The n-dimensional pair of pants is defined to be the complement of n + 2 generic hyperplanes in CPn. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism A1 algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with n+2 generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (Cn+2,W), where W = z1...zn+2. We show that the endomorphism A1 algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.

We give existence results for simple closed curves with prescribed geodesic curvature on S2, which correspond to periodic orbits of a charge in a magnetic field.

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We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.