We study Legendrian embeddings of a compact Legendrian submanifold L sitting in a closed contact manifold (M,ξ) whose contact structure is supported by a (contact) open book OB on M. We prove that if OB has Weinstein pages, then there exist a contact structure ξ′ on M, isotopic to ξ and supported by OB, and a contactomorphism f:(M,ξ)→(M,ξ′) such that the image f(L) of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of OB.

Let Λ be a numerical semigroup, C⊆An the monomial curve singularity associated to Λ, and T its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in Λ in terms of the codimension and the maximum degree of the equations of T, when T is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.

A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.

We consider surfaces of geometric genus 3 with the property that their transcendental cohomology splits into 3 pieces, each piece coming from a K3 surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of (N−12)×(N−12) inside triangles with vertices (0,0),(0,λβ−−√) and (λ/β−−√,0) with respect to β>0, for fixed λ≥0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N+σ)×(N+τ), for σ,τ>−1. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, σ=τ=−12, is a critical case.