Sonstige Vorträge

Abstract: The answer that I will propose is a higher categorical structure in symplectic geometry that includes all Fukaya categories and geometric functors between them. Its highly involved algebraic and analytic details are the topic of joint work with Nate Bottman. However, its basic structure can be understood as a natural -- yet not previously studied -- extension of the category of topological spaces and continuous maps. So the goal of this talk is to make broadly accessible the two underlying ideas: The notion of quilts (collections of maps related by ``seam conditions'') and the string diagram approach to constructing and making sense of 2-categories.

The Prym varieties introduced by Mumford in the 70's opened new perspectives in the study of the moduli space of abelian varieties: Prym varieties are abelian varieties arising from covers between curves. They generalize Jacobians and provide a larger class of abelian varieties which can be understood in terms of 1-dimensional geometric objects. For instance, it is known that the general principally polarized abelian variety of dimension less than or equal to 5 is the Prym variety of some unramified double cover. The Prym map between the moduli space of covers and the moduli space of abelian varieties assigns to each covering its Prym variety. In this talk I will discuss the geometry of the fibres of the Prym map for double covers in low genera, the global injectivity of the Prym map for ramified double covers, as well as the most recent results on the general injectivity of the Prym map for cyclic covers.

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