Minicourse I: Graphs and their automorphism groups (abstract,
slides)
Lecturers:
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.
Minicourse II: Symmetries in Graphs with Python and Sage (abstract,
slides)
Lecturer:
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.
Minicourse III: Monstrous Moonshine (abstract,
slides)
Lecturer:
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.
Minicourse IV: Existence and conjugacy of Hall subgroups. Contemporary progress and open problems
(abstract,
slides)
Minicourse V: Synchronizing finite automata: a problem everyone
can understand but nobody can solve (so far) (abstract,
slides)
Lecturer:
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.
