Klavdija Kutnar (University of Primorska, Koper),

Dragan Marušič (University of Primorska, Koper)

This course is organized in the frame of the international cooperation between Slovenia and Russia in 2014-2015.

In mathematics we usually tend to study structures that admit a certain degree of symmetry. In graphs the degree of symmetry is given by the automorphism group which is the group of all adjacency preserving permutations of its vertex set. In this course we introduce various families of graphs with a rather large degree of symmetry such as vertex-transitive graphs, edge-transitive graphs and arc-transitive graphs. We review some of the methods for constructing such graphs and present some results from the rich theory that has developed in the last few decades. We also present some open problems in the area.

It contains 8 lectures.

Tomaž Pisanski (University of Ljubljana, Ljubljana, University of Primorska, Koper)

This course is organized in the frame of the international cooperation between Slovenia and Russia in 2014-2015.

In this course we will learn basics of Python and Sage that will enable participants to start exploring non-trivial questions about symmetries of graphs. We will construct some bi-Cayley graphs, such as Haar graphs, rose-window graphs, I-graphs and their generalizations. We will also analyze some existing censuses of graphs and related structures, such as maps and polytopes. Each participant is expected to posses basic skills of computer programming and have his or her own lap-top available. Each registered participant will receive a handout, including relevant references and tutorials, prior to the beginning of this minicourse. A list of questions, ranging from simple exercises that will enable participants to recall the learned skills, to non-trivial mathematical problems will be distributed.

It contains 4 lectures.

Nadezhda Timofeeva (Yaroslavl State University, Yaroslavl)

A starting point was the paper of 1979 by J.H. Conway and S.P. Norton entitled "Monstrous Moonshine". It comprises several seeming-coincidences relating the Monster group (in that time its existence was only conjectured) to modular forms. Since this original paper many more connections of modular forms to sporadic simple groups were discovered. They all are collectively referred to as Moonshine. In 1998 R. Borcherds won the Fields medal in part for his work where he proved the original conjectures of J.H. Conway and S.P. Norton. The proof opened connections of the representation theory and the theory of modular forms to mathematical physics. In my lectures I will try to explain the basic notions and to describe the key moments of the Moonshine in its classical version. If the time permit, I will sketch some generalizations and say several words about open problems.

It contains 2 lectures.

Evgeny Vdovin (Sobolev Institute of Mathematics, Novosibirsk, NSU, Novosibirsk)

In the lectures we plan to discuss general methods for answering to the following problems: whether given finite group possesses a $\pi$-Hall subgroup for a set of primes $\pi$, and how many classes of conjugate $\pi$-Hall subgroups the group has. One of the main technical tool is the notion of a group of induced automorphisms and the inclusion to the wreath product with this group (see [1], and theorem 3 from this paper). We recommend the attendants to read paper [2] also (at least the main part without Appendix).

It contains 2 lectures.

1. E.P.Vdovin, Groups of induced automorphisms and their application to studying the existence problem for Hall subgroups // Algebra and logic, 2014, 53, 5, 418--421. doi:10.1007/s10469-014-9301-x

2. E.P.Vdovin, D.O.Revin, Theorems of Sylow type, Russian Math. Surveys vol. 66 (2011), \No 5 829--870.

Mikhail Volkov (UrFU, Yekaterinburg)

Most current mathematical research, since the 60s, is devoted to fancy situations: it brings solutions which nobody understands to questions nobody asked (quoted from Bernard Beauzamy, "Real life Mathematics", Irish Math. Soc. Bull. 48 (2002), 43-46). This provocative claim is certainly exaggerated but it does reflect a really serious problem: a communication barrier between mathematics (and exact science in general) and human common sense. The barrier results in a paradox: while the achievements of modern mathematics are widely used in many crucial aspects of everyday life, people tend to believe that today mathematicians do "abstract nonsense" of no use at all. In most cases it is indeed very difficult to explain to a non-mathematician what mathematicians work with and how their results can be applied in practice. Fortunately, there are some lucky exceptions, and one of them has been chosen as the present course's topic. It is devoted to a mathematical problem that was frequently asked by both theoreticians and practitioners in many areas of science and engineering. The problem, usually referred to as the synchronization problem, can be roughly described as the task of determining the simplest way to restore control over a device whose current state is not known - think of a satellite which loops around the Moon and cannot be controlled from the Earth while "behind" the Moon. While easy to understand and practically important, the synchronization problem turns out to be surprisingly hard to solve even for finite automata that constitute the simplest mathematical model of real-world devices. This combination of transparency, usefulness and unexpected hardness should, hopefully, make the course interesting for a wide audience.

Among other things, the course will present a recent major advance in the theory of synchronizing finite automata: Avraam Trahtman's proof of the so-called Road Coloring Conjecture by Adler, Goodwyn, and Weiss. The conjecture that admits a formulation in terms of recreational mathematics arose in symbolic dynamics and has important implications in coding theory. The proof is elementary in its essence but clever and enjoyable.

It contains 3 lectures.