Title: The Weil Conjectures
Speaker: Stefan Patrikis
Abstract: The Riemann zeta function organizes the integer primes in a way that encodes their distribution. E. Artin defined an analogous "zeta function" associated to any collection of (multivariable) polynomial equations with coefficients in the integers mod p (a prime), which encodes how many simultaneous solutions the equations have over Z/p and each of its finite field extensions. The Weil conjectures deepen this analogy, asserting that the fundamental properties of the Riemann zeta function (both proven and conjectural!) admit analogues for these Artin zeta functions. Just as remarkably, the statements and proofs of the Weil conjectures reveal a glimpse of what Grothendieck called "the profound identity between geometry and arithmetic," in this case astonishing parallels between the topology of algebraic varieties over the complex numbers and the solution counts of equations over finite fields. I will give an elementary introduction to this circle of ideas.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn Semester. Pre-candidacy PhD students can sign up for this lecture series by registering for two credit hours of Math 6193 with Professor Nimish Shah.
