Achill Schürmann
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Achill Schürmann has a total of 9 co-authors. Here are the top co-authors by number of shared publications:
- Henry Cohn
Microsoft Research New England One Memorial Drive 02142 Cambridge MA USA
5 shared publications - Abhinav Kumar
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
4 shared publications - Frank Vallentin
Delft Institute of Applied Mathematics, Technical University of Delft, Delft, The Netherlands
2 shared publications - Mathieu Dutour Sikiric
Rudjer Boskovic Institute, Bijenicka 54, 10000 Zagreb, Croatia
2 shared publications - Noam D. Elkies
Department of Mathematics, Harvard University, Cambridge, USA
2 shared publications Brandon Ballinger
Google, Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043
1 shared publication
Latest Publications
- Open Access
Point configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
Open AccessPoint configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
- Open Access
Ground states and formal duality relations in the Gaussian core model
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.
Open AccessGround states and formal duality relations in the Gaussian core model
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.
- 23 June 2010
- Published by Elsevier in Advances in Mathematics
- Vol. 225 (5) , 2546-2564
Perfect, strongly eutactic lattices are periodic extreme
- 23 June 2010
- Published by Elsevier in Advances in Mathematics
- Vol. 225 (5) , 2546-2564
Perfect, strongly eutactic lattices are periodic extreme
- Open Access
- 3 June 2010
- Published by Springer Nature in Discrete & Computational Geometry
- Vol. 44 (4) , 904-911
The Contact Polytope of the Leech Lattice
- 3 June 2010
- Published by Springer Nature in Discrete & Computational Geometry
- Vol. 44 (4) , 904-911
Open AccessThe Contact Polytope of the Leech Lattice
- Open Access
- 23 March 2010
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 138 (8) , 2863-2872
Point configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R 3 \mathbb {R}^3 , and his classification is equivalent to the converse for R 3 \mathbb {R}^3 . In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
- 23 March 2010
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 138 (8) , 2863-2872
Open AccessPoint configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R 3 \mathbb {R}^3 , and his classification is equivalent to the converse for R 3 \mathbb {R}^3 . In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
- Open Access
The contact polytope of the Leech lattice (complete version)
The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.
Open AccessThe contact polytope of the Leech lattice (complete version)
The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.
- 14 December 2009
- Published by American Physical Society (APS) in Physical Review E
- Vol. 80 (6) , 061116
Ground states and formal duality relations in the Gaussian core model
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.
- 14 December 2009
- Published by American Physical Society (APS) in Physical Review E
- Vol. 80 (6) , 061116
Ground states and formal duality relations in the Gaussian core model
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.
- 1 January 2009
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 18 (3) , 257-283
Experimental Study of Energy-Minimizing Point Configurations on Spheres
In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.
- 1 January 2009
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 18 (3) , 257-283
Experimental Study of Energy-Minimizing Point Configurations on Spheres
In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.