Algebraic Methods in Combinatorics
Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, well-developed tools.
One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications.
One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications.
Lecturer
Benjamin Sudakov
Date
19th September ~ 19th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 15:20 - 16:55 | Online | ZOOM 12 | 815 762 8413 | BIMSA |
Prerequisite
It might be helpful to have basic knowledge of combinatorics, linear algebra and polynomials, but all relevant notions will be defined in the class.
Syllabus
This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include (but are not limited to):
Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others.
Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others.
Audience
Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Benny Sudakov received his PhD from Tel Aviv University in 1999. He had appointments in Princeton University, the Institute for Advanced Studies and in University of California at Los Angeles. Sudakov is currently professor of mathematics in ETH, Zurich. He is the recipient of a Sloan Fellowship, NSF CAREER Award, Humboldt Research Award, is Fellow of the American Math. Society and was invited speaker at the 2010 International Congress of Mathematicians. He authored more than 300 scientific publications and is on the editorial board of 14 research journals. His main scientific interests are combinatorics and its applications to other areas of mathematics and computer science.