Topics in Topology A (22/23)
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WMMA03405 
Vaknaam  Topics in Topology A (22/23) 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester II a 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Topics in Topology A (tweejaarlijks 22/23)  
Leerdoelen  At the end of the course, the student is able to: 1. Reproduce key properties and examples of topological invariants of knots and their relation in other fields of mathematics. 2. Apply key concepts, such as basic representation theory to solve problems in knot theory. 3. Study a subject from the field of lowdimensional topology at research level by her/himself guided by the lecturer. 4. Give a professional oral presentation on a specific subject in topology and explain the subject to his/her peers (which includes answering questions). 

Omschrijving  We will give an introduction to algebraic methods in knot theory and lowdimensional topology. Take a piece of string and tie a knot. Can you tell what kind of knot it is just by looking at it? What are its properties? We will turn any picture of a knot into a computation in such a way that the final answer only depends on the knot itself. Such a computation is called a knot invariant. Knot invariants are central to lowdimensional topology and serve to connect this field with other areas of mathematics in unexpected ways. Guided by the close analogies between pictures of knots and the representation theory of algebras we will provide a general framework for knot invariants. Famous examples such as the Jones polynomial and the Alexander polynomial come up naturally this way. The relevant algebras turn out to be Hopf algebras with close ties to the theory of Lie groups and quantum physics. Dealing with such algebras directly is challenging but can done either through studying their representations or by turning linear maps into power series. 

Uren per week  
Onderwijsvorm  Bijeenkomst (S), Hoorcollege (LC), Opdracht (ASM)  
Toetsvorm 
Opdracht (AST), Practisch werk (PR), Presentatie (P)
(Presentation 25% Practical Work 10% (Active participation in the seminar by regular attendance and asking questions and making remarks in the discussions. Also suggesting additional material and solving exercises in class.) Biweekly homework 65%) 

Vaksoort  master  
Coördinator  dr. R.I. van der Veen  
Docent(en)  dr. R.I. van der Veen  
Verplichte literatuur 


Entreevoorwaarden  Knowledge assumed: Introduction to metric and topological spaces, linear algebra, group theory, multivariable analysis. Analysis on manifolds and, advanced algebraic structures are recommended but not required. 

Opmerkingen  
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