School of Mathematics

Period map: past, present and the future

Gao Chen
Member, School of Mathematics
September 26, 2017
I will start from a simple statement: the radius of a sphere determines the shape of it. This simple statement has been generalized to the study of Riemann surfaces, K3 surfaces, Hyperkähler manifolds and other important objects. Recently, the conjecture about the generalization to the subject of special holonomy has drawn many attentions.

\(2^\infty\)-Selmer groups, \(2^\infty\)-class groups, and Goldfeld's conjecture

Alex Smith
Harvard University
September 14, 2017
Take \(E/Q\) to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of \(E\) have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to \(2^k\)-Selmer groups for any \(k > 1\).

String topology coproduct: geometric and algebraic aspects

Manuel Rivera
University of Miami
May 11, 2017
The string topology coproduct is an intersection type operation, originally described by Goresky-Hingston and Sullivan, which considers transverse self-intersections on chains of loops in a smooth manifold and splits loops at these intersection points. The geometric chain level construction of string topology operations involves deforming chains to achieve certain transversality conditions and these deformations introduce higher homotopy terms for algebraic compatibilities and properties.