Conformally Kähler Einstein-Maxwell metrics on blow-ups.
Conformally Kähler Hermitian metrics of constant Riemannian scalar curvature and J-invariant Ricci are called conformally Kähler Einstein-Maxwell metrics. In this talk, we discuss deformations and possible constructions of such metrics on blow-ups. This is a joint work in progress with Abdellah Lahdili.
Geometric flows have emerged as important tools in the application of analysis to geometry. The most basic example of a geometric flow is the curve shortening flow. At its simplest, this is a flow of plane curves, in which a plane curve is deformed in the direction of its normal vector with speed given by the curvature. It has been studied extensively in R2, especially when the flowing curve is closed. In this talk, we study the curve-shortening flow in hyperbolic 2-space. We will prove a general theorem characterizing the curves that evolve in a self-similar manner. Analogous results for curves in the 2-sphere have recently been obtained by Santos dos Reis and Tenenblat (2019) and for curves in the plane by Halldorsson (2012). Time permitting, we will discuss curves that flow so as to remain self-similar up to an overall scaling, including curves that extend to infinity and have a well-defined renormalized length. This is joint work with Ran Xie and should be accessible to undergraduates who have had a first course in curves and surfaces.
J-holomorphic curves or pseudoholomorphic curves have been of great interest to mathematics since the discovery that J-holomorphic curves allow to construct invariants of symplectic manifolds by Gromov and that those invariants are related to topological superstring theory by Witten. A crucial step towards Gromov–Witten invariants is the compactification of the moduli space of J-holomorphic curves via stable maps which was first proposed by Kontsevich and Manin. In this talk, I want to report on a supergeometric generalization of J-holomorphic curves and stable maps where the domain is a super Riemann surface. Super Riemann surfaces have first appeared as generalizations of Riemann surfaces with anti-commutative variables in superstring theory. Super J-holomorphic curves and super stable maps couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and might ultimatively lead to a supergeometric generalization of Gromov-Witten invariants. Based on arXiv:2010.15634 & arXiv:1911.05607, joint with Artan Sheshmani and Shing-Tung Yau.