Walk through Combinatorics, Fall 2018

Crossroads
Highly non-parallel roads

When:

Mondays, Wednesdays, Fridays 9:30

Where:

Posner Hall 147 [map]

What:

If you ever stumbled upon a mathematical object that is finite and is fun to play with, you must have wandered into the land of combinatorics (also known as discrete mathematics). You might have walked there through one of the many well-trodden roads from logic and computer science, or were yanked through some of the recently-discovered wormholes from number theory or topology. Even if you have not been there, you still might want a guide.

In this course we will explore this land, and learn of paths that connect its different parts. We will focus on three areas:

  1. Ramsey theory, where we learn how to find order in the utter chaos. Here the main results are the pigeonhole principle, Ramsey's theorem, theorems of van der Waerden and Hales–Jewett. Much of the time will be devoted to their applications.
  2. Enumerative combinatorics, where we learn how to count not only trees, and forests, but also walks, and cycles. Here we encounter clever bijections, Möbius inversion formula, generating functions, Cauchy's theorem as a tool in asymptotic analysis, as well as a multitude of combinatorial objects to count.
  3. Probabilistic method, where we learn how use the power of randomness to conjure mathematical objects that are both simple and exotic. Our main helpers will be linearity of expectations, second moment method, alteration tricks, Lovász local lemma, and concentration inequalities. We will also see and use the famed Szemerédi regularity lemma.

Resources:

Due to the variety of topics, there is no single book that covers everything. Some good written resources covering parts of the course are

Links to additional resources will be posted as the course progresses.

More fun:

More fun can be had at my office hours on Wednesdays 1:30–2:30pm and Fridays 2:30–3:30pm in Wean 6202. I am also available by appointment.

Course activities:

Mastery of combinatorics requires practice. Hence, there will be regular homeworks. For your own good, you are strongly encouraged to do as much homework as possible individually. Collaboration and use of external sources are permitted, but must be fully acknowledged and cited. All the writing must be done individually. Failure to do so will be treated as cheating. Collaboration may involve only discussion; all the writing must be done individually. The homeworks will be returned one week after they are due.

Students are expected to fully participate in the class. Discussions during the lectures are encouraged.

The homework will count for 40% of the grade. There will a mid-term on October 15, which will count for 25% of the grade. There will be a final exam (on December 11 at 5:30pm) which will count for the remaining 35% of the grade.

Homework must be submitted in LaTeX via e-mail. I want both the LaTeX file and the PDF that is produced from it. The filenames must be of the form lastname_discr_homeworknumber.tex and lastname_discr_homeworknumber.pdf respectively. Pictures do not have to be typeset; a legible photograph of a hand-drawn picture is acceptable.

The homework must be submitted by 9:30am of the day it is due. For each minute that it is late, the assignment grade will be reduced by 10%.

There will be opportunities for extra credit. Unusually insightful solutions, and other achievements will be appropriately rewarded.

Staying sane and healthy:

This is a graduate course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.

Lectures:

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