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Universally optimal distribution of points on spheres

- Henry Cohn, Abhinav Kumar
- Mathematics
- 19 July 2006

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).… Expand

The sphere packing problem in dimension 24

- Henry Cohn, Abhinav Kumar, S. Miller, D. Radchenko, M. Viazovska
- Mathematics
- 21 March 2016

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… Expand

K3 surfaces and equations for Hilbert modular surfaces

- N. Elkies, Abhinav Kumar
- Mathematics
- 16 September 2012

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… Expand

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

- Henry Cohn, Abhinav Kumar, S. Miller, D. Radchenko, M. Viazovska
- Mathematics
- 13 February 2019

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… Expand

Optimality and uniqueness of the Leech lattice among lattices

- Henry Cohn, Abhinav Kumar
- Mathematics
- 16 March 2004

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… Expand

K3 Surfaces Associated with Curves of Genus Two

- Abhinav Kumar
- Mathematics
- 24 January 2007

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… Expand

Optimal simplices and codes in projective spaces

- Henry Cohn, Abhinav Kumar, Gregory T. Minton
- Mathematics
- 14 August 2013

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… Expand

Ground states and formal duality relations in the Gaussian core model.

- Henry Cohn, Abhinav Kumar, A. Schürmann
- PhysicsPhysical review. E, Statistical, nonlinear, and…
- 11 November 2009

TLDR

Elliptic fibrations on a generic Jacobian Kummer surface

- Abhinav Kumar
- Mathematics
- 9 May 2011

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… Expand

The densest lattice in twenty-four dimensions

- Henry Cohn, Abhinav Kumar
- Mathematics
- 17 June 2004

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… Expand

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