- Alp Bassa (Sabanci University, Istanbul)
- Kazim Buyukboduk (Koc University, Istanbul)
- Safak Ozden (Mimar Sinan University, Istanbul)

- Ayhan Günaydın (Mimar Sinan Fine Arts University)
- Sonat Suer (Bilgi University)

10.00 - 12.00 Sonat Suer

13.30 - 15.30 Ayhan Günaydın

- Ayhan Günaydın (Mimar Sinan Fine Arts University) Title: Implementations of Pila-Wilkie counting theorem on number theoretic problems
- Sonat Suer (Bilgi University) Title: A Fast Introduction to O-minimality

Abstract: Assuming the knowledge of the statement of Pila-Wilkie Theorem on counting rational points of sets definable in o-minimal structures, we present proofs of certain number theoretic problems of Diophantine nature. The main example is a (re)-proof of the Manin-Mumford conjecture by Pila and Zannier. After presenting this proof, we shall focus on more general problems of André-Oort type. (This is an expository talk, and none of the results are due to the speaker.)

Abstract: An ordered first order structure M is called order minimal, or o-minimal for short, if every definable subset of M can be defined using the order alone. Any real closed field is o-minimal by a theorem of Tarski. A more recent example is the real field together with restricted analytic functions and the full exponential function. In any o-minimal structure M one can decompose definable subsets of M^n into well behaved definable subsets called cells, and using this, one can define notions of dimension and Euler characteristics. In this respect, o-minimality is an axiomatic approach to Grothendieck's hoped-for notion of tame topology. In the first part of the talk, we will sketch the proof of the cell decomposition theorem. The second part will be about uniformaization theorems in o-minimal structures, setting the stage for the following talk by Ayhan Günaydın.

Galatasaray University: Barlas Tolan

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