Continued Fractions, Hyperbolic Geometry, and Quadratic Forms

June 18 - June 28

TIME: MR 10:10 am - 12:05 pm

INSTRUCTOR: SVETLANA KATOK

In this mini-course we explored connections between continued fractions, hyperbolic geometry, and closed geodesics on the modular surface. Particular topics included: theory of minus and plus continued fractions; hyperbolic geometry (geodesics, isometries, hyperbolic area); modular group and its fundamental region; arithmetic coding of closed geodesics; conjugate matrices and Gauss Reduction Theory in the matrix language.

Galois Theory

July 2 - July 12

TIME: MR 1:25 pm - 3:30 pm

INSTRUCTOR: VICTOR NISTOR

The course covered the following topics: Review of groups and rings. Field theory: minimal polynomial, field extensions, splitting fields, algebraically closed fields, extensions of embeddings. Galois theory in characteristic zero: the Galois group, the Galois corespondence between intermediate extensions and subgroups of the Galois group, normal subgroups correspond to Galois extensions.