### Research interests

- Computational discrete mathematics and its applications.
- Abstract algebra, combinatorics, optimisation and search.
- Open-source mathematical software development.
- Parallel computing in a variety of settings.
- Recomputable scientific experiments.

**Group Rings**

In my PhD thesis, I proved that the normalized unit group of the modular group algebra of a 2-group of maximal class *G* has a section isomorphic to the wreath product *C _{2 }*wr

*G’*of the cyclic group of order 2 and the derived group

*G’*of

*G*, giving for such groups a positive answer on a question formulated by Aner Shalev. Later I gave a construction of the wreath product

*C*wr

_{2 }*G’*for another class of 2-groups.

Later I became also interested in the **Modular Isomorphism Problem**.

My current interests in the area of group rings are concentrated around torsion units of integral group rings of finite groups. The long-standing conjecture of Hans Zassenhaus **(ZC-1)** says that every torsion unit in the integral group ring of the finite group *G* is rationally conjugate to an element in *G*. Wolfgang Kimmerle proposed to relate **(ZC)** with some properties of graphs associated with groups. The Gruenberg – Kegel graph (or the prime graph) of *G* is the graph with vertices labelled by the prime divisors of the order of *G *with an edge from *p* to *q* if and only if there is an element of order *pq* in the group *G*. Then Kimmerle’s conjecture **(KC)** asks whether *G* and the notmalized unit group of its integral group ring have the same prime graph.

Jointly with Victor Bovdi, we started the program of verifying **(KC) **for sporadic simple groups. Currently we are able to report on the checking **(KC)** for the following 13 out of 26 sporadic simple groups:

- Mathieu groups
,*M*_{11},*M*_{12},*M*_{22},*M*_{23}*M*_{24} - Janko groups
,*J*_{1},*J*_{2}*J*_{3} - Higman-Sims group
*HS* - McLaughlin group
*McL* - Held group
*He* - Rudvalis group
*Ru* - Suzuki group
*Suz*