Explore a 53-minute lecture from the Fields Institute's 2022-2023 Matroids - Combinatorics, Algebra and Geometry Seminar, delivered by Swee Hong Chan from Rutgers University. Delve into the world of combinatorial atlas for log-concave inequalities, starting with examples of binomial coefficients and permutations with k inversions. Examine matroid conditions and Mason's Conjecture from 1972, including its solution and refinements. Investigate graphical matroids, matroid contraction, and parallel numbers. Learn about equality conditions in Mason's Conjecture and study the atlas example of simplified matroids. Explore hyperbolic inequalities, irreducibility conditions, inheritance conditions, and subdivergence conditions. Conclude with the bottom-to-top principle for hyperbolic inequalities, gaining a comprehensive understanding of this complex mathematical topic.
Combinatorial Atlas for Log-Concave Inequalities
Overview
Syllabus
Intro
Example: binomial coefficients
Example: permutations with k inversions
Matroids: Conditions
Mason's Conjecture (1972)
Solution to Mason (3)
Warmup: graphical matroids refinement
Refinement for different matroids
Matroid contraction
Parallel number
Refinement for Mason (3)
When is equality achieved?
Equality conditions
Atlas example: matroid (simplified)
Hyperbolic inequality
Irreducibility condition
Inheritance condition
Subdivergence condition
Bottom-to-top principle for hyperbolic inequalities
Taught by
Fields Institute
