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The BBVA Foundation Frontiers of Knowledge Award in the Basic Sciences category goes, in this sixteenth edition, to Claire Voisin and Yakov Eliashberg for their outstanding contributions to two faces of geometry, algebraic and symplectic.

The spaces in algebraic geometry are defined by solutions to algebraic equations using complex numbers; those in symplectic geometry are generalizations of the phase space which describes the state of a physical system. Both types of geometry involve spaces of high dimensions which are difficult to visualize and which necessitate new mathematical techniques to understand and study.

Voisin’s results in algebraic geometry have shed new light on the role of Hodge theory and led the subject into new realms of activity by her work on hyperkähler manifolds. Eliashberg’s extensive results in symplectic and contact geometry gave in particular a topological characterization of Stein manifolds and the identification of a dichotomy between flexible and rigid aspects of contact manifolds. The two areas of mathematics have been reinvigorated in recent years because they relate to quantum field theories studied in theoretical physics. Their roles are interchanged in that context by mirror symmetry.

Voisin and Eliashberg have played essential roles in developing these different aspects of geometry, in particular by adapting concepts from either side, crossing the boundary between the two disciplines. Their work has inspired a high level of international research activity in both areas of mathematics.