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Universally optimal distribution of points on spheres
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points).
The sphere packing problem in dimension 24
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and
K3 surfaces and equations for Hilbert modular surfaces
We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the
Universal optimality of the $E_8$ and Leech lattices and interpolation formulas
We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they
Optimality and uniqueness of the Leech lattice among lattices
We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore
K3 Surfaces Associated with Curves of Genus Two
It is known ([10, 27]) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the
Optimal simplices and codes in projective spaces
We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto
Ground states and formal duality relations in the Gaussian core model.
The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations that suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.
Elliptic fibrations on a generic Jacobian Kummer surface
We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the
The densest lattice in twenty-four dimensions
In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in R^24. Complete details will appear elsewhere, but here we