Benjamin Sudakov: Catalogue data in Autumn Semester 2019
|Prof. Dr. Benjamin Sudakov
Institut für Operations Research
ETH Zürich, HG G 65.1
|+41 44 632 40 28
|Seminar in Theoretical Computer Science
|A. Steger, B. Gärtner, M. Ghaffari, M. Hoffmann, J. Lengler, D. Steurer, B. Sudakov
|Presentation of recent publications in theoretical computer science, including results by diploma, masters and doctoral candidates.
|The goal is to introduce students to current research, and to enable them to read, understand, and present scientific papers.
|Prerequisites / Notice
|This seminar takes place as part of the joint research seminar of several theory groups. Intended participation is for students with excellent performance only. Formal minimal requirement is passing of one of the courses Algorithms, Probability, and Computing, Randomized Algorithms and Probabilistic Methods, Geometry: Combinatorics and Algorithms, Advanced Algorithms. (If you cannot fulfill this restriction, because this is your first term at ETH, but you believe that you satisfy equivalent criteria, please send an email with a detailed description of your reasoning to the organizers of the seminar.)
|Algebraic Methods in Combinatorics
|2V + 1U
|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. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas.
|The students will get an overview of various algebraic methods for solving combinatorial problems. We expect them to understand the proof techniques and to use them autonomously on related problems.
|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.
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.
The course website can be found at
|Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely.
|Prerequisites / Notice
|Students are expected to have a mathematical background and should be able to write rigorous proofs.