# Advanced Topics in Logic

# Math 512

# Spring 2010

Course Meeting: 1:00 MWF 316 Taft Hall

Call Number: 28297

Instructor: David Marker

Office: 312 SEO/ 411 SEO

Office Hours: (in 312 SEO) By appointment

phone: (312) 413-3044

e-mail: marker@math.uic.edu

course webpage:
http://www.math.uic.edu/~marker/math512s10

## THE FIRST CLASS MEETING WILL BE WED JAN 20

### Description

This will be a seminar style course investigating a circle of recent themes involving algebra, number
theory and model theory. Although the course will not be self contained, I hope to give students enough
background to appreciate and understand the material I am talking about. The model theoretic
prerequisites will be minimal but some familiarity with basic algebraic geometry will be assumed.
Some of the material I hope to cover includes:
- Ax's proof of Schanuel's Conjecture for function fields and Kirby's extension to semiabelian varieties
- Zilber's Conjecture on Intersection of tori, generalizations and applications
- Pila and Wilkie's work on rational points on definable sets in o-minimal structures and applications.

All of these ideas play interesting roles in number theoretic finiteness theorems. Below are some of the most important
papers in the area. Though they covers **far** more than we will cover in the class.
There will be no homework.
### References

- Ax, J., On Schanuel's conjectures. Ann. of Math. (2) 93 1971 252--268.
- Bombieri, E.; Masser, D.; Zannier, U. On unlikely intersections of complex varieties with tori. Acta Arith. 133 (2008), no. 4, 309--323.
- Kirby, J., The theory of the exponential differential equations of semiabelian varieties
- Masser, D. and Zannier, U., Torsion anomalous points and families of elliptic curves. C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 491--494.
- Pila, J., Rational points of definable sets and results of André-Oort-Manin-Mumford type, Int. Math. Res. Not. IMRN 2009, no. 13, 2476--2507.
- Pila, J. and Wilkie, A., The rational points of a definable set, Duke Math. J. 133 (2006), no. 3, 591--616.
- Pila, J. and Zanier, U.,
Rational points in periodic analytic sets and the Manin-Mumford conjecture,Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), no. 2, 149--162.
- Zilber, B., Intersecting varieties with tori