# Combinatorics and Hodge theory

강연자 허준이 미국 프린스턴대 교수, 한국 고등과학원 석학교수 2022-07-27

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.

 제목+내용제목내용댓글이름닉네임태그
1. ### Birational Geometry of varieties with effective anti-canonical divisors

Fano varieties are fundamental objects in algebraic geometry. These can be considered as the unique output of the -K -minimal model program on the varieties with effective anticanonical divisors. Thus the initial models should encode the in...
Category수학강연회 소속연세대학교 강연자최성락
2. ### Contact instantons and entanglement of Legendrian links

We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Ha...
Category수학강연회 소속IBS-CGP /POSTECH 강연자오용근
3. ### <학부생을 위한 ɛ 강연> Self-Supervised Learning in Computer Vision

In recent years, artificial intelligence has made remarkable progress in developing algorithms that can learn from vast amounts of carefully labeled data. This paradigm of supervised learning has made great success in training specialist mo...
Category수학강연회 소속인하대학교 강연자현윤석
4. ### <학부생을 위한 강연> 수학과 보험산업

Category수학강연회 소속라이나생명 강연자유신옥
5. ### Counting number fields and its applications

It is a fascinating and challenging problem to count number fields with bounded discriminant. It has so many applications in number theory. We give two examples. First, we compute the average of the smallest primes belonging to a conjugacy ...
Category수학강연회 소속UNIST 강연자조재현
6. ### 2022-2 Rookies Pitch: Algebraic Geometry (박현준)

CategoryBK21 FOUR Rookies Pitch 소속KIAS 강연자박현준
7. ### 2022-2 Rookies Pitch: Geometric Topology (김승원)

CategoryBK21 FOUR Rookies Pitch 소속성균관대학교 강연자김승원
8. ### 2022-2 Rookies Pitch: Probability Theory (이중경)

CategoryBK21 FOUR Rookies Pitch 소속수리과학부 강연자이중경
9. ### 2022-2 Rookies Pitch: Probability Theory (변성수)

CategoryBK21 FOUR Rookies Pitch 소속KIAS 강연자변성수
10. ### Towards Trustworthy Scientific Machine Learning: Theory, Algorithms, and Applications

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, h...
Category수학강연회 소속KAIST 강연자신연종
11. ### Elliptic equations with singular drifts in critical spaces

Category수학강연회 소속서강대학교 강연자김현석
12. ### Contact topology of singularities and symplectic fillings

For an isolated singularity, the intersection with a small sphere forms a smooth manifold, called the link of a singularity. It admits a canonical contact structure, and this turns out to be a fine invariant of singularities and provides an...
Category수학강연회 소속순천대학교 강연자권명기
13. ### 2022-2 Rookies Pitch: Representation Theory(이신명)

CategoryBK21 FOUR Rookies Pitch 소속수리과학부 강연자이신명
14. ### 2022-2 Rookies Pitch: Representation Theory(허태혁)

CategoryBK21 FOUR Rookies Pitch 소속QSMS 강연자허태혁
15. ### 2022-2 Rookies Pitch: Harmonic Analysis (오세욱)

CategoryBK21 FOUR Rookies Pitch 소속고등과학원 강연자오세욱
16. ### 2022-2 Rookies Pitch: Harmonic Analysis (함세헌)

CategoryBK21 FOUR Rookies Pitch 소속수학연구소 강연자함세헌
17. ### Combinatorics and Hodge theory

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...
Category특별강연 소속미국 프린스턴대 교수, 한국 고등과학원 석학교수 강연자허준이
18. ### 2022-1 Rookies Pitch: Harmonic Analysis (Kalachand Shuin)

CategoryBK21 FOUR Rookies Pitch 소속BK21 강연자Kalachand Shuin