**Meetings:** Monday, Wednesday and Friday 12:10pm to 1:00pm.

We will meet at Veihmeyer Hall 116 (or ONLINE if needed, see ZOOM info on CANVAS).

**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 or Zoom. MWF 2:00pm-3:00pm or by appointment.

I am here to help you! I care that you enjoy and learn this lovely Mathematics!

**Description and objectives:** This 4-unit course is offered under the header Algebraic Combinatorics. This time I focus on
* Geometric and Topological Combinatorics*.

These are techniques from Topology and Geometry (convex, algebraic, tropical, differential,etc) used to solve combinatorial problems. We will also consider

combinatorial questions about geometric objects such as PL-Manifolds, Oriented Matroids, Polytopes, Simplicial Complexes, arrangements of hyperplanes, Graphs, etc.

This is a graduate class, thus I will point to applications (CS,Economics,Optimization,etc) and open problems for research.

** Topics and Schedule:** The course starts with topological/algebraic topics, and progressively goes more and more polyhedral.
Each topic is about 5 lectures.

- Topic 1: Polyhedral + simplicial Complexes from Combinatorics: from Graphs (clique, matching complexes), Posets (order complexes),

and Finite Sets (Nerve complexes). The Kruskal-Katona theorem on f-vectors of simplicial complexes. A pinch of PL-topology. - Topic 2: Counting faces of polytopes and spheres, the upper bound theorem. A pinch of Commutative Algebra and Algebraic Geometry.
- Topic 3: Sperner ' s lemma=Brouwer ' s fixed-point theorem, Tucker ' s lemma=Borsuk Ulam theorem, games and fair division problems,

Ham-Sandwich theorem, Gerrymandering and geometry. A pinch of algebraic topology.*NOTE: Final projects selected by students by the end of this unit. YOU CAN FIND SEVEN IDEAS IN CANVAS (see files)* - Topic 4: A crash course on Combinatorial Convexity: Convex Sets, Helly's theorem, Caratheodory's and Tverberg ' s theorems. Geometric Ramsey Theorems.
- Topic 5: Polyhedra part I. Hyperplane Arrangements, Polytopes and Polyhedra, Weyl-Minkowski, Counting cells on arrangements, Zonotopes, Minkowski sums,

Simplicial Polytopes. Triangulations and subdivisions of Polytopes and configurations. Volumes and Lattice points. A pinch of toric and tropical geometry. - Topic 6: Polyhedra part II: Realization spaces of polytopes, Bizarre Polytopes, Folkman-Lawrence Oriented matroid characterization. Universality theorems,

Graphs of Polytopes, planarity and Steinitz theorem, Hirsch conjecture. A pinch of real algebraic geometry.

**References**:
I use many references in the course. I aim to give you an expert's view of the field and to make easier to read papers by

supplementing the necessary background, adding examples, exercises, etc. Here are a few of the books I plan
to use (more or less

in Chronological order). I also use my own writings, and you will create notes.

''Topological Methods", Chapter 34 in Handbook of Combinatorics, by A. Bjorner, Elsevier.

''Combinatorics: Set systems, Hypergraphs, Families of vectors, and Combinatorial Probability'' by B. Bollobas, Cambridge.

''Combinatorics and Commutative Algebra'' by R. P. Stanley, Birkhauser.

''A Course in Topological Combinatorics'' by M. de Longueville, Springer.

''Algebraic Combinatorics on Convex Polytopes'', T. Hibi, Carslaw publications

''The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg'' by J.A. De Loera et al., AMS Bulletin

''Lectures on Discrete Geometry'' by J. Matousek, Springer

'' Lectures on Polytopes'', G.M. Ziegler, Springer

''Convex Polytopes'', B. Grunbaum, Springer

''Ideals, Varieties, and Algorithms'' Cox, Little, and OShea, Springer Verlag

''Undergraduate Commutative Algebra'' Reid, London Math Soc Student texts, Cambridge Univ Press

''Combinatorial Commutative Algebra'' Miller and Sturmfels, Springer Verlag

''Computational Algebraic Geometry'' Schenck, London Math Soc Student texts,
Cambridge Univ Press

''Groebner bases and Convex Polytopes'' Sturmfels, AMS university lecture series

''Triangulations: Structures for Algorithms and Applications'', by J.A. De Loera, J. Rambau, and F. Santos, Springer.

**Pre-requisites:** Math 245 is **NOT** a pre-requisite (that course has counting as a focus). I require the mathematical maturity a second year graduate student

that has completed all requirements.
Some algebraic/combinatorial topology, basic algebraic geometry, and commutative algebra will be needed, but I will

cover via lecture/videos/or exercises. The Most important pre-requisite: Willingness to work through a lot of exercises and dive passionately into the material.

**Format of course**: The course consists of about 30 lectures, I intend to cover 6 topics. Participants will form study
groups, work out LOTS of problems,

and collaborate/discuss. Mini-exams will be submitted in teams. With exception of first week online, the course is expected to be in person, but I will record

and post videos of all
my lectures and with your help make notes available so that everyone can follow from a distance. Auditors (even those outside UC Davis)

are welcome
but I will *not* grade their work.

*MY AXIOM: Learning=Doing!* Therefore, I promise to provide plenty of opportunity for you to learn.

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

** (1) (60 points) I will assign 6 home-exams **, approximately one every 10 days (or 4 lectures), with about 15-25 problems.

Each mini-exam is worth 15 points and covers one topic of the course. I drop the lowest 2 scores,

Mini-exams are submitted in teams of students.

**(2) (15 points) Latex Notes for one Topic ** Teams write clean careful
LaTEX notes and solutions for exercises of one of the units.

No more than
2 teams can typeset the same unit.

**(3) (25 points) Final Project+ Presentation ** Teams of students will explore a topic of geometric-topoogical combinatorics

of their choice. They read an advanced paper(s) and prepare a 45 mins
oral presentation.
long.