An LMS online lecture course in Poisson structures.
Tag - Differential geometry
In the lectures we will cover the basic concepts of modern differential geometry. Differential geometry studies smooth manifolds, that is, geometric objects that, roughly speaking, locally look like ℝn (and whose global topological properties are not too weird, see for example the so-called 'long line'). You already know examples such as the n-sphere Sn, or smooth surfaces in ℝ3 from analysis. On smooth manifolds we will study a number of constructions and structures, such as vector fields, metrics and various curvature concepts. In addition, smooth manifolds are suitable as spaces for ordinary and partial differential equations, which allow different global topological properties compared to regions in ℝn (e.g. PDEs on the Klein bottle or on the real-projective spaces ℝPn). A focus of this lecture will be submanifolds and induced geometric structures. We will also look at some topics from the perspective of the calculus of variations, e.g. geodesics as critical points of the energy functional. This lecture is also expressly suitable for students of physics courses, as differential geometry represents a fundamental theoretical basis for many modern theories in physics (especially ART, gauge theories such as Yang-Mills -Theory, SuSy, SuGra,...).
We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.
Mirror symmetry predicts that the moduli space of complex structures/special Lagrangians on one Calabi-Yau is dual to the moduli space of complexified forms/stable bundles on the mirror Calabi-Yau. However, the precise definition of a complexified Kähler form/stable bundle has remained mysterious. I will discuss these notions in the setting of Strominger-Yau-Zaslow mirror symmetry, the connection to fully non-linear PDEs and algebro-geometric stability.
We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. This construction will provide one with a general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface threefolds (recently analyzed in full detail by Tanaka and Thomas) and such nested Hilbert schemes. This construction (via applying Mochizuki’s wallcrossing techniques) enables one to obtain a relations between the generating series of Seiberg-Witten invariants of the surface, the Vafa-Witten invariants and some modular forms.
We generalize the classical light bulb theorem to four dimensions. I.e. a smooth 2-sphere in S2 × S2 that is transverse to S2 × 0 and homologous to 0 × S2 is smoothly isotopically standard. We discuss generalizations to spheres in other spaces and applications.
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