Ergodic Ramsey Theory: A Dynamical Approach to Static Theorems
Vitaly Bergelson will give a series of lectures on Ergodic Ramsey Theory. The course will take place on January 25 and 26 in Prague and from January 29 till February 2 in pension Kavalir in Borova Lada. You can find the schedule of the talks below, or download it in a printable booklet form. See also the list of recommended literature below. You can register via e-mail nesetril@kam.
Programme
In Prague:
Thu, Jan 25 | 14:00 S5 | (65th Mathematical Colloquium) Ergodic Theorems Along Polynomials: From Combinatorial Applications to Challenges for Physicists |
Fri, Jan 26 | 10:40 S5 | The Early Results of Ramsey Theory and Their Modern Counterparts |
The bus to Borova Lada departs on Sunday, Jan 28 at 2pm from Malostranske namesti. In Borova Lada:
Mon, Jan 29 | Three Main Principles of Ramsey Theory |
Partition Ramsey Theory and Topological Dynamics | |
Tue, Jan 30 | Density Ramsey Theory and Furstenberg's Correspondence Principle |
Stone-Cech Compactification and Hindman's Theorem | |
Wed, Jan 31 | IP Sets and Ergodic Ramsey Theory |
The Nilpotent Connection | |
Thu, Feb 1 | Ergodic Ramsey Theory and Amenable Groups |
Progressions in Primes: Green-Tao Theorem | |
Fri, Feb 2 | What Next? Open Problems and Conjectures |
The bus back from Borova Lada is scheduled on Saturday, Feb 3 around noon.
Vitaly Bergelson studied in Russia and in Israel. He was awarded Landau Prize for his dissertation thesis (created under the supervision of prof. H. Furstenberg at the university in Jerusalem). Currently, he is a full professor at University of Ohio in Columbus, where he has been employed since 1984. Prof. Bergelson is an acknowledged researcher in ergodic theory and its applications, especially in combinatorial number theory. Together with his student A. Leibman, he proved e.g. the well-known polynomial versions of Van der Waerden and Szemeredi Theorems. Vitaly Bergelson is a renowned lecturer and a popular teacher. In 2006, he presented an invited talk at International Congress of Mathematicians in Madrid and Mordell Lecture in Cambridge.
Ergodic Theorems Along Polynomials: From Combinatorial Applications to Challenges for Physicists
Various recurrence and convergence results obtained in recent years indicate that dynamical systems exhibit regular behavior along polynomial times. While these results were mainly motivated by applications to number theory and combinatorics, such as the polynomial extension of Szemeredi's theorem on arithmetic progressions, this phenomenon also deserves attention from the point of view of potential applications to physics. For example, the Poincare recurrence theorem, as well as convergence theorems of the von Neumann and Birkhoff type, hold along any sequence of the form p(n), n=1,2,... where p(n) is a polynomial with integer coefficients satisfying p(0) = 0, and it would be of interest to give a physical interpretation of these facts. After reviewing some known results, we will discuss the intriguing dichotomy between the theorems related to polynomial and exponential behavior. The last part of the talk will be devoted to open problems and conjectures.Some literature:
- H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number
- R. McCutcheon, Elemental Methods in Ergodic Ramsey Theory
- Papers downloadable from Vitaly Bergelson's pages:
- Ergodic Ramsey Theory -- an update, Ergodic Theory of Z^d-actions (edited by M. Pollicott and K. Schmidt), London Math. Soc. Lecture Note Series 228 (1996), 1-61
- Ergodic theory and diophantine problems, Topics in symbolic dynamics and applications (edited by F. Blanchard, A. Maass and A. Nogueira), London Math. Soc. Lecture Note Series 279 (2000), 167-205.
- The multifarious Poincare recurrence theorem, Descriptive set theory and dynamical systems (edited by M. Foreman, A. Kechris, A. Louveau, B. Weiss), London Math. Soc. Lecture Note Series 277 (2000), 31-57.
- Minimal idempotents and ergodic Ramsey theory, Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, 2003.
- Combinatorial and Diophantine Applications of Ergodic Theory (with appendices by A. Leibman and by A. Quas and M. Wierdl), Handbook of Dynamical Systems, vol. 1B, B. Hasselblatt and A. Katok, eds., Elsevier (2005), pp. 745-841.
- Recurrence for semigroup actions and a non-commutative Schur theorem (jointly with R. McCutcheon), Topological dynamics and applications Contemporary Mathematics, 215 (1998), 205-222
- Squarefree numbers, IP sets and ergodic theory (jointly with I. Ruzsa), "Paul Erdos and his Mathematics I", Bolyai Society Mathematical Studies, 11, Budapest (2002), 147-160.
- Simultaneous Diophantine approximations and VIP systems (jointly with I. Haaland and R. McCutcheon), Acta Arithmetica 116 (2005) no. 1, 13--23.
- Multiplicatively large sets and ergodic Ramsey theory, Israel Journal of Mathematics 148 (2005), 23-40.
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