In recent decades, combinatorics has matured into a major branch of mathematics, demonstrating important connections with probability theory, number theory, geometry, computer science, and other fields. This two-week summer program in July 2011 will introduce fifteen U.S. PhD student participants to some core areas of modern combinatorics through an internationally mentored program of study at the Eötvös Loránd University, in Budapest, Hungary.
The program will be led by U.S. principal investigator, Professor Bela Bollobas of the University of Memphis, in cooperation with faculty in the Department of Computer Science at Eötvös University. As a foundation for the student-centered activities, there will be four lecture series: "Random Geometric Graphs" by Professor Paul Balister of the University of Memphis, "Percolation" by Professor Bela Bollobas, "Algebraic and Geometric Methods in Additive Combinatorics" by Professor Gyula Karolyi of Eötvös University, and "Ramsey Theory" by Professor Imre Leader of the University of Cambridge. The lecturers will present fundamental and recent results as well as open problems central to advancing these topics--topics which have important applications to internet and wireless technology and offer insights into the science of neural networks.
The daily schedule will consist of morning lectures and afternoon workshops. Throughout this summer program in combinatorics, the students will present papers on topics related to the courses, give informal seminars, and work in small groups. There will be ample opportunity to attend seminars given by established mathematicians. In the summer school, the junior U.S. participants will interact with international student counterparts and senior researchers, not only from Eötvös University, but also from the Alfréd Rényi Institute and the Computer and Automation Institute of the Hungarian Academy of Sciences (SZTAKI), as well as the University of Cambridge and other centers of learning in Europe. Overall, the well integrated program will promote learning in both theoretical and applied contexts. Broader impacts include an early career introduction for the U.S. PhD students to an international professional network consisting of renowned European experts in combinatorics.
Hungary has a long tradition of excellence in mathematics research and education. A select group of students was able to experience the Hungarian mathematical tradition and the country’s rich culture through the Memphis-Budapest Summer School in Combinatorics which took place 7 - 21 August 2011 at the Rényi Institute in Budapest, Hungary. These students enjoyed access to Eötvös University and the close-by Mathematical Institute of the Hungarian Academy of Sciences, the two institutions known for having educated more than half of Hungary's highly acclaimed mathematicians. Under the direction of Professor Béla Bollobás of the University of Memphis, a select group of twenty-seven research students and postdocs from the universities in the US joined a small group of young researchers from all over the world and participated in the summer school. Fifty-six students applied for travel support provided by the National Science Foundation and twenty-seven students received varying levels of that support. Additional funding was provided by the Chair of Excellence in Graph Theory and Combinatorics at the University of Memphis and from Dr. Bollobás. These students partook in a rigorous series of four excellent courses taught by some of the most renowned scholars in combinatorics: Branching Processes, Paul Balister, University of Memphis. This course gave a basic, but rigorous, treatment of branching processes, and the probabilistic tools needed to describe them. Branching processes can be used to model dynamic processes or events. Percolation, Béla Bollobás, University of Memphis and University of Cambridge. This course addressed percolation, a physical process that describes for a system, a transition from one state to another. Examples can be found not only in physical phenomena, but also in biological and ecological ones (evolution), and also in economic and social ones. The course introduced the basic concepts of percolation theory and some fundamental results concerning them. Additive Combinatorics, Gyula Károlyi, Eötvös University. This course introduced some basic concepts and results of the theory via simple yet powerful algebraic and geometric methods. Ramsey Theory, Imre Leader, University of Cambridge. This course was concerned with the general question of whether, in a large amount of disorder, one can find regions of order. A typical example is van der Waerden's theorem, which states that whenever we partition the natural numbers into finitely many classes there is a class that contains arbitrarily long arithmetic progressions. At the core of these courses lies not only the inherent combinatorial interest in the solutions to problems, but also in their applicability to other areas of mathematics, engineering and computer science and development of new mathematical tools. These courses provided research experiences and introduced students to challenging problems which provide framework for practical applications.In addition, students worked collaboratively during problem sessions and several students presented talks on their current, related research. This program strengthened the University of Memphis graduate program in combinatorics, which is one of the strongest in the nation. It reinforced existing and developing collaborative alliances between the University of Memphis and this major research center, the Renyi Institute. It also fostered the cultivation of energy and ideas in this and the up-and-coming generation of researchers in one of the most dynamic areas of mathematics. This was an excellent opportunity for graduate students with interest in combinatorics to be introduced to problems suitable for further investigation. Collaborations were formed among the young researchers and between senior researchers and the students that might not otherwise have happened. In addition to learning the theory and application of much maths, the participants had ample opportunity to enjoy the attractions of Budapest, and to experience the European style of life.
