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

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!
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,...,hd) that turn out to either be unimodal, that is, rising then falling, or more generally, log-concave, meaning hi2 ≥ hi-1 hi+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  
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,
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.)
Potential papers for student talks
(to be augmented):
Long-standing problems and questions
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