Talk

This is a joint work with Vincent Bouchard, Dmitry Noshchenko and Nitin Chidambaram.

A matched pair of Lie algebroid representations is equivalent to two seemingly different objects; the bicrossproduct Lie algebroid and the double Lie algebroid of the matched pair. In similar manner, a Lie bialgebroid is equivalent to a ‘cotangent double Lie algebroid’, and to a ‘bicrossproduct' Courant algebroid.

This Minicourse is based on the speaker's monograph "a mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory" (to appear in MPIM preprint series).  In this talk I plan to review the fundamental structure of the Connes--Kreimer renormalization Hopf algebra of Feynman diagrams and combinatorial Dyson--Schwinger equations. Then I explain a recent application of infinite graph theory to Quantum Field Theory which leads to build a renormalization machinery for infinite formal expansions of Feynman diagrams.

The following is joint work with Rachel Newton. In the spirit of work by Yongqi Liang, we relate the arithmetic of rational points to that of zero-cycles for the class of Kummer varieties over number fields. In particular, if X is any Kummer variety over a number field k, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on X over all finite extensions of k, then the Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of any odd degree on X.