Reading Seminar: Hodge Theory in Combinatorics

This reading seminar is aimed to understand the recent works of June Huh et al. on combinatorics and in particular its interaction with algebraic geometry: the Poincaré duality, the Hard Lefschetz theorem and the Hodge-Riemann relations and their applications to the elementary combinatorics of graphs and matroids. In the end we should understand the works of Karim-Huh-Katz in [1] and Huh-Wang in [2] below.


Homepage http://www.jliumath.com/seminars/rs2018fall.html
Organizers
Information
  • The first meeting to assign topics will be on September 4.
  • The first talk will be on September 18.
  • There is no talk on October 2 because of the national holiday.
  • There are no talks on November 6 and November 20.
  • There is no talk on January 1 because of the holiday.
Place South building （南楼）913
Time Tuesday 10:00 am -- 12:00 am
Planning
  • Sept. 18, Classical Hodge theory in algebraic geometry (Yunfei Gao)
    The talk should start with a reminder on the classical Hodge theory in algebraic geometry, in particular the Poincaré duality, the hard Lefschetz theorem and the Hodge-Riemann relations. One can follow the exposition in Part II of [7].
  • Sept. 25 and Oct. 9, Matroid, characteristic polynomial and realizability (Xiangfei Li)
    The goal of the talk is aimed to explain the basic concepts and properties of matroids introduced by Whitney. We should understand the notion of realizability and characteristic polynomial of matroids. One can follow Section 1 in [5] and Section 3 in [2], see also Section 2.6 in [6].
  • Oct. 9, Bergman Fan and piecewise linear functions (Rongji Kang)
    Introduce the Bergman fan \(\Sigma\) of a matroid \(M\) and discuss the piecewise linear functions on \(\Sigma\). This talk should show that the ample cone and its ambien vector space depend only on the given order filter and the combinatorial geometry of \(M\) ([2] Proposition 4.8). One can follow Section 4 of [2].
  • Oct. 16, Chow ring and degree map (Yanglong Zhang)
    The goal of this talk is aimed to introduce the Chow ring for any unimodualr fan and to prove the characterization of realizability of motroids via Chow equivalence ([2] Theorem 5.12). The speaker should also introduce the degree map over the Chow ring of a Bergman fan.
  • Oct. 23, Surjectivity of the sum of the pullback homomorphism and Gysin homomorphisms (Xueqing Wen)
    The talk is aimed to introduce the so-called matroidal flip and to construct homomorphisms associated: the pullback homomorphism and the Gysin homomorphisms. We should understand the proof of the surjectivity of the sum of the pullback homomorphism and Gysin homomorphisms ([2] Proposition 6.10). One can follow the first part of Section 6 in [2].
  • Oct. 30, Decomposition and Poincaré duality (Jianping Wang)
    Continued with the previous talk, to show that the sum of the pullback homomorphism and Gysin homomorphisms is indeed an isomorphism ([2] Theorem 6.18). Moreover, the talk should prove the Poincaré duality for Chow rings ([2] Theorem 6.19). This is the last part of Section 6 of [2].
  • Nov. 13, Hard Lefschetz property and Hodge-Riemann relations I: preparation (Wenyao Hu)
    The talk is aimed to give the relations between the the Hard Lefschetz theorem and the Hodge-Riemann relations for matroids. These results will be used in the proof of the Hard Lefschetz theorem and the Hodge-Riemann relations in the next talk. Basically, one can follow Proposition 7.15 and Proposition 7.16 in [2].
  • Nov. 27, Hard Lefschetz property and Hodge-Riemann relations II: statements and proof (Feng Shao)
    The talk should start with the fact that the property HR is preserved by a matroidal flip for particular choices of ample classes. Combining it with the results given in the previous talk, the speaker should give a proof of the Hard Lefschetz theorem and the Hodge-Riemann relations for matroids ([2] Theorem 8.8). One can follow Section 8 in [2].
  • Dec. 4, Log-concavity conjecture for matroids I (Jian Xiao)
    This is a preparation for the proof of the log-convacity conjecture for matroids. We introduce the basic notion of graphs and its chromatic polynomial, and discuss some basic properties. Then we give a brief introduction of hyperplane arrangements and their characteristic polynomials. The relation of these two kinds of polynomials are given without proof.
  • Dec. 11, Log-convacity conjecture for matroids II (Jian Xiao)
    The goal of this talk is to explain how to use the Hodge-Riemann relation to prove the log-convacity conjecture for realizable matroids ([3]). The proof of the general case is also briefly discussed ([2]).
  • Dec. 18, Decomposition theorem for intersection complexes (Xiaoyu Su)
    In this talk, we give an overview of the proof of the main theorem in [1]. In particular, we focus on the decomposition theorem ingredient of the proof. Then we give a rough introduction of the proof of the decomposition theorem based on [8].
  • Dec. 25, "Top-heavy" conjecture for realizable geometric lattices I (Jie Liu)
    In this talk, we introduce the necessary notions for understanding [2]. We give the definition of finite geometric lattices and show that there exists an one-to-one correspondence between geometric lattices and simple matroids. Then we explain the "top heavy" conjecture for geometric lattices.
  • Jan. 8, "Top-heavy" conjecture for realizable geometric lattices II (Jie Liu, Qizheng Yin)
    In this talk, we explain the statement of the main theorem in [1] and discuss the construction of toric varieties used in the proof of main theorem. Finally, one gives the proof and discusses its possible generalization to any matroids.
References
  1. Huh, June; Wang, Botong. Enumeration of points, lines, planes, etc. Acta Math. 218 (2017), no. 2, 297--317.
  2. Karim, Adiprasito; Huh, June; Katz, Eric. Hodge theory for combinatorial geometries. Ann. of Math. (2) (2018), no. 188, 1--72.
  3. Huh, June. Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs. J. Amer. Math. Soc. 25 (3) (2012) 907–927.
  4. Huh, June; Katz, Eric. Log-concavity of characteristic polynomials and the Bergman fan of matroids. Math. Ann. 354 (2012) 1103–1116.
  5. Chambert-Loir, Antoine. Relations de Hodge-Riemann et matroïdes, d’après Adiprasito, Huh et Katz. no. 1144, Séminaire Bourbaki, Mars 2018.
  6. Huh, June. Combinatorial applications of the Hodge-Riemann relations. Proceedings of the International Congress of Mathematicians 3 (2018), 3079--3098.
  7. Voisin, Claire. Hodge theory and complex algebraic geometry. I. Translated from the French original by Leila Schneps. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2002. x+322 pp.
  8. Beilinson, Alexander; Bernstein, Joseph; Deligne, Pierre; Gabber, Ofer. Faisceaux pervers. Actes du colloque “Analyse et Topologie sur les Espaces Singuliers”. Partie I. Astérisque, Paris: Société Mathématique de France (SMF), 2018, 100, vi + 180