This course will cover modern topics about finite fields:
solutions to polynomial equations, Fourier methods, the sum-product phenomenon, pseudorandomness, expansion, algorithms, and applications to combinatorics, coding theory and theoretical computer science.
graduate level combinatorics, probability, linear algebra, mathematical maturity
Please note: the instructor and course information changes from semester to semester for this course number. Specifics for each semester below.
Bhargav Narayanan Peruvemba
Topics in Combinatorics
This is a topics course in discrete mathematics. Instead of pursuing a single overarching theme, we shall look at a few different problems whose solutions illustrate important techniques in the field. The focus will be on introducing tools from algebra, analysis and probability, and showing how these can be brought to bear on combinatorial problems.
No textbook covers the requisite material. We will borrow from a mixture of research papers, surveys and lecture notes.
The course will be more or less self-contained, but will assume familiarity with undergraduate algebra and probability.*
Algebraic gems in discrete mathematics and theoretical computer science
In the last few decades, algebraic methods have proven to be extremely powerful in several areas of discrete mathematics and theoretical computer science. Many of these important advances in the field have relied on very simple properties of polynomials. In this course we will see many interesting and often surprising applications of linear algebra and polynomials to combinatorics, discrete geometry, complexity theory and algorithm design. We will develop all the algebraic tools that we need along the way.
Basic combinatorics/discrete math, basic linear algebra, mathematical maturity
The term "Extremal Combinatorics" covers many of the most significant discrete developments of recent (and less recent) years, and many of the most interesting open problems. We'll sample some of these, trying to emphasize the wide range of ideas, methods and extra-combinatorial machinery that come into play.
None; relevant books will be on reserve.
There are no formal prerequisites but it will help to have at least seen the combinatorics sequence, 642.582-3. I'll try to fill in background as we go along. Check with me if in doubt.