the proof of the K(\pi,1) conjecture (Arnol'd, Brieskorn, Pham, Thom, van der Lek) for the double affine hyperplane arrangement of type A; 2) the canonical construction of the Morton-Samuelson braid skein algebra of the punctured torus; 3) the mapping class group of the 2-dimensional torus outer action on the double affine Hecke algebra. This is joint work with Eve Roller.
Speaker: Leonid Petrov (Virginia)
Title: Random Fibonacci Words
Abstract: Fibonacci words are words of 1's and 2's, graded by the total sum of the digits. They form a differential poset YF which is an estranged cousin of the Young lattice powering irreducible representations of the symmetric group. We introduce families of "coherent" measures on YF depending on many parameters, which come from the theory of clone Schur functions (Okada 1994). We characterize parameter sequences ensuring positivity of the measures, and we describe the large-scale behavior of some ensembles of random Fibonacci words. The subject has connections to total positivity of tridiagonal matrices, Stieltjes moment sequences, orthogonal polynomials from the (q-)Askey scheme, and residual allocation (stick-breaking) models.
Speaker: Alejandro Morales (Montreal)
Title: The generalized Pitman-Stanley flow polytope
Abstract: In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. This polytope is well-studied due to its connections to parking functions, lattice path matroids, generalized permutahedra/polymatroids, and flow polytopes. Its lattice points correspond to plane partitions of skew shape with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries 0,1,...,m. Since then, this generalization has been untouched. We study this polytope and show that it can also be realized as a flow polytope of a grid graph. In this talk I will discuss characterizations of its vertices and give formulas for the number of vertices and faces as well as old and new formulas for the number of lattice points and volume in terms of rectangular Standard Young Tableaux. The new formulas come from the volume polynomial formulas of flow polytopes in terms of vector partition functions of Baldoni and Vergne and lattice point formulas of Stanley of marked order polytopes.
This is joint work with Maura Hegarty, William Dugan, and Annie Raymond.
Speaker: Siddhartha Sahi (Rutgers)
Title: Simultaneous elections make single-party sweeps more likely
Abstract: In a country with multiple elections, it may prove economically expedient to hold some or all of them on a common polling date. We show that such a decision will bring about a systemic change at the political level; an increase in the simultaneity of polling increases the likelihood that a single party wins all the elections.
Our result holds under fairly general conditions and we discuss its applicability to the two most common real world electoral systems, namely "first-past-the-post" (most voters) and "party list proportional representation" (most countries). In the course of our analysis, we obtain a slight generalization of the Harris correlation inequality for monotone functions on the Boolean lattice.
This is joint work with Pradeep Dubey (Center for Game Theory, Stony Brook).
The talk will be elementary, self-contained, and suitable for a general audience.
Speaker: Hong Chen (Rutgers)
Title: Differential Operators for Macdonald's Hypergeometric Functions
Abstract: In his widely circulated 1980s manuscript (now available as [arXiv:1309.4568]), Macdonald introduced hypergeometric functions with a parameter \alpha (the Jack parameter) and posed foundational questions about their properties. In this talk, I will present a general construction of differential operators pDq that characterize Macdonald’s hypergeometric functions pFq for arbitrary p and q. These operators generalize the ones studied by Constantine--Muirhead, Fujikoshi, and Macdonald.
Speaker: Daoji Huang (IAS)
Title: Affine Robinson--Schensted correspondence via growth diagrams
Abstract: The Robinson--Schensted correspondence is one of the most fundamental tools in algebraic combinatorics. Besides the usual introduction as a combinatorial algorithm, this correspondence can be encoded in Viennot's shadow line construction and equivalently by Fomin's growth diagrams, whose geometric interpretation, which connects to Springer theory, is given by van Leeuwen. Motivated by Kazhdan--Lusztig theory, Shi introduced the analogue of the Robinson--Schensted correspondence for the affine Weyl group of type A via an insertion algorithm. We generalize Fomin's growth diagram and Viennot's shadow line construction to the affine setting, recover and refine Shi's algorithm, and give geometric interpretations in the style of van Leeuwen. This is ongoing joint work with Sylvester Zhang.
Speaker: Darij Grinberg (Drexel)
Title: The random-to-random shuffles and their $q$-deformations
Abstract: Consider a random shuffle acting on a deck of $n$ cards as follows: Uniformly at random, we select $k$ out of our $n$ cards, remove them from the deck, and then move them back to $k$ uniformly random positions. This shuffle--the so-called "$k$-random-to-random shuffle''-- is a Markov chain that is given by a certain element of the group algebra of the symmetric algebra. A celebrated result of Dieker, Saliola and Lafrenière says that this shuffle is diagonalizable with all eigenvalues rational. Earlier, it was observed by Reiner, Saliola and Welker that two such shuffles for different $k$'s always commute. Both results are deep and hard.
I will discuss a new approach to these shuffles that has resulted in simpler proofs as well as a $q$-deformation -- i.e., a generalization into the Hecke algebra of the symmetric group. Along the way, some properties of the Hecke algebras have been revealed, as well as some general results about integrality of eigenvalues.
Joint work with Sarah Brauner, Patricia Commins and Franco Saliola.
Special event time: 1:30 PM
Speaker: Bhargav Narayanan (Rutgers)
Title: Elementary symmetric polynomials under the fixed point measure
Abstract: I’ll talk about a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, Ayush Khaita, Ishan Mata and I recently proved that any n non-negative real numbers a_1, a_2, \dots, a_n satisfy the inequality
(1/n!) \sum_{f \in S_n} \prod_{i:i=f(i)} a_i >= (1/(n choose 2)) \sum_{{j,k} \in ([n] choose 2)} \sqrt{a_j a_k}.
This bound is sharp, and equality is attained if and only if a_i = 1 for all 1 \le i \le n.
Where does this inequality come from? Can this inequality be deduced from existing machinery in the vast (primarily algebraic) literature around symmetric polynomials? These are some of the questions that I will answer in my talk.