Point Counting over Finite Fields and the Cohomology of Moduli Spaces of Curves

Explore the intricate relationship between point counting over finite fields and the cohomology of moduli spaces of curves in this illuminating 1-hour 8-minute lecture by Sam Payne. Delve into the world of algebraic geometry, examining solution sets of polynomial equations in different contexts. Compare the topology of solution sets over complex numbers with point counting over finite fields. Discover how cohomological comparison theorems and trace formulas for Frobenius action connect these fundamental invariants for polynomials with integer coefficients. Learn about recent breakthroughs in point counts over finite fields and the cohomology of moduli spaces of curves, which have resolved longstanding questions in algebraic geometry and validated predictions from the Langlands program. Gain insights into this fascinating area of mathematical research presented at the Colloque des sciences mathématiques du Québec (CSMQ) on November 3, 2023.