SFNS Seminar: Park -- Analogy between polynomials and integers as global sections

Speaker: June Park, The University of Sydney
Abstract: In this talk, I will start from the simple example of x^2 + y^2 = 1 over Q
(Pythagorean triple numbers) and look at the analogous problem of Hom_n(P^1,P^1) over k
(Space of rational functions).  This will be fun and we will see the idea of "global
sections of sheaves over a global field" while reviewing classical works of V.
Arnol’d, J.  Milnor, M.  Atiyah, G.  Segal.  We will work with motives (Cruder
topology via Grothendieck ring of varieties) by the work B.  Farb and J.  Wolfson.
Afterward, we will review some basics on elliptic curves and consider the fact that each
E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1,1} of stable
elliptic curves, which in turn corresponds to a rational curve on Mbar_{1,1}.  We can
ask what is the Mordell-Weil group E(K) for a given E/K and we will consider the
question of boundedness of rank answered by the work of D.  Ulmer.  In the end, I will
explain the exact counting formula for all elliptic curves over k(t) via motivic methods
along wi