Math 612 Arithmetic and Geometry of Linear Algebraic Groups Semester: Spring 2015

Description of course: This course approaches the subject of linear algebraic groups from the perspective of applications to number theory and algebraic geometry. Specific instances could include the Hasse-Minkowski theorem and the classification of classical groups over number fields; and the theorems of Tsen and Lang on the cohomological properties of function fields with applications to families of projective homogeneous spaces. Following the approach of Serre, we highlight the theory of cohomological invariants for related algebraic structures, such as quadratic, symplectic, and hermitian forms, central simple algebras, octonion algebras, and Jordan algebras. The requisite background on algebraic geometry and Galois cohomology is introduced throughout the course.

Expected background: Familiarity with algebra and basic Galois theory. Some previous experience with basic algebraic geometry, representation theory, or algebraic number theory is helpful, but not essential.

Grading: Your grade will be based on the take-home final exam.