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.

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.