Abstract:
When a complex surface X admits a nowhere vanishing holomorphic 2-form, it determines a (holomorphic) symplectic structure on X. We study the symplectic geometry of such a symplectic structure when X is an elliptic surface. When the elliptic fibration is nonisotrivial, we define a factorization of Kodaira's functional invariant, which determines the symplectic geometry of a nonisotrivial elliptic fibration. This leads to a classification of isogenies of nonisotrivial symplectic elliptic fibrations with a fixed source. We also classify isogenies of symplectic elliptic fibrations with a fixed target in terms of germs of singular fibers.
As an application, we prove that a symplecto-biholomorphic map between germs of fibers of nonisotrivial elliptic K3 surfaces can be extended to compositions of isogenies of K3 surfaces. This is a joint work with Guolei Zhong.
Introduction:
Jun-Muk Hwang is the founding director of the Center for Complex Geometry at the Institute for Basic Science (IBS) in Korea, established in September 2020. He received the Korea Science Prize in 2001 and the Ho-Am Prize in 2009. He was an invited speaker at the ICM2006 and a plenary speaker at ICM 2014.
Hwang's research focuses primarily on complex algebraic geometry, with a particular emphasis on using the theory of rational curves to study the geometry and classification of projective algebraic varieties. His work has profoundly revealed the intrinsic relationships among Fano varieties, homogeneous manifolds, and projective varieties with a rich structure of rational curves. Hwang's work combines geometric intuition with algebraic rigor. His achievements have not only advanced the field of complex geometry but also deeply intersected with representation theory, differential geometry, and mathematical physics. As a leading figure in Korean complex geometry, he has cultivated an active research team, making the Center for Complex Geometry a major international hub in this field.
Abstract
Enumerative geometers and their colleagues in representation theory and mathematical physics are very excited about the new perspectives on old and new problems offered by the nascent field of 3-dimensional mirror symmetry. While most formulations or explanations of what 3-dimensional mirror symmetry is require a lot of prerequisites and a high level of abstraction, some of its core predictions can be easily cast in the language that people like P.L. Chebyshev, C.G. Jacobi, and I.G. Macdonald would have no problem grasping. This is what I will try to do in this talk, which I hope will be accessible to the general mathematical audience.
Biography
Professor Andrei Okounkov, born in Moscow, Russia, is a world-renowned mathematician, who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions.
He obtained his PhD from Lomonosov Moscow State University in 1995, under the supervision of Alexandre Kirillov. Since then he held professorships at world-leading academic institutions including the University of California, Berkeley, the University of Chicago, and Princeton University. He is now Samuel Eilenberg Professor of Mathematics at Columbia University.
Professor Okounkov is a member of the US National Academy of Sciences (2012), the American Academy of Arts and Sciences (2016), the Royal Swedish Academy of Sciences, and was elected as a Foreign Member of the Chinese Academy of Sciences in 2023.
Professor Okounkov received many important prizes including European Mathematical Society Prize (2004), and the Fields Medal (2006). He is a ICM Plenary speaker in 2018.