Congling Qiu, MIT

Title: Deformation among diagonals and symmetry of a curve

Abstract: The concept of deformation is ubiquitous in topology and geometry. Perhaps unexpectedly, in number theory, deformation among algebraic submanifolds of an algebraic manifold also plays a key role, as it provides a way to extract finite and meaningful counts from the infinite set of such submanifolds. Moreover, these counting results are expected to be governed by intriguing analytic functions that generalize the Riemann Zeta function. This deep connection is the formidable Beilinson–Bloch conjecture, a broad generalization of the Millennium Birch–Swinnerton-Dyer conjecture. A key early example in this direction is the deformation of diagonals in the self-product of a curve, introduced by Gross and Schoen in the 1990s. In this talk, I will focus on this example, beginning with more general historical context and then discussing surprising recent progress in both geometry and number theory, where the symmetry of the curve plays a crucial role.