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We prove a quantitative bi-Lipschitz non-embedding theorem for the Heisenberg group with its Carnot–Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans–Linial semidefinite relaxation of the sparsest cut problem.

Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on Y. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.

In this paper the centroaffine Minkowski problem, a critical case of the L p-Minkowski problem in the n+ 1 dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.