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.
In this talk I will explain a construction of equivariant version of Lagrangian Floer homology with compact group action. I will explain some of the ideas to construct it and its (potential) applications.
Modelling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to a degenerate elliptic non-linear PDE. A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry.