coneill «at»
Mathematical Sciences
Building, Room 3149
Department of Mathematics
University of California Davis
One Shields Ave
Davis, CA 95616
Math 485: Topics in Combinatorics (Spring 2016)
Lectures: Wed 9:00am - 10:00am, Room 605A.
Office hours: by appointment.
We will cover a variety of topics in combinatorics: Catalan numbers, partially ordered sets, matroids, polytopes, Ehrhart theory, and non-unique factorization. Other topics will be covered by request as time permits.
January 17, 2016: Course webpage has been created, and the syllabus has been uploaded.
January 17, 2016: A lecture schedule has been uploaded.
January 23, 2016: The meeting time and location for the course have been updated.
March 28, 2016: This webpage, including the lecture schedule, has been updated.
The course syllabus.
A tentative week-by-week schedule (subject to change without notice).
Recommendations for Further Reading
Enumerative combinatorics (volume 1 and volume 2), by Richard Stanley. An outstanding overview of enumerative combinatorics from one of the pioneers of the area.
Matroids, by James Oxley. A thorough, well-written introduction to matroids.
How do you measure primality?, by Christopher O'Neill and Roberto Pelayo. A gentle and example-driven introduction to the omega-primality factorization invariant. Several open problems are given.
Arithmetic congruence monoids: a survey, by Paul Baginski and Scott Chapman. A detailed survey paper on factorization theory in ACMs. Several examples are given, and some open problems are posed.
Factorization invariants in numerical monoids, by Christopher O'Neill and Roberto Pelayo. A survey paper on several factorization invariants and related results for numerical monoids. Some open problems are given.
Polytopes, by Gunter Ziegler. A high-level but thorough introduction to polytopes. Touches on both geometric and combinatorial aspects.
Computing the continuous discretely, by Matthias Beck and Sinai Robins. An introduction to polytopes and lattice point enumeration.
Combinatorial reciprocity theorems, by Matthias Beck and Raman Sanyal. Gives several in-depth examples of Ehrhart theory and lattice point enumeration in action.
The Dehn-Sommerville Relations and the Catalan Matroid, by Anastasia Chavez and Nicole Yamzon. Relates face vectors of polytopes and Dyck paths.
Homework Assignments
Each week, I will assign one or more problems on the current topic, due the following week. I will post links to the assignments below. If you have questions about course material or homework problems, do not hesistate to ask me in class, stop by my office hours, or email me.
Week 1: Organizational meeting
Week 2: Review of Math 431
Selection problems from last semester.
Week 3: Catalan Numbers
Handout (see me if you need a new copy).
Week 4: Introduction to Posets
Problems 5, 7, 8, 10, 25, 34, 35, and 36 in Chapter 16 of Bona.
Week 5: Posets and Möbius Functions
Problems 6, 29, 30, 42, 43, and 45 in Chapter 16 of Bona.
Week 6: Introduction to Matroids
Homework problems.
Week 7: Class cancelled
Week 8: Generalizing Structures to Matroids
Homework problems.
Week 9: Matroid Duality and the Tutte Polynomial
Homework problems.
Week 10: Non-unique Factorization
Homework problems.
Week 11: Class cancelled
Week 12: Introduction to Polytopes
Compute the face poset of any polytope of dimension at least 3, such as a platonic solid.
Week 13: Exploring Face Numbers of Polytopes
Homework problems.