Information for potential students – University of Copenhagen

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Department of Mathematical Sciences > Research > Topology, Functional Analysis and Algebra > Topology > Information

Information for potential students of the topology group

Recommended course program 2014/15:

4th year program: A B C
Block 1  AlgTop  (Alg3, AlgGeo) (FunkAn, Riemann S)
Block 2   HomAlg  (Geom2)  GeomGroup
Block 3 

Buildings (IntroOpAlg)

 AlgTop 1.5 (GeomLie)
Block 4 AlgTop 2  (KomAlg)  K-theory


5th year program: A B C
Block 1  (Alg3, AlgGeo) CatTop (FunkAn)
Block 2   (Approx, XM) (Geom2) (GeomGroup)
Block 3  MS THESIS         
Block 4 MS THESIS    

OLD recommended course program 2013/14:

4th year program: A B C
Block 1  AlgTop  (Alg3) (FunkAn)
Block 2   HomAlg  Geom 2  
Block 3  (ModAlg)  AlgTop 1.5 (Lie Groups, IntroOpAlg)
Block 4 AlgTop 2 (AlgTal)  (RepChGr, vNAlg) K-theory


5th year program: A B C
Block 1  CatTop (K-theory 2, Alg3) (FunkAn)
Block 2   (Approx) (KK-theory, Geom2) TopTopics
Block 3  MS THESIS         
Block 4 MS THESIS    

OLD recommended course program 2012/2013:

4th year program:  A B C
Block 1                 AlgTop  (Alg3) (Lie Alg, FunkAn)
Block 2    HomAlg  Geom 2  
Block 3  (GeomRie, KomAlg)  AlgTop 1.5 (IntroOpAlg, AlgTrans)
Block 4 AlgTop 2  (AdvAlgGeo, EllKurv, vNAlg)  K-theory

5th year program: A B C
Block 1 RatHom (Alg3, NCG-TBA) (LieAlg, FunkAn)
Block 2 (Approx) StableHom* (Geom2)  CatTop**
Block 3 MS THESIS    
Block 4 MS THESIS    

* = PhD course on Stable homotopy theory by Markus Szymik (prereq: AlgTop2).
** = Officially canceled.

For a diagram plan of courses at the department of mathematical sciences 2012/13, please see here (slight modifications may occur). The official list can be found at SIS.

Structural description

If you are a bachelors or beginning masters student enrolled at the University of Copenhagen, interested in learning about algebraic topology or working with the topology group this will tell you how to get started.  (For students currently enrolled at other universities interested in pursuing PhD studies with our group, please see here instead.)

As a bachelor student you should take the bachelor course " Topology (Top) " which introduces point-set topology, usually following the book "Munkres: Topology".

In fall of the first year of your masters studies,  you are suggested  to follow the course " Algebraic Topology (AlgTop) " in block 1 (Sept-Nov) and the course " Homological Algebra (HomAlg) " in block 2 (Nov-Jan) along with other courses suiting your direction of specialization, such Geometry 2, Algebra 3, or introduction to operator algebras. The course "Algebraic Topology (AlgTop)"  introduces algebraic topology, often following the book "Algebraic Topology" by Allen Hatcher ,  with the main focus being on homology theory (Chapter 2). Both AlgTop and HomAlg can also be taken by advanced undergraduates.

In block 4 we usually offer " Algebraic Topology II (AlgTopII) ", which is a more thorough introduction to homotopy theory (see Ch 4 of Hatcher's book Algebraic Topology, as well as Ch 1 of Hatcher's online book: Spectral Sequences in Algebraic Topology).  Often a course " Algebraic Topology 1.5 (AlgTop1.5) " is offered in block 3 covering topics around cohomology (as presented in Ch 3 of Hatcher's online book) and a course " K-theory " is offered in block 4 as well.

For 2nd year graduate students and PhD students we aim to offer a course " Categories and Topology (CatTop) " in block 1.  This course builds on AlgTopII, and covers the basics of simplicial homotopy theory and constructions such as homotopy limits and colimits, as e.g., explained in "Bousfield-Kan: Homotopy limits, Completions and Localizations", with an emphasis which varies from year to year. A supplementary advanced topics course TopTopics is aimed offered in block 2. in fall 2012 is was on "Rational Homotopy Theory", and "Stable homotopy theory" and in fall 2013 on "Group actions".

These courses can be followed and supplemented by additional courses, whose exact schedule and contents varies from year to year. Topics can include:

Group cohomology (see eg the book by Kenneth Brown: Group Cohomology)
Topological K-theory (see eg Atiyah's book & Hatcher's online prebook)
Algebraic K-theory (see eg Weibel's online prebook)
Stable homotopy theory (see eg Adams book: Stable homotopy theory ).
Simplicial homotopy theory (Goerss-Jardine)
Model categories (book by Hovey; Dwyer-Spalinski paper, etc)
Equivariant homotopy theory (see eg "Transformation group" by tom Dieck)
Manifold topology, high or low
Homotopy fixed points and unstable homotopy theory (eg book by Lionel Schwartz)

There are also ample opportunities for bachelors theses, masters theses, and PhD studies.

A list of past students/theses can be found here.
A list of suggestions for bachelor projects can be found here.

Topologists often need serious algebra, and students wishing to specialize in topology are also encouraged to seek out courses in topics such as algebraic geometry,Lie theory, and representation theory, offered at the Department of Mathematical Sciences.

General information about current courses at the Department of Mathematicsl Sciences can be found through the "Study Information System (SIS)" .

You are most welcome to drop by any member of the topology group to hear more. See you!