Mathematics Graduate Student Colloquium

MGSC Website: http://math.uci.edu/~mgsc/

Cohomology on Complements of Singular Hypersurfaces in P3 of A-D-E singularities

Matthew Cheung
Thursday, May 21, 2020
1:00 pm - 1:50 pm
Virtual via Zoom

Talk Abstract:

The number of solutions to a given hypersurface in projective space over finite fields is encoded in a generating function called the zeta functions of an algebraic variety over a finite field. Hence, to know the number of solutions to a given hypersurface, it suffices to determine the zeta function. The zeta function is given by the determinant of a matrix given by the actino of Frobenius on certain cohomology groups. Hence, the first step is to understand these cohomology groups. Counting points on smooth hypersurfaces has been well-studied so we move to singular hypersurfaces. In this talk, we explain about De Rham Cohomology on the Complement of a singular hypersurface of type A-D-E singularities over the complex numbers. We then explain how this result helps in determining the number of points over our singular hypersurface over finite fields and problems that still need to be answered.

About the Speaker:

Matthew is a 3rd year graduate student working on number theory and algebraic geometry. In his spare time, he likes watching anime and playing anime games.

Refreshments:

Pizza will be served after the talk.