# Algebra - Mathematical Sciences - Department of Mathematical Sciences

# Algebra

The group's research activity is mostly oriented towards theoretical studies of algebraic questions, but members of the group also work on more applied topics related to cryptography.

## Research activity

The research activity within the theoretical side of the algebra group consists mainly of

- representation theory of algebras
- homological algebra, including Hochschild cohomology
- triangulated and derived categories
- Lie algebras
- commutative algebra
- algebraic geometry
- connections between geometry and representation theory

A part of the group has been working on topics related to **cluster algebras**, which was introduced a few years ago and have many interesting connections to different parts of algebra and other areas of mathematics. Hochschild cohomology, Koszul algebras, support varieties of module categories and triangulated categories (and connections between these) are all active research topics today.

### Cryptography

The cryptographic research consists mostly of **cryptographic protocol analysis**.

One project studies anonymous communications and payment, which also involves theoretical work on how to prove protocols secure and how to increase confidence in the correctness of those proofs.

Another more applied line of research relates to electronic voting and identification, where we have contributed analysis of (to-be-)deployed systems.

Members of the group have also worked on the application of** elliptic curves** to cryptography, most recently on the use of bilinear pairings in cryptography.

## Contact

## Seminars

## Graduate courses in algebra

#### Bachelor and Master Level Courses

- MA3201 Rings and Modules
- MA3202 Galois Theory
- MA3203 Ring Theory
- MA3204 Homological Algebra
- TMA4150 Algebra
- TMA4160 Cryptography
- TMA4185 Coding Theory

#### PhD Level Courses

## Projects

### Current projects

- Secure, Usable and Robust Cryptographic Voting Systems (NFR, 2018–2022)
- Representation theory via subcategories (NFR, 2016–2020)

### Completed projects

- Clusters, combinatorics and computations in algebra (NFR, 2013–2018)
- Triangulated Categories in Algebra (NFR, 2013–2017)
- Combinatorial and Categorical Methods in Representation Theory (NFR, 2009–2015)
- Privacy-preserving Seamless Digital Infrastructures (NFR, 2008–2011)
- HoGeMetAlg (NFR, 2005–2009)
- LieGrits (EU, 2004–2008)

## Conferences and workshops

**Upcoming**

**Previously held**

- Conference for Idun Reiten, 10-13. May 2017
- GAP Days 2015, Fall
- Cluster categories and cluster tilting
- Abelsymposium 2011: Algebras, Quivers and Representations
- Homological and geometric methods in algebra, 10-14. August 2009
- Perspectives in Auslander-Reiten Theory, 10-12. May 2007
- Conference for Idun Reiten, 1-2. March 2002
- Nordfjordeid, summer school 2001
- ICRA VIII, 1996

## Recent publications

Aslak Bakke Buan

**From classical tilting to two-term silting**

*Science China Mathematics*

Lars Winther Christensen, Sergio Estrada and Peder Thompson

**Homotopy categories of totally acyclic complexes with applications to the flat-cotorsion theory**

*Contemporary Mathematics*

Katriel Cohn-Gordon, Cas Cremers, Kristian Gjøsteen, Håkon Jacobsen and Tibor Jager

**Highly Efficient Key Exchange Protocols with Optimal Tightness**

*Lecture Notes in Computer Science*

Karin Erdmann, Øyvind Solberg and Xingting Wang

**On the structure and cohomology ring of connected Hopf algebras**

*Journal of Algebra*

Steffen Oppermann, Chrysostomos Psaroudakis and Torkil Utvik Stai

**Change of rings and singularity categories**

*Advances in Mathematics*

Øyvind Solberg, Henning Krause, Aslak Bakke Buan and Nicole Snashall

**Support varieties—an axiomatic approach**

*Mathematische Zeitschrift*