Oberseminar Algebra und Geometrie

Winter term 2022/23: Combinatorics and Commutative Algebra

Chaired by I. Halupczok, H. Kammeyer and 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 classic`Combinatorics 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

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.
28.10.22	Guest Talk: An Introduction to A¹-enumerative geometry (Sabrina Pauli) 4.45pm in HS 5H - tea/coffee from 4.15pm in 25.22.0053 Abstract: A classical result in enumerative geometry states that there are always 27 lines on a smooth cubic surface when counting over an algebraically closed field. However, when counting lines defined over a non-algebraically closed field, the number of lines depends on the choice of cubic surface. Results from A¹-homotopy theory allow to get an invariant "count" over an arbitrary base field. In the talk I will give a brief introduction to A¹-homotopy theory, explain how this new way of counting works and give some illustrating examples.
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 (David) 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 (Giada) 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	CANCELLED
02.12.22	CANCELLED
09.12.22	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).
16.12.22	7. The face ring of a simplicial complex II (Iker) Cover Chapter II, Sections 5-8. Keywords: Gorenstein face rings; Gorenstein Hilbert functions; canonical modules of face rings; Buchsbaum complexes.
13.01.23	8. Further aspects of face rings I (Luca) Cover Chapter III, Sections 1-2. Keywords: simplicial polytopes, toric varieties and the $g$-theorem; shellable simplicial complexes.
20.01.22	9. Further aspects of face rings II: Matroid complexes and Stanley’s conjecture (Moritz) Cover Chapter III, Section 3. Keywords: matroid complexes, level complexes and doubly Cohen-Macauley complexes. Additional source: A. Constantinescu and M. Varbaro, h-vectors of matroid complexes, Combinatorial methods in topology and algebra, 203–227, Springer INdAM Ser., 12, Springer, Cham, 2015.
27.01.23	10. Further aspects of face rings III (Immanuel) Cover Chapter III, Section 4. Keywords: balanced complexes, order complexes and flag complexes.
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.

Oberseminar Algebra und Geometrie

Winter term 2022/23: Combinatorics and Commutative Algebra

Aims and Content - a short description

Schedule

Literature

Archive