Moduli Spaces, Motives, Periods, and Scattering Amplitudes

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Recently, new ideas in pure mathematics have had a strong impact on theoretical physics, and vice versa. In particular, the ideas of one of the most sophisticated areas of mathematics, the so-called theory of motives, which is part of algebraic geometry, found application in the calculation of scattering amplitudes from the data observed in experiments. On the other hand, the general notion of quantization that originated in theoretical physics has found concrete realizations in many of areas of pure mathematics. Using these new insights, this project will investigate several concrete questions in number theory, algebraic geometry, and representation theory. This project involves research in several related topics. In one direction, the project will study scattering amplitudes in quantum field theory by using theory of mixed motives and theory of cluster varieties. In a second direction, the project will further develop the on-shell approach to scattering amplitudes, with the goal of finding effective ways to calculate scattering amplitudes. In a third direction, the project will study a quantum field theory approach to mixed Hodge theory and develop quantum Hodge field theory. In a fourth direction, the project will study Donaldson-Thomas invariants of three-dimensional Calabi-Yau categories relevant to representation theory and geometry. In a fifth direction the investigator will continue work on moduli spaces of local systems on two-dimensional surfaces and its quantization, and on the relationship with representation theory and mirror symmetry. In particular, the investigator will study the hyperkähler structure of the moduli spaces of local systems on topological surfaces, and moduli spaces of non-commutative local systems on surfaces. In a final direction, the theory of hyperbolic three-dimensional manifolds can be viewed as the study of certain local systems on three-dimensional manifolds with values in one of the simplest complex Lie groups. The project aims to develop a similar theory for all complex reductive Lie groups, as well as its quantum analog.