Conference: Motives, Quantum Field Theory and Pseudodifferential Operators

Trustees Of Boston University, Boston

Investigators

Abstract

On June 2-14, 2008, Boston University will host the conference Motives, Quantum Field Theory, and Pseudodifferential Operators. This conference will feature both introductory and advanced lectures on the rapidly developing relationships among these established fields. Since the fundamental tools in these research areas are quite different, there is a need for experts in different aspects of the conference to both present overviews of their subjects and point out the interactions among these topics. Speakers and participants have been invited from the US and Europe. Already over 30 top mathematicians from the three conference fields have agreed to attend, including Spencer Bloch, Alain Connes, Alexander Goncharov, Dirk Kreimer, and Richard Melrose. The attendance for this conference is expected to be roughly 75 -- 100 mathematicians and mathematical physicists. The three fields of motives, quantum field theory, and pseudodifferential operators are well established and active as independent fields, but until the past decade they have been viewed as essentially disjoint enterprises. The theory of motives was developed by Grothendieck in the 1960s as an arithmetic algebraic geometric setup incorporating diverse cohomology theories. Perturbative quantum field theory first appeared around 1945 to explain laboratory results in quantum electrodynamics and other field theories, and depends heavily on the calculation of Feynman integrals. Finally, the theory of pseudodifferential operators, dating from the 1960s, is a framework to treat all classical differential, Green's and heat operators in one setting. Until recently, connections among these apparently very different fields were limited mainly to the appearance of Green's operators in quantum field theory. The unexpected appearance of multiple zeta values as values of Feynman integrals indicated a deep connection between quantum field theory and motives, the usual setting for multiple zeta functions. In addition, the Hopf algebra structure encoding the complicated combinatorics of Feynman integrals, uncovered by Connes and Kreimer, appears both in the theory of multiple zeta values and symbol calculations for pseudodifferential operators. Further research has shown that the connections are much more than coincidences, and the work of Bloch, Connes, Esnault, Kreimer, Marcolli and others points towards the possibility of a noncommutative geometry explanation or a direct motives/quantum field theory connection. This is the first conference on this topic in the US. Since junior faculty and graduate students are encouraged to attend with partial NSF support, the schedule will include introductory lectures on the three conference areas and on Hopf algebras and multiple zeta functions. Other speakers will give research level talks on the conference areas, with a particular emphasis on their interactions.