14 | IHES Annual Report 2020

Maxim KONTSEVICH Mathematics, permanent professor since 1995,

AXA-IHES Chair for Mathematics.

With Gourab Bhattacharya, MAXIM KONTSEVICH introduced a notion of Kaehler metric for a noncommutative algebra over complex numbers. For such a metric , one can define moment map equations for Hermitian norms on left modules. This leads to a unified algebraic for many equations in mathematical physics, including Hermitian Yang-Mills equation, monopole equation, Nahm equation, etc. Together with Alexander Odesskii, they proved a new formula linking p-curvatures of differential equations depending on small parameter and eigenvalues of the monodromy along certain small loops surrounding several critical points.

Together with Graeme Segal, they proposed a new axiomatics for unitary quantum field theories on curved background, based on a holomorphicity in the space of complex-valued metrics. Together with Yan Soibelman, they continued a vast project in algebraic analysis which can be called Holomorphic Floer theory . In particular, they developed a notion of analytic wall-crossing structure. Also, they found a new approach to quantum mechanics via deformation quantization, leading to a new broad class of resurgent series, and non-perturbative quantization of arbitrary complex polynomial Hamiltonians. Finally, together with Ludmil Katzarkov, Maxim Kontsevich proposed a new theory of dimension spectra associated with Frobenius manifolds, as a part of a general project linking quantum cohomology and birational geometry. In contrast with previous applications to lower-dimensional geometry (e.g. the case of cubic 4-folds), the new theory has the potential to be applicable in arbitrary dimensions. One of hypothetical corollaries is that any odd-dimensional smooth Fano variety which is a complete intersection of sufficiently high degree, is not rational.

SCIENTIFIC ACTIVITY

DISTINCTIONS

Otto Hahn Medal (1992) EMS Prize, 1st European Congress of Mathematics (1992) Henri Poincaré Prize, Daniel Iagolnitzer Foundation (1997) Fields Medal (1998) Crafoord Prize (2008) Shaw Prize (2012) Breakthrough Prize in Fundamental Physics (2012) Breakthrough Prize in Mathematics (2014) Doctor Honoris Causa, Aarhus Universitet, Denmark (2014) Doctor Honoris Causa, Universität Wien (2015) Academia Europaea, member Académie des sciences de Paris, member National Academy of Sciences, foreign member London Mathematical Society, honorary member Simons Collaboration in Homological Mirror Symmetry (3 years starting in September 2019) ERC Grant "ReNewQuantum" (5 years starting in September 2019)

Editeur of: Compositio Mathematica International Mathematical Research Papers Letters in Mathematical Physics Les Publications Mathématiques de l IHES Selecta Mathematica (New Series) Communications in Number Theory and Physics Journal of Noncommutative Geometry