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We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

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We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows ($X$,μ,$T$_{$t$}) and$f$∈$L$^{$p$}($X$,μ), there is a set$X$_{$f$}⊂$X$of probability one, so that for all$x$∈$X$_{$f$}, $$\lim_{s\downarrow0}\int_{s<|t|<1/s}e^{i\theta t} f(\textup{T}_tx)\,\frac{dt}t\quad\text{exists for all}\ \theta.$$ The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.

We study derived categories arising from quivers with potential associated to a decorated marked surface S_Delta, in the sense taken in a paper by Qiu. We prove two conjectures from Qiu’s paper in which, under a bijection between certain objects in these categories and certain arcs in S, the dimensions of morphisms between these objects equal the intersection numbers between the corresponding arcs.