We study BPS states of 5d N=1 SU(2) Yang-Mills theory on S^1×R^4. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface F_0. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d N=2 Seiberg-Witten theory, recovering spectral networks in the limit.

This is the note for a series of lectures that the author gave at the Centre de Recerca Matem´atica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

We obtain two-sided sub-Gaussian estimates of heat kernels for strongly local Dirichlet forms on intervals, equipped with self-similar measures generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. We first give a framework for heat kernel estimates on intervals, and then consider examples of self-similar measures to illustrate this phenomenon. These examples include the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. The walk dimensions obtained under these new metrics are strictly greater than 2 and are closely related to the spectral dimension of fractal Laplacians.

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