Let X be a smooth variety (e.g., affine space) over a finite field

(e.g., the integers modulo a prime). In the course of proving the last

of Weil's conjectures on zeta functions of varieties over finite field,

Deligne studied a certain category of representations of the fundamental

group of X which carry information about these zeta functions. He also

made a far-reaching conjecture to the effect that such objects always

look as if they "come from geometry".

We will state the conjecture, describe some of its more concrete consequences, and discuss some

results of various authors (L. Lafforgue, V. Lafforgue, Deligne,

Drinfeld, T. Abe, Abe-Esnault, and the speaker) which very recently have

led to a resolution of this 40-year-old open problem.

### Penn Mathematics Colloquium

Wednesday, March 13, 2019 - 3:30pm

#### Kiran Kedlaya

UCSD