ICMS-Sofia and IMSA-Miami


Consortium Distinguished Lecture Series

November 20, 2023, 17:00, Sofia time

ICMS-Sofia, Room 403 and via Zoom

Bubbling symplectic structures on moduli

Tony Pantev, University of Pennsylvania

Tony Pantev received his Ph.D. in 1994 from the University of Pennsylvania. He was a C.L.E. Moore Instructor at MIT, a Sloan Research Fellow, and has held visiting positions at the Isaac Newton Institute in Cambridge, England, the Centro de Investigación en Matemáticas in Guanajuato, Mexico, Ohio State University and the Institute for Advanced Studies in Princeton. He is a professor at the mathematics department of the University of Pennsylvania which he joined in 1997.

Pantev’s research interests include algebraic and differential geometry, Hodge theory, and mathematical physics. Together with Katzarkov, Toen, and Simpson he has obtained fundamental results in non-abelian Hodge theory, that led to the proof of the Shafarevich conjecture for varieties with linear fundamental groups. Together with Donagi he proved Langlands duality for Hitchin systems, a result that applies directly to mirror symmetry. Elaborating on this work, Pantev, jointly with Arinkin, and Block proved the existence of quantization for Fourier-Mukai transforms for general analytic manifolds. In a different direction, Pantev together with Katzarkov and Kontsevich developed the foundations of non-commutative geometry and non-commutative Hodge theory and studied the non-commutative aspects of the mirror correspondence. In an ongoing project with Toen, Vaquie, and Vezzosi, Pantev is exploring a major conceptual advance in derived geometry which was realized through the new notion of a shifted symplectic and Poisson structures.

Pantev has published over 50 peer-reviewed articles, one book, and has edited 3 proceedings volumes. He has supervised 14 PhD dissertations, 8 MSc students, and has mentored 6 postdocs. organized over 25 conferences, workshops, and schools on algebraic geometry, mirror symmetry, and mathematical physics. He serves on the editorial board of Advances in Mathematics, European Journal of Mathematics, and Research in Mathematical Sciences.

Abstract: I will describe a new geometric method for constructing and controlling shifted symplectic structures on the moduli of vector bundles along the fibers of a degenerating family of Calabi-Yau varieties. The method utilizes bubbling modifications of the boundaries of limiting moduli spaces to extend the symplectic structure on the general fiber to a relative symplectic structure defined on the whole family. As a proof of concept we show that this produces a universal relative symplectic structure on the moduli of Gieseker Higgs bundles along a semistable degeneration of curves. We also check that the construction works globally over the moduli stack of stable curves and show that the Hitchin map has the expected behavior in the limit. This is a joint work with Oren Ben-Bassat and Sourav Das.