# 401-3055-64L  Algebraic Methods in Combinatorics

 Semester Herbstsemester 2021 Dozierende B. Sudakov Periodizität 2-jährlich wiederkehrende Veranstaltung Lehrsprache Englisch

### Lehrveranstaltungen

NummerTitelUmfangDozierende
401-3055-64 VAlgebraic Methods in Combinatorics2 Std.
 Mi 10:15-12:00 IFW A 36 »
B. Sudakov
401-3055-64 UAlgebraic Methods in Combinatorics1 Std.
 Mo 12:15-13:00 ML F 34 » 13:15-14:00 ML F 34 »
B. Sudakov

### Katalogdaten

 Kurzbeschreibung 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. Lernziel 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. Inhalt 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 https://moodle-app2.let.ethz.ch/course/view.php?id=15757 Skript Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely. Voraussetzungen / Besonderes Students are expected to have a mathematical background and should be able to write rigorous proofs.

### Leistungskontrolle

 Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird) Leistungskontrolle als Semesterkurs ECTS Kreditpunkte 6 KP Prüfende B. Sudakov Form Sessionsprüfung Prüfungssprache Englisch Repetition Die Leistungskontrolle wird in jeder Session angeboten. Die Repetition ist ohne erneute Belegung der Lerneinheit möglich. Prüfungsmodus schriftlich 180 Minuten Hilfsmittel schriftlich Students are allowed to bring ONLY a printed copy of the lecture notes with no extra writing (highlighting and blank post-its are allowed). Diese Angaben können noch zu Semesterbeginn aktualisiert werden; verbindlich sind die Angaben auf dem Prüfungsplan.

### Lernmaterialien

 Hauptlink Moodle of the course Moodle-Kurs Moodle-Kurs / Moodle course Es werden nur die öffentlichen Lernmaterialien aufgeführt.

### Gruppen

401-3055-64 UAlgebraic Methods in Combinatorics
GruppenG-01
 Mo 12:15-13:00 ML F 34 »
G-02
 Mo 13:15-14:00 ML F 34 »

### Einschränkungen

 Keine zusätzlichen Belegungseinschränkungen vorhanden.

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