# Information for potential students of the topology group

**Recommended course program 2018/19: **

4th year program: | A | B | C |

Block 1 | AdVec | (Alg3) | AlgTop |

Block 2 | (FunkAn) | Geom2 | HomAlg |

Block 3 |
(IntroRep, IntroOpAlg) |
AlgTop 1.5 (EllCurv) | (ComAlg, Diffun) |

Block 4 | (AlgNT) | AlgTopII | (AlgGeo, K-theory) |

_________

5th year program: | A | B | C |

Block 1 | (Approx) | CatTop | (AlgGeo2, AnMan) |

Block 2 | (FunkAn, TopicsOA) | (TopicsAlgNT, NCG) | TopTopics |

Block 3 | MS THESIS | ||

Block 4 | MS THESIS |

# 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 "**A****lgebraic 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 Functional Analysis. 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.

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). 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).

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,** which is a "seminar course" where participants take turn presenting research papers to the group. This is meant as a course presenting students to research, and can lead MS students to MS theses in block 3+4 and at the same time give PhD students a general background in research topics more advanced than the regular course sequence.

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 UCPH course catalogue.

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