Constant mean curvature (CMC) tori in \(S^3\), \(R^3\) or \(H^3\) are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes. I will address these periodicity questions for the spherical and Euclidean cases, using Whitham deformations, which I will explain.

The IP theorem, which asserts that IP = PSPACE (Lund et. al., and Shamir, in J. ACM 39(4)), is one of the major achievements of complexity theory. The known proofs of the theorem are based on the arithmetization technique, which transforms a quantified Boolean formula into a related polynomial. The intuition that underlies the use of polynomials is commonly explained by the fact that polynomials constitute good error correcting codes. However, the known proofs seem tailored to the use of polynomials, and do not generalize to arbitrary error correcting codes.

If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript \(\mathrm{BC}\). Let \(p:M\to S\) be a proper holomorphic submersion of complex manifolds. Let \(F\) be a holomorphic vector bundle on \(M\) and let \(Rp_{*}F\) be its direct image.