Algebraic & Enumerative Combinatorics - Connor Baetz, Arnav Kumar, & Ron Cherny

Title: The ASEP and alternate multi-line queues

Abstract: The asymmetric simple exclusion process (ASEP) is a parametrised Markov process on a state space of particle permutations and comes from statistical physics. It is a generalisation of the simplest system that displays more than two distinct "phases", analogous to solids, liquids, and gasses in matter. The problem at hand looks at the ASEP on a loop with a fixed number of particles of a given type. Its was proved by James Martin that the stationary distribution can be computed by taking the sum of weights over a certain combinatorial class called the multi-line queues (MLQs), which he introduced. He also introduced a modification of them, the alternative multi-line queues (AMLQs), that have a slightly simpler weight scheme and conjectured that they also may be used to compute the stationary distribution of the ASEP on a loop. However, despite being widely recognised as likely being true, the result was only known for the case when N, the number of distinct particle types, is 2. This talk will sketch a proof of his conjecture for arbitrary N.

Title: Dimension of posets and random graph orders

Abstract: A poset P = (X,P) is a set X equipped with a partial order P.

The dimension of P is the minimum number of linear orderings on X required so that their intersection is P. We investigate two problems regarding the dimension of posets. The first problem is a conjecture by Bollobas and Brightwell from 1997 that the poset with a unicyclic cover graph has dimension at most 3. The second problem was proposed by Erdos in 1991 about the dimension of the random partial order obtained by taking the transitive closure of the random graph G(n,p), and the random bipartite graph B(n,n,p). The random graph order is a type of classical sequential growth model that physicists use to model relations of spacetime events in the Minkowski space, and thus finds its applications in the causal set approach to quantum physics.

Title: A Combinatorial Case of The Gerstenhaber Problem

Abstract: The well-known Cayley Hamilton theorem tells us that the unital algebra generated by a single $n \times n$ matrix has dimension at most $n$. In 1961 Gerstenhaber proved that the unital algebra generated by a pair of commuting $n \times n$ matrices has dimension at most $n$. The analogous statement for triples of pairwise commuting matrices remains an open problem, named the Gerstenhaber Problem. In this talk, we introduce the problem and reformulate it in commutative algebraic language, then we narrow our focus to a special class of matrices that arise from plane partitions.