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Mini Course by Vitaly
Bergelson Ergodic Ramsey Theory: Dynamical Methods at the service of Number
Theory and Combinatorics |
August 20~24, 2007
Ajou
University & Korea Institute for Advanced Study
The lectures will be devoted to the presentation of some of the numerous
and multifaceted connections which exist between Ergodic
Theory, Combinatorics and Number Theory. We will start with reviewing some
applications of the Poincare recurrence theorem to
such diverse algebraic results as Hilbert's irreducibility theorem and Dickson-Schur theorem on the solvability of Fermat equation over
finite fields. We will move then to the discussion of Furstenberg's ergodic approach to Szemeredi's
theorem on arithmetic progressions. Furstenberg's proof was a starting point of
new exciting developments and we will discuss some of them, including the
polynomial Szemeredi theorem, multiple recurrence
theorems for general groups and the role of dynamical systems on nil-manifolds
in the study of multiple recurrence. Some of the recent results rely heavily on
methods of topological algebra in the Stone-Cech compactifications and we will review some of these methods.
We will also discuss the ergodic underpinnings of
recent spectacular theorem of Green and Tao on arithmetic progressions in
primes and its recent polynomial extension by Tao and Ziegler. Finally, we will
give a review of some natural open problems and promising directions of
research.¡¡
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Prelude |
621 Paldal-Kwan,
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Monday |
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Young Ho Ahn ( Introduction
to Ergodic Theory I |
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Lunch |
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Dong Han Kim ( Introduction
to Ergodic Theory II |
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Mini Course by Bergelson |
Room
1423, KIAS |
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Tuesday |
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Ergodic Ramsey Theory I: Overview of some
classical results in Ramsey theory |
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Ergodic Ramsey Theory II: Partition Ramsey
theory and topological dynamics |
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Lunch |
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Ergodic Ramsey Theory III: Furstenberg¡¯s ergodic approach to Szemeredi¡¯s
theorem on arithmetic progressions |
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Wednesday |
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Ergodic Ramsey Theory IV: Polynomial
extensions of Szemeredi theorem |
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Ergodic Ramsey Theory V: Stone-Cech compactification of
integers and Hindman¡¯s theorem |
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Lunch |
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Ergodic Ramsey Theory VI: IP ergodic theory versus Cesaro ergodic theory |
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Thursday |
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Ergodic Ramsey Theory VII: Bohr compactification of integers and sumsets |
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Ergodic Ramsey Theory VIII: New trends: multiple
recurrence in nilpotent and amenable set-up |
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Lunch |
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HKK's 2n+3 Theorem for Quadratic Forms (What the Ergodic
theory improved.) |
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Colloquium |
Room
1423, KIAS |
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Thursday |
Ergodic Ramsey
Theory and Patterns in Primes (See Abstract Below) |
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Epilogue |
Room
1423, KIAS |
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Friday |
Open problems and
conjectures |
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Colloquium: Ergodic Ramsey Theory and Patterns in Primes Abstract Szemeredi's theorem on arithmetic
progressions states that any set of positive upper density in N contains
arbitrarily long arithmetic progressions. Furstenberg's ergodic
proof of Szemeredi's theorem has revealed the
dynamic content of Szemeredi's theorem and has paved
the way to numerous spectacular generalizations and extensions, most of which
do not have as yet a conventional proof. We will survey some of important
developments and conclude with the discussion of the recent work of Tao and
Ziegler in which they merge the theorem of Green and Tao on arithmetic
progressions in primes with the polynomial extension of Szemeredi's
theorem due to Bergelson and Leibman. Note: This colloquial talk is also given in 621 Paldal-Kwan at Ajou University
on August 29, 2007. |
References
H. Furstenberg: Recurrence in Ergodic Theory and Combinatorial Number Theory
P. Walters: An Introduction to Ergodic Theory
W. Parry: Topics in Ergodic
Theory
K. Petersen: Ergodic
Theory
T. Tao and V.H. Vu: Additive Combinatorics
Organizers: Kyewon Koh Park (
Soon-Yi Kang (Korea
Institute for Advanced Study)
Local organizer: Poo-Sung Park (Korea Institute for
Advanced Study)
Sponsors: Korea
Institute for Advanced Study
Department
of Mathematics (BK 21 Program), Ajou University