Title: Inequalities for Riemannian surfaces
Speaker: Grigory Papayanov
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Abstract:
If f: X -> Y is a holomorphic map between two compact hyperbolic Riemannian surfaces, then degree of f is less or equal than (g(X)-1)/(g(Y)-1). This is very easy to prove using either Schwarz lemma or Riemann-Hurwitz formula. What if f is not holomorphic, but only continious? Then the formula (called Kneser's inequality) still holds, but it becomes harder to prove it by conventional topological tools. I want to tell how to prove this inequality by putting an L^1-norm on the space of singular chains of a surface. If time permits, I will tell how to use these methods to prove Milnor-Wood inequality: if a vector bundle over a surface $X$ admits a flat connection, then its Euler number is less or equal than g(X)-1.Date: Friday, February 15, 2019