# 2014 Phd Summer School in Discrete Mathematics and Symmetries of Graphs and Networks IV

from 29 June 2014 to 5 July 2014
Europe/Ljubljana timezone
 Minicourse 1: Construction techniques for graph embeddings Lecturer: Mark Elingham, Vanderbilt University, USA Mathematicians have been trying to construct embeddings of specific graphs in surfaces since at least the 1890s.  However, until the 1960s the construction techniques were usually fairly ad hoc, although some general ideas such as `schemes of cyclic sequences' had emerged.  This changed with the development of current graphs by Gustin and others in the 1960s, which provided a unified framework for many earlier constructions and played an important role in the proof of the Map Colour Theorem.  Fifty years later we have a number of useful general tools for constructing embeddings of graphs.  These lectures will survey tools of various kinds.  We will look at algebraic methods such as current, voltage and transition graphs; surgical tools such as the diamond sum and adding handles or crosscaps around a vertex; lifting constructions due to Bouchet and his collaborators; and techniques that use objects from design theory, such as latin squares, to construct embeddings. Lecture notes (draft version) Additional exercises Minicourse 2: Combinatorial designs Lecturer: Mariusz Meszka, AGH University of Science and Technology, Poland   Combinatorial design theory rapidly developed in the second half of the twentieth century to an independent branch of combinatorics. It has deep interactions with graph theory, algebra, geometry and number theory, together with a wide range of applications in many other disciplines. Most of the problems are simple enough to explain even to non-mathematicians, yet the solutions usually involve innovative techniques as well as advanced tools and methods of other areas of mathematics. The most fundamental problems still remain unsolved. This series of lectures is intended to provide a solid introduction to the major topics and concepts: block designs, Latin squares, difference sets, Hadamard matrices, Room squares, resolvable designs. Basic methods, constructions, results and open research problems will be followed by various examples and exercises. Lecture notes (draft version) Minicourse 3: Symmetric key cryptography and its relation to graph theory Lecturer: Enes Pasalic, University of Primorska, Slovenia   Modern cryptology relies on many scientific disciplines such as information theory, probability theory, discrete mathematics among others. In addition, many public cryptosystems are based on some  hard graph theoretic problems  such as graph coloring for instance. While not directly derived from the concepts related to graphs, the most important cryptographic properties of certain discrete structures may be defined and analyzed in the graph theoretic framework which might give at least different insight at these structures. We will give a short survey of cryptography with the emphasis on these discrete structures being basic primitives in the so-called symmetric key cryptography. Booolean functions, vectorial mappings over finite structures and permutations over finite fields, as the most important representatives of these structures, will be considered in real-life encryption schemes. Their cryptographic properties will be stated both in a classical way using some suitable tools in cryptology and these will be then translated in the graph theoretic language. The students will also get a brief insight in the state-of-the-art research in this direction.   Lecture notes (draft version) Slides Minicourse 4: Some topics in the theory of finite groups Lecturer: Primož Moravec, University of Ljubljana, Slovenia The theory of finite groups plays a central role in group theory and has several applications in other branches of mathematics, including discrete mathematics and cryptography. The theory culminated with the classification of finite simple groups in 1983, and has developed afterwards into several different directions such as the theory of groups of prime power order, invariant theory, and many others. This mini course will address some topics of the above theory. These will include advanced applications of Sylow's theory, techniques of building new groups from old, basic theory of finite p-groups, and problems regarding enumeration of finite groups. Lecture notes (draft version) Problem set Slides DISCLAIMER: These are draft versions of the lecture notes. A revised version of the lecture notes will be available in printed and electronic version by the end of 2014.