# Galatasaray University · Department of Mathematics

Çırağan Cad. No:36 34357 Ortaköy / İstanbul · Phone: +90 212 227 4480 · Fax: +90 212 260 5345

# Istanbul Number Theory Meetings

# March 16, 2013

# Galatasaray University, Istanbul, Turkey

## Organizers

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

## Speakers

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

## Program - 16.03.2013

10.00 - 12.00 Sonat Suer

13.30 - 15.30 Ayhan Günaydın

## Abstracts

- 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.

## Yer

Galatasaray University: Barlas Tolan

## Research Activities at GSU

- Journees Arithmetiques (2019)
- Department Seminars
- Geometry and
Arithmetic Around Teichmüller Theory

- CIMPA/TÜBİTAK/GSU Summer School:

Algebraic Geometry and Number Theory (2014) - Istanbul Toric and Tropical Geometry Meetings (2013)
- İstanbul Number Theory Meetings (February 16, March 16 2013)
- İstanbul Workshop on Teichmüller Theory (October 11-14, 2012)
- İstanbul Workshop on Fixed Point Theory and Applications (October 11-14, 2012)
- Functoriality and Correspondence

(March 30 - April 27, 2011) - CIMPA/TÜBİTAK Summer School:

Commutative Algebra and Applications to Combinatorics and Algebraic Geometry

(September 12-25, 2010). Co-organized with Feza Gürsey Institute - GTEM/TÜBİTAK Summer School:

Geometry and Arithmetic around Galois Theory

(June 08-19, 2009). - GTEM/TÜBİTAK Summer School:

Geometry and Arithmetic of Moduli Spaces of Coverings

(June 09-20, 2008). - CIMPA Summer School:

Arrangements and Local Systems and Singularities

(June 11-22, 2007). - EMS Summer School:

Arithmetic and Geometry Around Quantization

(June 05-15, 2006). - CIMPA Summer School:

Arithmetic and Geometry Around Hypergeometric Functions

(June 13--25, 2005).

## Venue

The conference venue is Mathematics Department of Galatasaray University, and its address is

Galatasaray
University

Department of Mathematics,

Çırağan Caddesi, No: 36,

34357 Ortaköy, Beşiktaş,

İstanbul / Turkey.

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