Tamagawa Numbers in the Function Field Case (Lecture 2)
February 5, 2014
In the previous lecture, we dened the Tamagawa measure associated to a connected semisimple algebraic
group G over the eld Q and formulated Weils conjecture: if G is simply connected
Weils Conjecture on Tamagawa Numbers (Lecture 1)
January 30, 2014
Let R be a commutative ring and let V be an R-module. A quadratic form on V is a map q : V R
satisfying the following conditions:
(a) The construction (v, w) q (v + w) q (v ) q (w) determin
Three Descriptions of the Cohomology of BunG(X ) (Lecture 4)
February 5, 2014
Let k be an algebraically closed eld, let X be a algebraic curve over k (always assumed to be smooth
and complete), and let G be a smooth ane group scheme over X . We let BunG (
Cohomological Formulation (Lecture 3)
February 5, 2014
Let Fq be a nite eld with q elements, let X be an algebraic curve over Fq , and let G be a smooth
ane group scheme over X with connected bers. Let d = dim(G). For each closed point x X , let deg(x)
de
Nonabelian Poincare Duality in Algebraic Geometry (Lecture 9)
February 22, 2014
In the previous lecture, we stated the following result:
Theorem 1 (Nonabelian Poincare Duality in Topology). Let M be a manifold of dimension n and let
(Y, y ) be a pointed s
-adic Cohomology (Lecture 6)
February 12, 2014
Our goal in this course is to describe (in a convenient way) the -adic cohomology of the moduli stack of
bundles on an algebraic curve. We begin in this lecture by reviewing the -adic cohomology of schemes; w
Acyclicity of the Ran Space (Lecture 10)
February 24, 2014
Throughout this lecture, we let k denote an algebraically closed eld, a prime number which is invertible
in k , and cfw_Z , Q , Z/ d Z. Suppose we are given an algebraic curve X over k and a smoot
Higher Category Theory (Lecture 5)
February 7, 2014
In the previous lecture, we outlined some approaches to describing the cohomology of the classifying
space of G-bundles BunG (X ) on a Riemann surface X . For example, we asserted that the cochain comple
Homology and Cohomology of Stacks (Lecture 7)
February 19, 2014
In this course, we will need to discuss the -adic homology and cohomology of algebro-geometric objects
of a more general nature than algebraic varieties: for example, the moduli stack BunG (X
Nonabelian Poincare Duality (Lecture 8)
February 19, 2014
Let M be a compact oriented manifold of dimension n. Then Poincare duality asserts the existence of
an isomorphism
H (M ; A) Hn (M ; A)
for any abelian group A. Here the essential hypothesis is tha