We will give a brief introduction to Schur polynomials and intersection theory and show how to use Section 10 of the Brändén-Huh paper to obtain log-concavity results for Schur polynomials.
https://arxiv.org/abs/1902.03719
https://arxiv.org/abs/1906.09633
I will continue to describe deterministic algorithms for approximately counting common bases of matroids within an exponential factor. This is based on AOVI and previous works of AO.
This is quick tutorial on bounding the mixing time of a finite Markov chain in terms of functional inequalities defining the spectral gap and the entropy constant of a Markov chain. The lecture will include some examples, including bounding the mixing time of the random transposition shuffle and (time permitting) that of the basis-exchange walk on a balanced matroid.
This is intended as a review lecture before Nima Anari’s lectures (during Nov. 4-6) on applications of Lorentzian polynomials, including recent breakthrough analyses of the basis-exchange walk on an arbitrary matroid.
We will discuss a deterministic, polynomial (in the rank) time approximation algorithm for counting the bases of a given matroid and for counting common bases between two matroids of the same rank. This talk follows the paper (https://arxiv.org/abs/1807.00929) of Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant.
Using what we have studied in the Brändén-Huh paper, we will go over the proof of the ultra-log-concavity version of Mason's conjecture.
I will discuss a proof of the statement that the support of a Lorentzian polynomial is M-convex, based on sections 3-5 of the Brändén—Huh paper.
I will show some operations that preserve Lorentzian property following Section 6 of https://arxiv.org/pdf/1902.03719.pdf
I will provide an introduction to Lorentzian Polynomials in the sense of https://arxiv.org/abs/1902.03719
