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GRAPHS AND GROUPS, REPRESENTATIONS AND RELATIONS (G2R2-2018)

1–22 AUGUST 2018

NOVOSIBIRSK

RUSSIA

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Who are we?

Novosibirsk State University is located at the heart of Akademgorodok, a world-famous research center of the Russian Academy of Sciences. It was founded in the 1960s in the Siberian taiga as the birth place for new ideas and technologies. Now it's a world-class research and educational cluster with 35 research institutes, innovative Technopark and the University at the heart of this well-balanced system.

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About the course

This course is for those who are interested in graph theory and group theory with applications in other fields of science especially those involving group actions on combinatorial objects. We provide basic knowledge and explore modern trends in graph theory, algebraic graph theory, algebraic combinatorics, group theory, representation of finite groups, association schemes. We explain approaches and methods of research. We show universal connections with problems in geometry and topology, computer science and networks, chemistry and biology.

The list of participants |

The list of participants of G2R2-Siberian Summer School

« Graphs and Groups, Representations and Relations»

« Graphs and Groups, Representations and Relations»

Program

If you wish to travel to Altai please contact the G2R2-organizers by email: g2@math.nsc.ru.

Cultural program

All local tours are free of charge.

Altai is a place where you can conquer high mountain peaks, breathe the crystal-clear air of coniferous forests, follow paths where no one has ever walked before, and see animals that only live here. The Altai Republic is famous for its rivers and lakes, for producing one of the best varieties of honey in Russia and for the most exciting adventures: Russian hunting, rafting on fast rivers, horseback riding and crossing a suspension bridge 15 meters above the Katun River. This is an ancient land where people still live true to their ethnic traditions.

Application requirements

Good command of English is required

final assessment

Students who successfully pass exams, will be given transcripts at the end of the course.

Important dates

Application Start

Application Deadline

Payment Deadline

Program Start

Application is over

Mandatory fees

APPLICATION FEE

Who may apply?

— All students who are currently enrolled at college or university level institutions

— Sufficient English proficiency

— All students who are currently enrolled at college or university level institutions

— Sufficient English proficiency

For all universities

$50

PROGRAM FEE

— Tuition fee

— Study materials

— Cultural activities

— Field trips

— Certificate

Travel, insurance, meals and personal daily expenses should be paid individually

— Study materials

— Cultural activities

— Field trips

— Certificate

Travel, insurance, meals and personal daily expenses should be paid individually

For all universities

$520

Optional fees

DORMITORY FEE

New dormitories are located within 3 minute walk from the University. Students live in block-type dormitories, a block is composed of one or two rooms, plus a bathroom equipped with a shower and a toilet. Single or double room accommodation is available. A furnished kitchen and a leisure room are located on each floor.

Shared Room

$20

RUSSIAN LANGUAGE COURSES

Russian Language Courses provide opportunities both to start learning Russian and/or to improve your language knowledge. Classes are held in groups of 9–15 students (or small groups of 3–8 students) in the afternoon: 4 academic hours per day, 40 hours and 2 ECTS in total.

Small group

$280

Large group

$200

Lecturers

Professor, University of Southampton, UK

Professor, University of West Bohemia, Czech Republic

Professor, Tohoku University, Japan

Professor, Ben-Gurion University of the Negev, Israel

The course considers some applications of group theory to geometry in dimensions 2 and 3. The main theme will be the study of the automorphism groups of compact Riemann surfaces, especially those surfaces uniformised by subgroups of finite index in triangle groups. By Belyi's Theorem, these are the compact Riemann surfaces which, when regarded as complex algebraic curves, can be defined over an algebraic number field. As such, they give a faithful representation of the absolute Galois group (the automorphism group of the field of algebraic numbers), a group of great complexity and importance in algebraic geometry. These surfaces include the Hurwitz surfaces, those attaining Hurwitz's upper bound of 84(g-1) for the size of the automorphism group of a compact Riemann surface of genus g>1. The course will consider the corresponding Hurwitz groups, the finite quotients of the (2,3,7) triangle group, and it will conclude with a brief look at the corresponding situation in dimension 3, where the normaliser of the Coхeter group [3,5,3] plays a similar role.

**Outline of the course: **

Lecture 1. Riemann surfaces and Fuchsian groups.

Lecture 2. Compact Riemann surfaces and their automorphism groups.

Lecture 3. Triangle groups and their quotients.

Lecture 4. Maps and hypermaps on surfaces.

Lecture 5. Dessins d'enfants, and Belyi's Theorem.

Lecture 6. The absolute Galois group, and its action on dessins.

Lecture 7. Hurwitz groups and surfaces.

Lecture 8. Hyperbolic 3-manifolds with large symmetry groups.

**Bibliography**

1. M. D. E. Conder, An update on Hurwitz groups, Groups, Complexity and Cryptology 2 (2010) 25-49.

2. E. Girondo and G. González-Diez, Introduction*to Compact Riemann Surfaces and Dessins d'Enfants*, LMS Student Texts 79, Cambridge University Press, 2012.

3. G. A. Jones, Bipartite graph embeddings, Riemann surfaces and Galois groups, Discrete Math. 338 (2015) 1801-1813.

4. G. A. Jones and D. Singerman,*Complex Functions: An Algebraic and Geometric Viewpoint*, Cambridge University Press, 1986.

5. G. A. Jones and J. Wolfart,*Dessins d'Enfants on Riemann Surfaces*, Springer, 2016. 6. G. A. Jones, *Highly Symmetric Maps and Dessins*, Matej Bel University, 2015.

Lecture 1. Riemann surfaces and Fuchsian groups.

Lecture 2. Compact Riemann surfaces and their automorphism groups.

Lecture 3. Triangle groups and their quotients.

Lecture 4. Maps and hypermaps on surfaces.

Lecture 5. Dessins d'enfants, and Belyi's Theorem.

Lecture 6. The absolute Galois group, and its action on dessins.

Lecture 7. Hurwitz groups and surfaces.

Lecture 8. Hyperbolic 3-manifolds with large symmetry groups.

1. M. D. E. Conder, An update on Hurwitz groups, Groups, Complexity and Cryptology 2 (2010) 25-49.

2. E. Girondo and G. González-Diez, Introduction

3. G. A. Jones, Bipartite graph embeddings, Riemann surfaces and Galois groups, Discrete Math. 338 (2015) 1801-1813.

4. G. A. Jones and D. Singerman,

5. G. A. Jones and J. Wolfart,

A covering between two graphs is a graph epimorphism which is locally bijective. Although the concept of coverings of topological spaces was well known in algebraic topology for a long time, the systematic combinatorial approach to graph coverings is related to the solution of the Heawood map colouring problem by Ringel and Youngs. Nowadays the concept of graph coverings forms an integral part of graph theory and has found dozen of applications, in particular, as a strong construction technique. The aim of the course is to explain foundations of the combinatorial theory of graph coverings and its extension to branched coverings between 1-dimensional orbifolds.

In the course we shall follow the attached plan.

Part 1. Graphs and Groups: graphs with semi-edges and their fundamental groups; actions of groups on graphs; highly symmetrical graphs; subgroup enumeration in some finitely generated groups; enumeration of conjugacy classes of subgroups.

Part 2. Graph coverings and voltage spaces: graph coverings and two group actions on a fibre; voltage spaces; permutation voltage space; Cayley voltage space, Coset voltage space; equivalence of coverings and T-reduced voltage spaces; enumeration of coverings.

Part 3. Applications of graph coverings: regular graphs with large girth; large graphs of given degree and diameter; nowhere-zero flows and coverings; 3-edge colourings of cubic graphs; Heawood map coloring problem.

Part 4. Lifting automorphism problem: classical approach; lifting of graph automorphisms in terms of voltages; lifting problem - case of abelian CT(p); case of elementary abelian CT(p).

Part 5. Branched coverings of graphs: definition and basic properties; Riemann Hurwitz theorem for graphs; Laplacian of a graph and the Matrix-Tree Theorem; Jacobians and harmonic morphisms; graphs of groups, uniformisation.

**Bibliography**

1. Gross, Jonathan L.; Tucker, Thomas W.*Topological graph theory,* Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1987. xvi+351 pp. ISBN: 0-471-04926-3

2. A. Mednykh, R. Nedela,*Harmonic morphisms of graphs*, Part I: Graph Coverings, Matej Bel University, 2015.

In the course we shall follow the attached plan.

Part 1. Graphs and Groups: graphs with semi-edges and their fundamental groups; actions of groups on graphs; highly symmetrical graphs; subgroup enumeration in some finitely generated groups; enumeration of conjugacy classes of subgroups.

Part 2. Graph coverings and voltage spaces: graph coverings and two group actions on a fibre; voltage spaces; permutation voltage space; Cayley voltage space, Coset voltage space; equivalence of coverings and T-reduced voltage spaces; enumeration of coverings.

Part 3. Applications of graph coverings: regular graphs with large girth; large graphs of given degree and diameter; nowhere-zero flows and coverings; 3-edge colourings of cubic graphs; Heawood map coloring problem.

Part 4. Lifting automorphism problem: classical approach; lifting of graph automorphisms in terms of voltages; lifting problem - case of abelian CT(p); case of elementary abelian CT(p).

Part 5. Branched coverings of graphs: definition and basic properties; Riemann Hurwitz theorem for graphs; Laplacian of a graph and the Matrix-Tree Theorem; Jacobians and harmonic morphisms; graphs of groups, uniformisation.

1. Gross, Jonathan L.; Tucker, Thomas W.

2. A. Mednykh, R. Nedela,

Lecturer: Professor Akihiro Munemasa, Tohoku University, Japan

In these lectures, we first introduce the theory of permutation representations of finite groups. The existence of a canonical basis of a permutation module makes it different from a general module, leading to numerical invariants such as Krein parameters. Krein parameters are an analogue of tensor product coefficients for irreducible representations, as seen by Scott's theorem. We then discuss multiplicity-free permutation representations in detail, giving a motivation to a more general concept of commutative association schemes. Lack of a group in the definition leads to slight discrepancy in theory, and a long standing conjecture about splitting fields.

**Outline of the course: **

Lecture 1. Transitive permutation groups and orbitals

Lecture 2. Permutation modules and the centralizer algebra

Lecture 3. Spherical functions and eigenvalues

Lecture 4. The holomorph of a group

Lecture 5. Krein parameters and Scott's theorem

Lecture 6. Association schemes as an abstract centralizer algebra

Lecture 7. Eigenmatrices of association schemes

Lecture 8. Splitting fields of association schemes

**Bibliography**

1. E. Bannai and T. Ito.*Algebraic Combinatorics I: Association Schemes,* Benjamin/Cummings, Menlo Park, 1984.

2. A. Munemasa. Splitting fields of association schemes, J. Combin. Theory, Ser.A, 57 (1991), 157-161.

Lecture 1. Transitive permutation groups and orbitals

Lecture 2. Permutation modules and the centralizer algebra

Lecture 3. Spherical functions and eigenvalues

Lecture 4. The holomorph of a group

Lecture 5. Krein parameters and Scott's theorem

Lecture 6. Association schemes as an abstract centralizer algebra

Lecture 7. Eigenmatrices of association schemes

Lecture 8. Splitting fields of association schemes

1. E. Bannai and T. Ito.

2. A. Munemasa. Splitting fields of association schemes, J. Combin. Theory, Ser.A, 57 (1991), 157-161.

Lecturer: Professor Mikhail Muzychuk, Ben-Gurion University of the Negev, Israel

The goal of my lectures is to give an introduction to the theory of coherent configurations with the main focus on a particular case of association schemes. The closely related objects like Schur rings and table algebras will be presented too. In my lectures I will talk about the structure and representation theories of association schemes. A connection between coherent configurations and permutation groups known as Galois correspondence will be discussed too. Some classical results and new developments in this area with their applications will be presented. I also will remind and discuss some open problems in this area.

**Outline of the course: **

Lectures 1-2-3. Coherent configurations. Association schemes. Schur rings. Table algebras (main definitions and basic properties)

Lecture 4. Galois correspondence between coherent configurations and permutation groups. Schurian coherent configurations and 2-closed permutation groups.

Lecture 5-6. Representation theory of coherent configurations (the semisimple case). Frame number. Applications of representation theory.

Lecture 7. Structure theory of association schemes. Closed subsets, quotients, normal and strongly normal closed subsets.

Lecture 8. Primitive association schemes.

**Bibliography**

1. Z. Arad, E. Fisman, M. Muzychuk, Generalized table algebras, Israel J. of Mathematics, 114(1999), pp. 29-60.

2. A. Hanaki and K. Uno, Algebraic structure of association schemes of prime order, J. Algebr. Comb. 23 (2006), pp.189–195.

3. D. Higman, Coherent algebras, Linear Algebra and Its Applications, 93 (1987), pp. 209-239.

4. B. Wiesfeiler, On construction and identification of graphs, LNM 558, Springer, 1974

5. P.-H. Zieschang, Algebraic approach to association schemes, LNM 1628, Springer, 1996.

Lectures 1-2-3. Coherent configurations. Association schemes. Schur rings. Table algebras (main definitions and basic properties)

Lecture 4. Galois correspondence between coherent configurations and permutation groups. Schurian coherent configurations and 2-closed permutation groups.

Lecture 5-6. Representation theory of coherent configurations (the semisimple case). Frame number. Applications of representation theory.

Lecture 7. Structure theory of association schemes. Closed subsets, quotients, normal and strongly normal closed subsets.

Lecture 8. Primitive association schemes.

1. Z. Arad, E. Fisman, M. Muzychuk, Generalized table algebras, Israel J. of Mathematics, 114(1999), pp. 29-60.

2. A. Hanaki and K. Uno, Algebraic structure of association schemes of prime order, J. Algebr. Comb. 23 (2006), pp.189–195.

3. D. Higman, Coherent algebras, Linear Algebra and Its Applications, 93 (1987), pp. 209-239.

4. B. Wiesfeiler, On construction and identification of graphs, LNM 558, Springer, 1974

5. P.-H. Zieschang, Algebraic approach to association schemes, LNM 1628, Springer, 1996.

Professor, Sobolev Insitute of Mathematics,

Novosibirsk State University

Novosibirsk State University

Program coordinator

Our contacts

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+7 (383) 363-42-92

summerschool@nsu.ru

summerschool@nsu.ru

International Student Recruitment Office

1 Pirogova str., Office 2325

Novosibirsk, Russia, 630090

1 Pirogova str., Office 2325

Novosibirsk, Russia, 630090