Search result: Catalogue data in Spring Semester 2017
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
|Electives: Pure Mathematics|
|401-4206-17L||Group Actions on Trees||W||4 credits||2V||N. Lazarovich|
|Abstract||As a main theme, we will explain how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure. After introducing the general theory, we will cover various topics in this general theme.|
|Objective||Introduction to the general theory of group actions on trees, also known as Bass-Serre theory, and various important results on decompositions of groups.|
|Content||Depending on time we will cover some of the following topics.|
- Free groups and their subgroups.
- The general theory of actions on trees, i.e, Bass-Serre theory.
- Trees as 1-dimensional buildings.
- Stallings' theorem.
- Grushko's and Dunwoody's accessibility results.
- Actions on R-trees and the Rips machine.
|Literature||J.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9|
C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
|Prerequisites / Notice||Familiarity with the basics of fundamental group (and covering theory).|
|401-4148-17L||Moduli of Maps and Gromov-Witten invariants||W||2 credits||4A||G. Bérczi|
|Abstract||Enumerative questions motivated the development of algebraic geometry for centuries. This course is a short tour to some ideas which have revolutionised enumerative geometry in the last 30 years: stable maps, Gromov-Witten invariants and quantum cohomology.|
|Objective||The aim of the course is to understand the concept of stable maps, their moduli and quantum cohomology. We prove Kontsevich's celebrated formula on the number of plane rational curves of degree d passing through 3d-1 given points in general position.|
1) Brief survey on moduli spaces: fine and coarse moduli.
2) Stable n-pointed curves
3) Stable maps
4) Enumerative geometry via stable maps
5) Gromov-Witten invariants
6) Quantum cohomology and quantum product
7) Kontsevich's formula
|Literature||The main reference for the course is:|
J. Kock and I.Vainsencher: Kontsevich's Formula for Rational Plane Curves
-Algebraic varieties: I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
-Moduli of curves: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag
-Moduli spaces (fine and coarse): Peter. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag
|Prerequisites / Notice||Some minimal background in algebraic geometry (varieties, line bundles, Grassmannians, curves).|
Basic concepts of moduli spaces (fine and coarse) and group actions will be explained mainly through examples.
|401-3056-00L||Finite Geometries I|
Does not take place this semester.
|W||4 credits||2G||N. Hungerbühler|
|Abstract||Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.|
|Objective||Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.|
|Content||Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design|
|Literature||- Max Jeger, Endliche Geometrien, ETH Skript 1988|
- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983
- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press
- Dembowski: Finite Geometries.
|401-3574-61L||Introduction to Knot Theory |
Does not take place this semester.
|Abstract||Introduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.|
|Objective||The aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.|
|Content||Definition of a knot and of equivalent knots.|
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)
|Literature||An extensive bibliography will be handed out in the course.|
|Prerequisites / Notice||Prerequisites are some elementary knowledge of algebra and topology.|
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