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

SIBERIAN SUMMER SCHOOL
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»
1 Aljohani Mohammed, Taibah University, Saudi Arabia
2 Baykalov Anton, The University of Auckland, New Zealand
3 Berikkyzy Zhanar, University of California, USA
4 Cho Eun-Kyung, Pusan National University, South Korea
5 Chen Huye, China Three Gorges University, China
6 Churikov Dmitry, Novosibirsk State University, Russia
7 Dogra Riya, Shiv Nadar University, India
8 El Habouz Youssef, University ibn Zohr, Morocco
9 Evans Rhys, Queen Mary University of London, UK
10 Fu Zhuohui, Northwestern Polytechnical University,China
11 Jin Wanxia, Northwestern Polytechnical University,China
12 Kim Jan, Pusan National University, South Korea
13 Kaushan Kristina, Novosibirsk State University, Russia
14 Khomyakova Ekaterina, Novosibirsk State University, Russia
15 Konstantinov Sergey, Novosibirsk State University, Russia
16 Kwon Young Soo, Yeungnam University, Korea
17 Lin Boyue, Northwestern Polytechnical University,China
18 Mattheus Sam, Vrije Universiteit Brussel, Belgium
19 Mednykh Ilya, Novosibirsk State University, Russia
20 Morales Ismael, Autonomous University of Madrid, Spain
21 Puri Akshay A, Shiv Nadar University, India
22 Qian Chengyang, Shanghai Jiao Tong University, China
23 Ryabov Grigory, Novosibirsk State University, Russia
24 Song Mengmeng, Northwestern Polytechnical University,China
25 Sotnikova Ev, Sobolev Institute of Mathematics, Russia
26 Smith Dorian, USA
27 Vuong Bao, Novosibirsk State University, Russia
28 Wang Guanhua, Northwestern Polytechnical University,China
29 Wang Hui, Northwestern Polytechnical University,China
30 Wang Jingyue, Northwestern Polytechnical University,China
31 Xiong Yanzhen, Shanghai Jiao Tong University, China
32 Xu Zeying, Shanghai Jiao Tong University, China
33 Yang Zhuoke, Moscow Institute of Physics and Technology, Russia
34 Yin Yukai, Northwestern Polytechnical University,China
35 Yu Tinzoe, Hebei Normal University, China
36 Zhang Yue, Northwestern Polytechnical University,China
37 Zhao Da, Shanghai Jiao Tong University, China
38 Zhao Yupeng, Northwestern Polytechnical University,China
39 Zhu Yan, Shanghai University, China
40 Zhu Yinfeng, Shanghai Jiao Tong University, China
Program
Arrival
August 1
Cultural program
Tuition
August 6-19
TimeTable
Exam
August 20-22
There is an alternative cultural program on the period of August, 3-5, for participants are desirous to take a 3-days Altai tour. The tour fee is 13200 Rub (except for the train from Novosibirsk to Biysk and back). The trip covers the most beautiful places of Altai region. All information and even a detailed map you can find via the link/
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.
3-Days trip to Altai
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
Undergraduates and postgraduates students from any institutions worldwide
Basic knowledge in graph theory, group theory, combinatorics, topology
Good command of English is required
final assessment
At the end of the course students will either pass a written examination or present a talk at the G2R2-conference. Everyone will be given a certificate of attendance.
Students who successfully pass exams, will be given transcripts at the end of the course.
Important dates
January 1
Application Start
May 25
Application Deadline
June 5
Payment Deadline
August 1
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
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
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
Per Course
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
Per course
Large group
$200
Per course
Lecturers
Gareth Jones
Professor, University of Southampton, UK
Roman Nedela
Professor, University of West Bohemia, Czech Republic
Akihiro Munemasa
Professor, Tohoku University, Japan
Mikhail Muzychuk
Professor, Ben-Gurion University of the Negev, Israel
Groups and symmetry in low dimensional geometry and topology (12 hours)
Lecturer: Gareth Jones, University of Southampton, UK
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.

Graph coverings and harmonic morphisms between graphs (12 hours)
Lecturer: Professor Roman Nedela, University of West Bohemia, Czech Republic
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.

Permutation representations of finite groups and association schemes (12 hours)
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.
Coherent configurations and association schemes: structure theory and linear representations (12 hours)
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.
Elena Konstantinova
Professor, Sobolev Insitute of Mathematics,
Novosibirsk State University
Program coordinator
Our contacts
Feel free to write and call us
+7 (383) 363-42-92
summerschool@nsu.ru
International Student Recruitment Office
1 Pirogova str., Office 2325
Novosibirsk, Russia, 630090
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