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