# Foundations of Combinatorics

### Fall, 2019 Prof. Sara Billey

Monday, Wednesday, Friday 12:30-1:20

### Syllabus

Summary: This three quarter topics course on Combinatorics includes Enumeration, Polytopes, and Graph Theory. Combinatorics has connections to all areas of mathematics, industry and many other sciences including biology, physics, computer science, and chemistry. We have chosen core areas of study which should be relevant to a wide audience. The main distinction between this course and its undergraduate counterpart will be the pace and depth of coverage. In addition we will assume students have a substantial knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research. New students are welcome to join in Winter or Spring with approval from the instructor depending on prior background.

The outline for this quarter is the following:

• Fall = Enumeration (Billey): Every discrete process leads to questions of existence, enumeration and optimization. This is the foundation of Combinatorics. In this quarter we will present the basic combinatorial objects and methods for counting various arrangements of these objects.
• Basic counting methods.
• Sets, multisets, permutations, and graphs.
• Inclusion-exclusion.
• Recurrence relations and integer sequences.
• Generating functions.
• Partially ordered sets.
• Complexity Theory

Winter = Discrete Geometry and Polytopes (Novik): Discrete Geometry deals with finite sets of points, lines, planes, circles, and many other seemingly simple geometric objects such as polytopes (defined as the convex hull of a finite set of points in R d ). It has deep connections to combinatorics, optimization, number theory, and computer science. It is also very rich in simple-to-state yet long-unsolved problems. Some examples include: does every d-dimensional centrally symmetric polytope have at least 3d faces? Can the number of i-dimensional faces of a polytope be smaller than both the number of its vertices and the number of its top-dimensional faces? These questions are still open except for a few small values of d. This course will be a sampler of a few of the topics in this vast field. We will start with classical theorems due to Radon, Helly, and Caratheodory as well as Dehn's solution to Hilbert's third problem. Then, depending on time and interest, we'll discuss some of the following topics:

• Various extensions of the Radon, Helly, and Caratheodory theorems, including fractional, colorful, and topological versions (and potentially touching on very recent exciting developments due to Florian Frick and others).
• Kahn-Kalai's counterexample to Borsuk's problem.
• Graphs of polytopes: "classical" results: Steinitz's theorem for 3-polytopes; Balinski's theorem; reconstructing simple polytopes from their graphs.
• Graphs of polytopes: last decade's results: Santos's counterexample to the Hirsch conjecture; positive results towards the polynomial Hirsch conjecture.
• Face numbers of simplicial polytopes: the Dehn-Sommerville relations; the upper and the lower bound theorems; the g-theorem and its consequences; face numbers of centrally symmetric polytopes.
• Volumes, high-dimensional polytopes, and high-dimensional paradoxes: the weak perfect graph conjecture; the Brunn-Minkowski inequality; almost spherical sections of the cube and the octahedron.

Spring = Graphs And Tropical Curves (Shokrieh): Graphs (1-dimensional simplicial complexes) appear in a variety of subjects including Computer Science, Physics and Chemistry, Social Sciences, Biology, Optimization, Knot Theory, Algebraic Geometry, Group Theory, and Number Theory. In this course we will discuss some fundamental results on both finite and metric graphs (aka abstract tropical curves). We will also present a few open problems along the way, some of which might be within reach for strong undergraduate or graduate students. Topics can include classical material and modern material: Fundamentals, Matching, Connectivity, Planarity, Trees, Coloring, Extremal Problems, Ramsey Theory, Random Graphs. Modern topics can include: Metric Graphs and Tropical Curves, Chip-Firing Games, Abel-Jacobi and Riemann-Roch Theory on Graphs, Potential Theory, and maybe even some recent applications in Algebraic Geometry and Number Theory. The exact emphasis on the material will be decided based on the target audience

Fall Textbooks:

Exercises: The single most important thing a student can do to learn combinatorics is to work out problems. This is more true in this subject than almost any other area of mathematics. Exercises will be assigned each week but many more good problems are to be found, with solutions, in your textbooks. Do as many of them as you can.

Problem sets: Grading will be based mostly on weekly problem sets due on Wednesdays. The problems will come from the text or from additional reading. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. Write up all of your solutions on your own! We will have weekly problem sessions to discuss the harder problems. The time will be determined in the first week of class.

Grading Problem Sets: Students will sometimes be asked to grade another students problem set. More instructions to come with the first assignment.

Optional final: The final exam will be held in the two hour time slot assigned for our class period to be determined. The exam will include both in class work on an open problem and a written component due 24 hours later. More details will be given in class.

Optional presentations: Several recently published research articles will be presented in class by students. You will have the option of doing a presentation sometime this year. If you do a presentation, your lowest homework grade will be dropped.

Computing: Use of computers to verify solutions, produce examples, and prove theorems is highly valuable in this subject. Please turn in documented code if your proof relies on it. If you don't already know a computer language, then try Python, SAGE, Maple, Mathematica or GAP. All are available for free on our math department servers.

### Tentative Schedule:

* Lecture 1: Introduction to combinatorial reasoning, counting functions, combinatorial objects. Generating functions. Read Chapter 1 of EC 1 and other sources. PS#1 handed out.

* Lecture 2: Generating functions. Bijective proofs. Catalan numbers.

* Lecture 3: Basic counting principles. Sets, multisets, * compositions, and combinations

* Lecture 4: 10 ways to describe permutations. PS#2 handed out.

* Lecture 5: 2 more permutation representations: RSK and cycle notation. Stirling numbers of the 1st kind.

### University Policy and Resources:

(there is some new suggested language that I support -- SB)
• Religious Accommodations Policy: “Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW's policy, including more information about how to request an accommodation, is available at Religious Accommodations Policy (https://registrar.washington.edu/staffandfaculty/religious-accommodations-policy/). Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form (https://registrar.washington.edu/students/religious-accommodations-request/).”

• Access and Accommodations: "Your experience in this class is important to me. If you have already established accommodations with Disability Resources for Students (DRS), please communicate your approved accommodations to me at your earliest convenience so we can discuss your needs in this course. If you have not yet established services through DRS, but have a temporary health condition or permanent disability that requires accommodations (conditions include but not limited to; mental health, attention-related, learning, vision, hearing, physical or health impacts), you are welcome to contact DRS at 206-543-8924 or uwdrs@uw.edu or disability.uw.edu. DRS offers resources and coordinates reasonable accommodations for students with disabilities and/or temporary health conditions. Reasonable accommodations are established through an interactive process between you, your instructor(s) and DRS. It is the policy and practice of the University of Washington to create inclusive and accessible learning environments consistent with federal and state law."

• Student Conduct and Academic Integrity Code: "The University of Washington Student Conduct Code (WAC 478-121) defines prohibited academic and behavioral conduct and describes how the University holds students accountable as they pursue their academic goals. Allegations of misconduct by students may be referred to the appropriate campus office for investigation and resolution. More information can be found online at https://www.washington.edu/studentconduct/"

• Safety: "Call SafeCampus at 206-685-7233 anytime – no matter where you work or study – to anonymously discuss safety and well-being concerns for yourself or others. SafeCampus's team of caring professionals will provide individualized support, while discussing short- and long-term solutions and connecting you with additional resources when requested."

Sara Billey