I will tell two interrelated stories illustrating fruitful interactions between combinatorics and Hodge theory.
The first is that of Lorentzian polynomials, based on my joint work with Petter Brändén.
They link continuous convex analysis and discrete convex analysis via tropical geometry, and they reveal subtle information on graphs, convex bodies, projective varieties,
Potts model partition functions, log-concave polynomials, and highest weight representations of general linear groups.
The second is that of intersection cohomology of matroids, based on my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang. It shows a surprising parallel between the theory of convex polytopes, Coxeter groups, and matroids. After giving an overview of the similarity,
I will outline proofs of two combinatorial conjectures on matroids, the nonnegativity conjecture for their Kazhdan-Lusztig coefficients and the top-heavy conjecture for the lattice of flats.
호지 구조와 조합론 사이의 상호작용을 보여주는 서로 밀접히 관련된 두 이야기를 들려드릴 계획이에요. 첫째는 해석과 조합을 잇는 로렌츠 다항식 이야기이고, 둘째는 기하와 조합을 잇는 매트로이드 교차 코호몰로지 이야기입니다. 선형 대수 이상의 수학에 익숙하신 분들께 추천해 드립니다.
