Location, time: 
MondayWednesdayFriday 12:20  1:10pm
online synchrounously via
Zoom

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

Office: Vincent Hall 256
Telephone (with voice mail): 6256682
Email: reiner@math.umn.edu

Office hours: 
3:354:25pm Tues, 1:253:20pm Thur,
at same Zoom ID
as lecture.

Discord server: 
Here's the class Discord server
set up by Galen DorpalenBarry (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
(h_{0},h_{1},...,h_{d})
that turn out to either be unimodal, that is, rising then falling,
or more generally, logconcave, meaning
h_{i}^{2} ≥ h_{i1} 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 StanleyReisner rings 
Notes batch 2

Fri Jan 29,
Mon Feb 1,
Wed Feb 3,
Fri Feb 5

Partitionability, shellability,
and CohenMacaulayness

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
logconcavity 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 wrapup,


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 5360 of these
in matroid notes.)

Sources: 
 Brion
 Fleming and Karu,
Hard Lefschetz theorem for simple polytopes
 Adiprasito, Huh and Katz,
Hodge Theory for Combinatorial Geometries
 Huh and Katz,
Logconcavity of characteristic polynomials and the Bergman fan of matroids
 Huh,
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
 Lenz,
Matroids and logconcavity
 Feichtner and Yuzvinsky,
Chow rings of toric varieties defined by atomic lattices
 Braden, Huh, Matherne, Proudfoot, Wang,
 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 crosspollination lecture series
 Adiprasito, Huh and Katz,
Hodge theory of matroids
 Adiprasito and Yashfe,
The partition complex: an invitation to combinatorial commutative algebra
 Ardila,
 Baker,
Hodge theory in combinatorics
 Bayer,
The cdindex: 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 TopHeavy Conjecture for vectors and matroids talk
 Eur,
 Francisco, Mermin and Schweig,
A survey of StanleyReisner theory
 Hopkins and others,
MathOverflow question/answers
 Huh,
 Katz,
Matroid theory for algebraic geometers
 Gaku Liu,
Polyhedral combinatorics lecture notes
 Ricky Liu,
Polytopes lecture notes
 Oxley,
What is a matroid?
 Proudfoot,
 Robles,
Linear structures of Hodge theory
 Stanley,
 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,
Logconcave 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,
Logconcavity of matroid hvectors and mixed Eulerian numbers
 Billera and Lee
A proof of the sufficiency of McMullen's conditions for fvectors 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 nonpartitionable CohenMacaulay simplicial complex
 Fulton and Sturmfels,
Intersection theory on toric varieties
 Gedeon, Proudfoot and Young,
KazhdanLusztig 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 finitedimensional Gorenstein algebras associated to matroids
 McMullen,
The polytope algebra
 Proudfoot,
Equivariant incidence algebras and equivariant KazhdanLusztigStanley theory
 Stanley,
Weyl groups, the Hard Lefschetz theorem, and the Sperner property

Longstanding problems and questions 
