Speaker 1: Faye Pasley Title: Determinantal Representations, the Numerical Range, and Invariance We study hyperbolic polynomials with symmetry and express them as the determinant of a Hermitian matrix with special structure. By properly modifying a construction of Dixon (1902), we show for every hyperbolic polynomial of degree $n$ invariant under the cyclic group of order $n$ there…

Speaker 1: Caprice Stanley Title: Markov Chain Mixing Time Abstract: Informally a Markov chain is a memoryless stochastic process and its mixing time is the time required for the chain to be near its stationary distribution. Depending on the context, analysis of mixing time can be of great importance. For example, random walk-based algorithms for…

We will talk about bijections between various families of combinatorial objects counted by OEIS A071356. These include certain underdiagonal lattice paths, pattern avoiding permutations, pattern avoiding inversion sequences, and posets. Connections will be made to combinatorial Hopf algebras using some lattice theory. OEIS A071356 is closely related to the Catalan numbers.