I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.
In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture.
I will report on recent progress towards understanding the growth of the torsion of the homology of subgroups of finite index in a given residually finite group G.
The cases I will consider are when G is amenable (joint work with P, Kropholler and A. Kar) and when G is right angled (joint work with M. Abert and T. Gelander).
I plan to discuss finite time singularities for the harmonic map heat flow and describe a beautiful example of winding behaviour due to Peter Topping.
Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n.
In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic. I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V). I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.
After some generalities on étale fundamental groups and anabelian geometry, I will explore some of the current results on the section conjecture, including those of Koenigsmann and Pop on the birational section conjecture, and a recent unpublished result of Mohamed Saidi which reduces the section conjecture for finitely generated fields over the rationals to the case of number fields.
Many hard problems in Diophantine geometry have analogues over function fields which are less hard. I will give some examples.
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