TBA

The celebrated Newlander-Nirenberg theorem states that on a smooth manifold, an almost complex structure $J$ is a complex structure if and only if it is integrable, namely, the Nijenhuis tensor $N_J$ vanishes. It was known from Hill and Taylor that if $J$ has Hölder regularity above $C^{1/2}$ then $N_J$ makes sense as a tensor with distributional coefficients. However $N_J$ is undefined for generic $C^{1/2}$ tensor due to the failure of multiplication for $C^{1/2}$ functions and $C^{-1/2}$ distributions.

In the talk, we will explore the integrability condition when $J$ has regularity below $C^{1/2}$. We give a necessary and sufficient condition for $J$ being a complex structure (at least) for $J\in C^{1/3+}$ using Bony's paradifferential calculus. If time permitted, I will also talk about how our method may be related to rough path theory in stochastic analysis and the Gromov's non-embedding problem in algebraic topology.

This is an in progress work joint with Gennady Uraltsev.

3d mirror symmetry predicts deep relationships between certain algebraic symplectic varieties. One such expectation is an "equivalence" between curve counts in a Higgs branch and those in the corresponding Coulomb branch. When it can be precisely formulated, this equivalence takes the form of an equality (after analytic continuation and change of variables) of meromorphic functions associated to the two branches. Bow varieties provide the largest currently known setting where the appropriate curve counts can be defined and their equivalence precisely formulated. In this talk, I will give an overview of these ideas and discuss my work with Tommaso Botta, in which we prove the duality of curve counts for finite type A bow varieties. Our proof combines geometric, combinatorial, and analytic arguments to eventually reduce to the case of the cotangent bundle of the complete flag variety. Time permitting, I will also discuss ongoing work to incorporate "descendant insertions" into the statements by using Hecke modifications of vector bundles.

TBA