Tuesday, March 5, 2019
03/05/2019 - 4:15pm
Let X be a closed, connected, hyperbolic surface of genus 2. Is it more likely for a simple closed geodesic on X to be separating or non-separating? How much more likely? In her thesis, Mirzakhani gave very precise answers to these questions. One can ask analogous questions for square-tiled surfaces of genus 2 with one horizontal cylinder. Is it more likely for a such a square-tiled surface to have separating or non-separating horizontal core curve? How much more likely? Recently, Delecroix, Goujard, Zograf, and Zorich gave very precise answers to these questions. Surprisingly enough, their answers were exactly the same as the ones in Mirzakhani’s work. In this talk we explore the connections between these counting problems, showing they are related by more than just an accidental coincidence.
03/05/2019 - 4:15pm
Abstract: Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the l-adic Tate module of any non-CM elliptic curve over a given number field. Recently in a series of papers Cadoret and Tamagawa established a definitive result regarding the uniform boundedness of the l-primary torsion for 1-dimensional abelian families. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension, in the case of abelian families parametrized by Picard modular surfaces over an imaginary quadratic field M. Namely, we establish a uniform irreducibility of Galois acting on the l-primary part of principally polarized Abelian 3-folds with multiplication by M, but without CM factors.
03/05/2019 - 6:00pm
Abstract: To define a complex holomorphic function, we take the definition of a real differentiable function and allow all the variables involved to be complex. As the miraculous theorems of complex analysis show, this quiet move from R to C introduces all sorts of rigidity. When we further require our functions to be injective, powerful connections between geometry and analysis arise.
Focusing on the Koebe function $k(z) = z + 2z^2 + 3z^3 + …,$ this talk provides an introduction to functions that are both holomorphic and injective (univalent). We will prove the Second Coefficient and Koebe One-Quarter theorems and discuss (without proof) de Branges’ Theorem and related statements.
If you have seen calculus and are willing to take some statements on faith, I hope this talk will excite you to study this magical area of math; for students who have seen complex analysis, I hope this talk will serve as an invitation to further study.