The theory of motives, initiated by Grothendieck, is one of the highlights of modern algebraic geometry. It was initially motivated by the existence of several different notions of cohomology for algebraic varieties and the quest for an underlying universal structure. It has since developed into a very complex and fascinating subject of pure mathematics, which is currently still at the center of much ongoing investigation and which, over the years, gave rise to many deep results in arithmetic and algebraic geometry. Given its original motivation and its historic development, the theory of motives appears very far remote from the world of theoretical physics. However, in recent years, explicit computations carried out by physicists working in perturbative quantum field theory, followed by an increasing number of precise mathematical results, have uncovered a deep connection between periods of motives (numerical invariant that, in an appropriate sense, measure the complexity of a motive) and amplitudes of Feynman diagrams in quantum field theory. This project is focused on the role of motives in various branches of theoretical physics, starting from the Feynman amplitudes, by investigating the relations between different models (the parametric Feynman integrals on one side, and the twistor based computations of Feynman amplitudes on the other), but also by using the algebraic varieties that occur in quantum field theory as a testing ground for new theories in algebraic geometry such as the so-called "geometry over the field with one element". Moreover, part of this project investigates the occurrence of motives and periods in statistical physics, through the algebro-geometric properties of the partition function of Ising models and their generalizations, the Potts models. Another part of this project deals with the development of a new theory of "noncommutative motives" and its role in string theory and quantum field theory.

This is an interdisciplinary project that bridges between a very abstract field of pure mathematics, the theory of motives, and concrete mathematical models in high-energy physics, statistical physics, and string theory. The research project described above will have the effect of importing new mathematical tools into some fast developing areas of physics, and will also, at the same time, lead to new mathematical results, and the further development of some new branched of pure mathematics, within the research areas of algebraic and arithmetic geometry, through a new input of ideas, motivation, and intuition from physical theories. The project has a strong educational component, with the direct involvement of several graduate and undergraduate students.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Andrew D. Pollington
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California Institute of Technology
United States
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