Seminar

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Elham Matinpour, Title: Morse Index and Stability of Singular Minimal Surfaces

Abstract:  The Morse index of a minimal surface reflects its stability within the ambient space, quantifying the number of independent directions in which the surface can be deformed to reduce its area. For smooth minimal surfaces, calculating the Morse index poses significant challenges, and for singular minimal surfaces, even less is known. In this talk, I will discuss some progress in understanding the Morse index of singular minimal surfaces, with introducing a one-parameter 

family of Y-singular surfaces, {Y_t}_{t∈(0,π]}, which are complete, two-sided minimal surfaces in R^3.Each surface Y_t in this family has a Morse index of two and a nullity of five. Additionally, I will show that as the parameter t approaches zero, the family converges to a union of a Y-catenoid and a plane. The Y-catenoid, a key limiting surface in this family, possesses a Morse index of one and a nullity of three.

Louis Dailly, Title: Uniformization of klt Fano pairs with Miyaoka–Yau equality

Abstract: In the early 20th century, Poincaré and Koebe classified the universal coverings of compact, connected Riemann surfaces, showing that there are only three possibilities: the projective space P1, the complex plane C, and the unit disk B1. In higher dimensions, it is generally not possible to fully describe the universal covering of a Kähler manifold X. However, if X satisfies certain geometric conditions (namely, if X is Kähler--Einstein and has constant holomorphic bisectional curvature), it can be covered by a projective space Pn, a complex space Cn, or a unit ball Bn, depending on the sign of its curvature. This talk aims to present singular versions of these results and explore possible generalizations that specifically describe quotients of projective spaces.