**August 06-19, 2018** - Novosibirsk, Russia

G2R2-Summer School is presented by four courses each of which contains eight 50-minutes
lectures.

G2R2-Summer School Timetable see here.

**Title:** Groups and symmetry in low dimensional geometry and topology (12 hours)

**Description:**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 Coxeter group [3,5,3] plays a similar role.

**Outline of the course:**

Lecture 1. Riemann surfaces and Fuchsian groups. (pdf)

Lecture 2. Compact Riemann surfaces and their automorphism groups. (pdf)

Lecture 3. Triangle groups and their quotients. (pdf)

Lecture 4. Maps and hypermaps on surfaces. (pdf)

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

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

Lecture 7. Hurwitz groups and surfaces. (pdf, Group notes)

Lecture 8. Hyperbolic 3-manifolds with large symmetry groups. (pdf)

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

2. E. Girondo and G. Gonzalez-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.

University of Southampton, UK

**Group and symmetry in low dimensional geometry and topology**

**Title:** Permutation representations of finite groups and association schemes (12 hours)

**Description:** 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 (pdf)

Lecture 2. Permutation modules and the centralizer algebra (pdf)

Lecture 3. Spherical functions and eigenvalues (pdf)

Lecture 4. The holomorph of a group (pdf)

Lecture 5. Krein parameters and Scott's theorem (pdf)

Lecture 6. Association schemes as an abstract centralizer algebra (pdf)

Lecture 7. Eigenmatrices of association schemes (pdf)

Lecture 8. Splitting fields of association schemes (pdf)

**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.

Tohoku University, Japan

**Permutation representations of finite groups and association schemes**

**Title:** Coherent configurations and association schemes: structure theory and linear representations (12 hours)

**Description:** 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:**

Lecture 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.

Lecture 1-2

Lecture 3-4

Lecture 5-6

Lecture 7-8

**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.

Ben-Gurion University of the Negev, Israel

**Coherent configurations and association schemes: structure theory and linear representations**

**Title:** Graph coverings and harmonic morphisms between graphs (12 hours)

**Description:** 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.

**Outline of the course:**

__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); 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 and uniformisation.

Graph Coverings 1 (pdf)

Graph Coverings 2 (pdf)

Graph Coverings 3 (pdf)

Graph Coverings 4 (pdf)

Graph Coverings 5 (pdf)

Graph Coverings 6 (pdf)

Graph Coverings 7 (pdf)

**Bibliography**

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

University of West Bohemia, Czech Republic

**Graph coverings and harmonic morphisms between graphs**

G2R2-summer school is included into the program of Siberian Summer Schools of Novosibirsk State University. In the frame of this program, G2R2-summer school is extended by a cultural program including local tours and a day trip to Altai. Novosibirsk State University awards 10 scholarships for this Summer School Program. At the end of the G2R2-summer school students will either pass a written examination or present a talk at the G2R2-conference. Everyone will be given a certificate of attendance. Please visit G2R2-Siberian Summer School to get more information.

**Aljohani Mohammed**, Taibah University, Saudi Arabia

**Baykalov Anton**, The University of Auckland, New Zealand

**Berikkyzy Zhanar**, University of California, USA

**Cho Eun-Kyung**, Pusan National University, South Korea

**Chen Huye**, China Three Gorges University, China

**Churikov Dmitry**, Novosibirsk State University, Russia

**Dogra Riya**, Shiv Nadar University, India

**El Habouz Youssef**, University ibn Zohr, Morocco

**Evans Rhys**, Queen Mary University of London, UK

**Fu Zhuohui**, Northwestern Polytechnical University, China

**Jin Wanxia**, Northwestern Polytechnical University, China

**Kim Jan**, Pusan National University, South Korea

**Kaushan Kristina**, Novosibirsk State University, Russia

**Khomyakova Ekaterina**, Novosibirsk State University, Russia

**Konstantinov Sergey**, Novosibirsk State University, Russia

**Kwon Young Soo**, Yeungnam University, Korea

**Lin Boyue**, Northwestern Polytechnical University, China

**Mattheus Sam**, Vrije Universiteit Brussel, Belgium

**Mednykh Ilya**, Novosibirsk State University, Russia

**Morales Ismael**, Autonomous University of Madrid, Spain

**Puri Akshay A.**, Shiv Nadar University, India

**Qian Chengyang**, Shanghai Jiao Tong University, China

**Ryabov Grigory**, Novosibirsk State University, Russia

**Song Meng Meng**, Northwestern Polytechnical University, China

**Sotnikova Ev**, Sobolev Institute of Mathematics, Russia

**Smith Dorian**, USA

**Vuong Bao**, Novosibirsk State University, Russia

**Wang Guanhua**, Northwestern Polytechnical University, China

**Wang Hui**, Northwestern Polytechnical University, China

**Wang Jingyue**, Northwestern Polytechnical University, China

**Xiong Yanzhen**, Shanghai Jiao Tong University, China

**Xu Zeying**, Shanghai Jiao Tong University, China

**Yang Zhuoke**, Moscow Institute of Physics and Technology, Russia

**Yin Yukai**, Northwestern Polytechnical University, China

**Yu Tinzoe**, Hebei Normal University, China

**Zhang Yue**, Northwestern Polytechnical University, China

**Zhao Da**, Shanghai Jiao Tong University, China

**Zhao Yupeng**, Northwestern Polytechnical University, China

**Zhu Yan**, Shanghai University, China

**Zhu Yinfeng**, Shanghai Jiao Tong University, China

G2R2-Siberian Summer School: August 1-22, 2018

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