34 | IHES Annual Report 2020
Alexander GONCHAROV Mathematics, Philip Schuyler Beebe professor, Yale University.
ALEXANDER GONCHAROV collaborated with Vladimir Fock and established geometric quantization of cluster Poisson varieties. Namely, given a rational Planck constant \hbar= ,he constructed on a complex cluster Poisson variety a gerbe and an analytic vector bundle with a connection to this gerbe of dimension sd, where 2d is the dimension of the generic symplectic leave. It is equivariant under the action of the cluster modular group. This construction has many applications in Geometry, Mathematical Physics, Representation Theory, such as:
a construction of a geometric variant of a tensor category structure on generic representations of the DeConcini-Kac quantum groups at roots of unity; a construction of new quantum invariants of threefolds, depending on the rational Planck constant ~, which include quantum deformations of the volume of hyperbolic threefolds; a conjecture linking the space of conformal blocks for the Wess-Zumino-Witten theory for a simple Lie group G to the geometric quantization of the higher Teichmüller spaces for G.
Gold Medal, 18th International Mathematical Olympiad (1976) European Mathematical Society prize (1992)
Editor of: Selecta Mathematica (New Series) Journal of Noncommutative Geometry
China (PR.) Cluster algebras 2020, School of Mathematical Sciences, Shanghai Jiao Tong University (17-21 and 24-28 August) The Second Motivic Chern Class and Cluster Varieties (video conference)
United States Homological Mirror Symmetry and Topological Recursion, Simons Collaboration on Homological Mirror Symmetry, University of Miami (27 January 1 February) Quantization of Moduli Spaces of Local Systems at Roots of Unity and its Applications (conference)
M-Seminar 2020, Kansas State University (3 September) The Second Motivic Chern Class and Cluster Varieties (video seminar)
Gretchen and Barry Mazur Chair