Thursday

Logarithmic Gromov--Witten (log GW) theory is about counting curves in a smooth complex projective variety (or closed symplectic manifold) which have prescribed tangency orders with a given simple normal crossings divisor. I will discuss recent work comparing algebraic and symplectic log GW moduli spaces, as well as ongoing work constructing virtual fundamental classes on symplectic log GW moduli spaces. Both of these are joint with Mohammad Farajzadeh-Tehrani.

I will describe an index-theoretic constraint for (differentiable) Lefschetz fibrations with 4-dimensional fibers. This constraint relates the index of a Dirac operator on the total space to the Seiberg–Witten invariants of the fibers and the monodromy of the fibration, and is obtained by studying the framed bordism class of suitable Seiberg–Witten moduli spaces. As an application, it yields obstructions to products of Dehn twist diffeomorphisms along self-intersection (-2)-spheres being smoothly isotopic to the identity in various 4-manifolds. This is based on joint work with Hokuto Konno, Jianfeng Lin, and Anubhav Mukherjee.

I will discuss singularity formation and compactness phenomena for Hermitian–Yang–Mills connections in higher dimensions, emphasizing their interaction with algebraic geometry. I will begin with the classical existence and compactness theory for Hermitian–Yang–Mills connections and briefly indicate some extensions, drawing in part on joint work with Richard Wentworth. I will then recall a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian–Yang–Mills connections, from joint work with Song Sun. In the final part of the talk, I will discuss recent work on bubbling in degenerating families of bundles, focusing first on point singularities on threefolds and then on bubbling at points on surfaces, with the surface case serving as a local model for bubbling along curves in threefolds.

Character varieties of Riemann surfaces provide an important class of symplectic manifolds. For example, given a punctured Riemann surface S, one can consider the space of all conjugacy classes of SU(2)-representations of the fundamental group of S such that the image of each small loop around a puncture lies in the conjugacy class of traceless elements in SU(2). One may also consider a variation in which some of these loops are instead mapped to a central element of SU(2). These character varieties are (possibly singular) symplectic manifolds. In the case that the character variety is smooth, taking pullback of the elements of the character variety with respect to any fixed non-trivial element of the mapping class of S determines a symplectomorphism, and it is natural to ask whether this symplectomorphism is given by a Hamiltonian isotopy. A version of this question was posed by Dostoglou and Salamon, and results in this direction were obtained by Seidel, Smith and Wehrheim–Woodward. In this talk I will discuss how Yang--Mills gauge theory and results on Heegaard splittings can be used to study this question. I will also explain how similar techniques can be used to study Lagrangian spheres in the Del Pezzo surface given by the projective plane blown up at 5 points. This is based on joint work with Chris Scaduto.

Results of W. Thurston provide an ample supply of examples, but if we impose a tautness condition, then the existence question for codimension-2 foliations (and, more generally, foliations with singularities modelled on the holomorphic case) appears to become very interesting. We will present some examples and questions, and just one simple uniqueness result, pointing to the fact that taut foliations are not completely flexible.