# Winter term 2022/23: Combinatorics and Commutative Algebra

Chaired by I. Halupczok, H. Kammeyer, B. Klopsch and M. Zibrowius.

Organised by B. Klopsch

All talks take place on Fridays at 12:30 pm in 25.22.03.73.

If you want to participate but did not yet receive any mails related to the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.

## Aims and Content - a short description

We decided to go through R.P. Stanley's modern classicCombinatorics and Commutative Algebra' (2nd edition). There are two key topics: non-negative integral solutions to linear equations and algebraic properties of face rings associated to simplicial complexes, each with a view toward applications in combinatorics.
Chapters I and II of the book are essentially self-contained - if one is willing to take on trust some standard background results in algebra, combinatoris and topology. Chapter III was added in the 2nd edition; here the exposition is less detailed, but Stanley provides numerous concrete references, which seminar speakers could follow up. The core of the book is based on a series of lectures that were originally given in the 1980s.
In the seminar, we will follow Stanley's book very closely. Speakers can, of course, use additional sources, if they wish. If your talk is based on Sections from Chapter III, you have more flexibility what to present in detail or what to skip, because those talks should be fairly independent from one another. For lack of time we will not quite reach the end of the book, but anyone interested should find it easy enough to also read the last two Sections.

## Schedule

Please volunteer for talks that have not yet been asigned to anyone!
 14.10.22 No Seminar - but: Norddeutsches Gruppentheorie-Kolloquium Please visit the NDGK webpage for details. All welcome! 21.10.22 1. Background: Combinatorics, commutative and homological algebra (Saba) Cover Chapter 0, Sections 1-2. 28.10.22 2. Background: Algebraic topology (Daniel) Cover Chapter 0, Section 3. Refer to any standard introductory text on algebraic topology to fill in additional details where you see fit. 04.11.22 3. Non-negative integral solutions to linear equations I (Florian) Cover Chapter I, Sections 1-4. Keywords: integer stochastic matrices (magic squares); graded algebras and modules; elementary aspects of non-negative integral solutions to linear equations; integer stochastic matrices again. 11.11.22 4. Non-negative integral solutions to linear equations II (NN) Cover Chapter I, Sections 5-7. Keywords: dimension, depth and Cohen-Macauley modules; local cohomology; local cohomology of modules associated to linear equations. 18.11.22 5. Non-negative integral solutions to linear equations III (NN) Cover Chapter I, Sections 8-13. Keywords: reciprocity; reciprocity for integer stochastic matrices; rational points in integral polytopes; free resolutions; duality and canonical modules; final look at linear equations. 25.11.22 6. The face ring of a simplicial complex I (Martina) Cover Chapter II, Sections 1-4. Keywords: elementary properties of the face ring; $f$-vectors and $h$-vectors of complexes and multi-complexes; Cohen-Macauley complexes and the Upper Bound Conjecture; homological properties of face rings. If you are interested, give a quick look also to Stanley's How the UBC was proved' in Ann. Comb. 18 (2014). 02.12.22 7. The face ring of a simplicial complex II (NN) Cover Chapter II, Sections 5-8. Keywords: Gorenstein face rings; Gorenstein Hilbert functions; canonical modules of face rings; Buchsbaum complexes. 09.12.22 8. Further aspects of face rings I (NN) Cover Chapter III, Sections 1-2. Keywords: simplicial polytopes, toric varieties and the $g$-theorem; shellable simplicial complexes. 16.12.22 9. Further aspects of face rings II (NN) Cover Chapter III, Section 3. Keywords: matroid complexes, level complexes and doubly Cohen-Macauley complexes. 13.01.23 10. Further aspects of face rings III (NN) Cover Chapter III, Section 4. Keywords: balanced complexes, order complexes and flag complexes. 20.01.23 11. Further aspects of face rings IV (NN) Cover Chapter III, Sections 5-6. Keywords: splines; algebras with straightening law and simplicial posets. 27.01.23 12. Further aspects of face rings V (NN) Cover Chapter III, Sections 7-8. Keywords: relative simplicial complexes; group actions. 03.02.23 Discussion of topics for the next semester (all of us)

## Literature

The main reference is:
[St] P. Stanley: Combinatorics and Commutative Algebra
The text contains many additional references.

## Archive

WS 2021/22: Superrigidity

SS 2021: Group cohomology

SS 2020 and WS 20/21: cancelled due to pandemic

WS 2019/20: Intersection theory

SS 2019: Knots and primes

WS 2017/18: Algebraic K-theory

SS 2017: Berkovich spaces

SS 2016: Modular Representation Theory

WS 15/16: The Milnor Conjectures

SS 2015: Rationality

WS 14/15: Essential Dimension

SS 2014: Varieties of Representations

Website edited by Benjamin Klopsch (2022)