Riemann Surfaces

Question 1

What “upgrades” on surface yield Riemann surface?

Answer 1

90 degree turn operator, measure of area

pg 1

Question 2

Discuss recovering geometry of H^2 from Spec R[x]

Answer 2

pg 2-4

Question 3

Discuss limit definition of exterior differentiation

Answer 3

pg 6

Question 4

Define Morse function - discuss existence

Answer 4

pg 8-9

Question 5

Discuss how to use Morse theory to classify surfaces

Answer 5

pg 10

Question 6

What is Uniformization? Proof?

Answer 6

Every compact surface of genus g >= 2 with a J-field admits a metric m of const. curvature = -1 s.t. R_m^pi/2 = J_p

pg 12

Question 7

Define closed, exact 1-forms. Locally what is relationship? Globally?

Answer 7

Locally closed = exact. Globally difference gives rise to cohomology

dw = infinitesimal stokes thm

pg13

Question 8

Discuss studying space by studying functions on space. Where does this work well? What problems arise for Riemann surfaces? Resolution?

Answer 8

Thm. (Gelfand) Let X be a compact metric space. Then out of Banach ring C(X, R), one can restore the topological space.

Studying (S, J) holomorphic functions on S = C. Not nearly enough to say anything interesting about the surface.

Fix: Look at 1 forms instead.
pg 16-19