Minicourse 1 (MC1)
Title: Using the GAP system and its packages for research in graph theory, design theory, and finite geometry (8 hours)
Lecturer: Professor Leonard Soicher, Queen Mary University of London, UK
Description: GAP is an internationally developed, open source, freely available system for computational algebra and discrete mathematics, and is used in research for studying groups and their representations, algebras, combinatorial structures, and more. See www.gap-system.org. GAP packages (included with GAP) provide additional specialist functionality to the main GAP system. In this course, I will give advanced mathematics students an introduction to GAP and its packages GRAPE (for computing with graphs together with groups acting on them), Digraphs (for computing with directed graphs), DESIGN (for constructing, classifying, partitioning, and studying block designs), and FinInG (for computation with finite incidence geometries). The emphasis will be on enabling the students to do computations relevant to their research interests in the areas of graphs and groups, designs, and finite geometries. The students are expected to have a background in basic graph theory and group theory, including permutation groups. Experience in computer programming is helpful, but not necessary.
Proposed outline of the course:
Lectures 1 and 2. Introduction to the GAP system. Topics to include: GAP statements, functions, and data structures, simple programming in GAP, and permutation groups in GAP.
Lectures 3 and 4. The GRAPE package. Topics to include: the structure of a graph in GRAPE, constructing and analysing graphs and subgraphs in GRAPE, determining regularity properties of graphs, automorphism groups and isomorphism testing for graphs, clique classification, and proper vertex colouring.
Lecture 5. An overview of the Digraphs package for computing with directed graphs.
Lectures 6 and 7. Overviews of the DESIGN package for the construction and analysis of block designs and the FinInG package for finite incidence geometry.
Lecture 8. Some research applications of GAP and the packages studied.
Minicourse 2 (MC2)
Title: Introduction to vertex algebras and vertex operator algebras. (8 hours)
Lecturer: Professor Atsushi Matsuo, The University of Tokyo, Japan
Description: We will give an overview of basics of vertex algebras (VA) and vertex operator algebras (VOA) so that the students will be able to understand advanced topics related to group theory and combinatorics. We will start with affine Lie algebras and Virasoro algebras, which give rise to important classes of VAs, and proceed to the definitions of VAs and VOAs and their basic properties. Among various examples, the lattice VAs and their modules will be treated in some detail. Some of more advanced topics on representation theory of VOAs will also be covered.
Proposed outline of the course:
Lecture 1. Affine Lie algebras and Virasoro algebra.
Lecture 2. Basics on vertex algebras.
Lecture 3. Vertex operator algebras.
Lecture 4. Lattice vertex algebras.
Lecture 5. Classification of simple modules.
Lecture 6. Determination of fusion rules.
Lecture 7. Twised modules and their applications.
Lecture 8. Modular invariance of characters.