MATH 246
Algebraic and Geometric Combinatorics

Meetings: Monday, Wednesday and Friday 1:10pm to 2:00pm.
We will meet at Social Sciences 70.

Instructor: Jesús A. De Loera.

The best way to contact me is via email
email: deloera@math.ucdavis.edu

URL: http://www.math.ucdavis.edu/~deloera/

Phone: (530)-554 9702

Office hours: Math Sci. Building 3228, 2:00pm-3:00pm or by appointment. Please feel free to ask questions! I am here to help you!

Description and objectives:

Algebraic & Geometric Combinatorics is the part of combinatorics that employs methods of abstract algebra OR methods from geometry/topology to solve combinatorial problems.
Sometimes we will also consider the application of combinatorial techniques to problems in algebra and geometry.

The kind of tools one uses are very diverse. For example, When a combinatorial problem has a lot of symmetry or there is a natural group action then the tools are typically from group theory and representation theory. But in this course I emphasize the use of tools from commutative algebra and convex geometry . A central theme will be the strong relationship between polynomials and combinatorics!!

Commutative Algebra is shortly the study of commutative rings and modules over them. We will cover the necessary algebraic (Groebner bases and monomial ideals, Nullstellensatz, primary ideal decomposition, Hilbert Functions, semigroup rings and toric ideals, graded rings and modules) and geometric basic knowledge (simplicial complexes, polytopes), but the main goal is to solve combinatorial problems!!

This course is suitable for first-year graduate students in mathematics. The main prerequisites are maturity with proofs and knowlegde of algebra at the level of Math 250A.

Text: The key reference we will use in the course is

``Combinatorics and Commutative Algebra'' by Richard P. Stanley, Birkhauser.

It is truly an extraordinary book, but it does not contain exactly the material I intend to cover and some parts are at a very advanced level that I aim to make easier to read by supplementing the necessary background, adding examples, exercises, etc. Here are a few of the extra books I plan to use:

``Groebner bases and Convex Polytopes'' Sturmfels, AMS university lecture series
``Combinatorial Commutative Algebra'' Miller and Sturmfels, Springer Verlag
``Algebraic Combinatorics on Convex Polytopes'' Hibi, Carslaw publications
``Graphs, Rings, and Polyhedra'', Gitler and Villarreal, SMM, Aportaciones Matematicas.
``Ideals, Varieties, and Algorithms'' Cox, Little, and OShea, Springer Verlag
``Undergraduate Commutative Algebra'' Reid, London Math Soc Student texts, Cambridge Univ Press
`` Computational Algebraic Geometry'' Schenck, London Math Soc Student texts, Cambridge Univ Press

I will also make some notes and slides available at the course web page.

Grading policy: There are a total of 100 points possible in this course:

I will assign many exercises each class, between 25 to 40 a week, for a total of about 250-300 exercises during the quarter.

(1) You can gain up to 65 points by writing clean careful LaTEX solutions for exercises (0.5, 1 or 2 points each), but there are some important rules :

(2) up to 35 more points can be gained by completing a final project. The project will consist of an investigation of an open problem in algebraic combinatorics chosen from a list of 3 topics I will provide.