UNIVERSITY OF MINNESOTA
SCHOOL OF MATHEMATICS

Math 8680: Topics in Combinatorics
Combinatorial rings and the Kähler package

Spring 2021

Location, time:

Monday-Wednesday-Friday 12:20 - 1:10pm
online synchrounously via Zoom

Instructor:

Victor Reiner (You can call me "Vic").

Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
E-mail: reiner@math.umn.edu

Office hours:

3:35-4:25pm Tues, 1:25-3:20pm Thur,
at same Zoom ID as lecture.

Discord server:

Here's the class Discord server set up by Galen Dorpalen-Barry (thanks, Galen!)

Study group?

Organizing some sort of study group would be wonderful, too!

Prerequisites:

We will assume knowledge of basic abstract algebra (groups, rings, fields, modules).
Familiarity with simplicial homology would help.
Exposure to some representation theory is less important, but might help.

Course content:

In combinatorics, there are many natural integers sequences (h₀,h₁,...,h_d) that turn out to either be unimodal, that is, rising then falling, or more generally, log-concave, meaning h_i² ≥ h_i-1 h_i+1. However, for many of them it has been hard to prove, in some cases requiring fundamental breakthroughs about the structure and properties of certain combinatorially defined (commutative) rings. Several of these rings turn out to satisfy what has become known as the Kähler package, originally proven for cohomology rings of smooth projective complex varieties, as a vast strengthening of Poincaré duality.

This class hopes to discuss some of these breakthroughs that resolved longstanding combinatorial conjectures, beginning with old ones from the 1980s regarding convex polytopes, and transitioning to those in the past few years regarding matroids. Matroids are an abstraction of the linear dependence properties of a set of vectors in a vector space.

Notes and videos:

Topic	Lecture Notes	Lecture videos
Overview	Notes batch 1	Wed Jan 20, Fri Jan 22, Mon Jan 25, Wed Jan 27
Simplicial complexes and Stanley-Reisner rings	Notes batch 2	Fri Jan 29, Mon Feb 1, Wed Feb 3, Fri Feb 5
Partitionability, shellability, and Cohen-Macaulayness	Notes batch 3	Mon Feb 8, Wed Feb 10, Fri Feb 12, Mon Feb 15
Polytopes, The Upper Bound Conjecture, and Poincaré duality	Notes batch 4 Notes batch 5	Wed Feb 17, Fri Feb 19, Mon Feb 22, Wed Feb 24, Fri Feb 26, Mon Mar 1, Wed Mar 3, Wed Mar 5
Loose ends: fans, systems of parameters, regular sequences, Gorensteinness	Notes batch 6	Mon, Mar. 8, Wed, Mar. 10, Fri, Mar. 12, Mon, Mar. 15, Wed, Mar. 17
Piecewise polynomials on simplicial fans	Notes batch 7	Fri, Mar. 19, Mon, Mar. 22, Wed, Mar. 24, Fri, Mar. 26,
The Kähler package for simplicial polytopes	Notes batch 8 Notes batch 9	Mon, Mar. 29, Wed, Mar. 31, Fri, Apr. 2, Mon, Apr. 12 Wed, Apr. 14
Matroids and log-concavity conjectures	Notes batch 10 Notes batch 11 Notes batch 12	Fri, Apr. 16, Mon, Apr. 19, Wed, Apr. 21, Fri, Apr. 23, Mon, Apr. 25, Mon, May 3 wrap-up,

Grading:

Students who are registered for the class and want to get an A should attend regularly, and either hand in some homework or give a talk.
The talk should be for 20 minutes on some suggested papers below, or a paper that you can convince the instructor is related.

Student speaker	Paper	Talk date
Libby Farrell	Fomin and Reading, Generalized cluster complexes and Coxeter combinatorics	Wed Apr 28 Talk video, slides
Ethan Partida	Kalai, A simple way to tell a simple polytope from its graph	Wed Apr 28 Talk video, slides
Sasha Pevzner	Stanley, Sec. 7,8 on Gorensteinness and canonical module (from his survey Invariants of finite groups and their applications to combinatorics)	Fri Apr 30 Talk video, slides
Mahrud Sayrafi	Fulton and Sturmfels, Intersection theory on toric varieties	Fri Apr 30 Talk video, slides
Shuo Zhang	McMullen, The polytope algebra	Mon May 3 Talk video, slides

The homework should be a total of 4 exercises from the first part of this list, suggested to be handed in by the end of Spring break. Here are some solutions.
The exercises in the rest of that list are more focused on the later part of the course having to do with matroids. One can find more exercises on matroid problems in pages 53-60 of these in matroid notes.)

Sources:

Brion
- Piecewise polynomial functions, convex polytopes and enumerative geometry
- The structure of the polytope algebra
Fleming and Karu, Hard Lefschetz theorem for simple polytopes
Adiprasito, Huh and Katz, Hodge Theory for Combinatorial Geometries
Huh and Katz, Log-concavity of characteristic polynomials and the Bergman fan of matroids
Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
Lenz, Matroids and log-concavity
Feichtner and Yuzvinsky, Chow rings of toric varieties defined by atomic lattices
Braden, Huh, Matherne, Proudfoot, Wang,
- A semi-small decomposition of the Chow ring of a matroid
- Singular Hodge theory for combinatorial geometries
Ardila, Denham and Huh, Lagrangian geometry of matroids
Huh and Wang, Enumeration of points, lines, planes, etc
Brändén and Huh, Lorentzian polynomials

Surveys/talks:

Adiprasito, Lefschetz, Hodge and combinatorics: an account of a fruitful cross-pollination lecture series
Adiprasito, Huh and Katz, Hodge theory of matroids
Adiprasito and Yashfe, The partition complex: an invitation to combinatorial commutative algebra
Ardila,
- The geometry of matroids
- Lagrangian Combinatorics of Matroids talk
Baker, Hodge theory in combinatorics
Bayer, The cd-index: a survey
Bayer and Lee, Combinatorial Aspects of Convex Polytopes
Billera and Björner Face numbers of polytopes and complexes
Bjorner, Homology and shellability of matroids and geometric lattices
Braden, The Top-Heavy Conjecture for vectors and matroids talk
Eur,
- ICERM two-lecture intro on Matroids Parts 1,2
- ICERM two-lecture intro on Hodge-Riemann relations and Lorentzian polynomials
Francisco, Mermin and Schweig, A survey of Stanley-Reisner theory
Hopkins and others, MathOverflow question/answers
Huh,
- Combinatorial applications of the Hodge-Riemann relations
- His ICM talk
- A shorter IAS talk
Katz, Matroid theory for algebraic geometers
Gaku Liu, Polyhedral combinatorics lecture notes
Ricky Liu, Polytopes lecture notes
Oxley, What is a matroid?
Proudfoot,
- The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
- His Fall 2019 class The wonderful geometry of matroids
Robles, Linear structures of Hodge theory
Stanley,
- How the Upper Bound Conjecture was proved
- The number of faces of simplicial polytopes and spheres
- An introduction to combinatorial commutative algebra
- Log-concave and unimodal sequences in algebra, combinatorics and geometry
- An introduction to hyperplane arrangments
- Combinatorics and commutative algebra, Progress in Mathematics 41, Birkhäuser, 1996
Wachs, Poset Topology: Tools and Applications

Potential papers for student talks
(to be augmented):

Adiprasito, Combinatorial Lefschetz theorems beyond positivity
Adiprasito, Papadakis and Petrotou, Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles
Anari, Gharan and Vinzant, Log-concave polynomials, Part I, Part II, Part III, Part IV
Backman, Eur and Simpson, Simplicial generation of Chow rings of matroids
Bastidas, The polytope algebra of generalized permutahedra
Berget, Eur, Spink and Tseng, Tautological classes of matroids
Berget, Spink and Tseng, Log-concavity of matroid h-vectors and mixed Eulerian numbers
Billera and Lee A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes
Björner and Ekedahl, On the shape of Bruhat intervals
Brändén, The space of Lorentzian polynomials is a ball
Clements and Lindström, A generalization of a combinatorial theorem of Macaulay
Dowling and Wilson, Whitney number inequalities for geometric lattices
Duval, Goeckner, Klivans, Martin, A non-partitionable Cohen-Macaulay simplicial complex
Fulton and Sturmfels, Intersection theory on toric varieties
Gedeon, Proudfoot and Young, Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
Greene, A rank inequality for finite geometric lattices
Hameister, Rao and Simpson, Chow rings of vector space matroids
Kalai A simple way to tell a simple polytope from its graph
Maeno and Numata, Sperner property and finite-dimensional Gorenstein algebras associated to matroids
McMullen, The polytope algebra
Proudfoot, Equivariant incidence algebras and equivariant Kazhdan-Lusztig-Stanley theory
Stanley, Weyl groups, the Hard Lefschetz theorem, and the Sperner property

Long-standing problems and questions

Charney-Davis-Gal conjecture on flag simplicial spheres
--see this survey by Hailung Zheng
Stanley's pure O-sequence conjecture for h-vectors of matroid complexes
--see page 59 in Stanley's paper, or this paper by Klee and Samper
Characterizing f-vectors, flag f-vectors or cd-indices of polytopes, even just for 4-dimensional polytopes
-- see this paper by Ziegler, or this paper by Adiprasito and Avohou
Which artinian algebras have the weak, strong Lefschetz properties?
-- see this survey by Migliore and Nagel
Nonnegativity of the Billera-Brenti total cd-index of a Bruhat interval
--see the paper by Billera and Brenti
Combinatorial invariance of Kazhdan-Lusztig polynomials
-- see Brenti's paper
Reading's conjecture that cd-indices are smaller for Bruhat intervals than for polytopes
-- see Reading's paper
Knutson and Miller's question on polytopality of subword complexes
-- see this preprint by Labbé
Five open problems on polytopes
-- see this blog post by Kalai
Easy-to-Explain but Hard-to-Solve Problems About Convex Polytopes
--see this talk by De Loera

Back to Reiner's Homepage.

UNIVERSITY OF MINNESOTA SCHOOL OF MATHEMATICS

Math 8680: Topics in Combinatorics Combinatorial rings and the Kähler package

Spring 2021

UNIVERSITY OF MINNESOTA
SCHOOL OF MATHEMATICS

Math 8680: Topics in Combinatorics
Combinatorial rings and the Kähler package