10,066 research outputs found
Inhomogeneous extreme forms
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two
reduction theories for lattices, well-suited for sphere packing and covering
problems. In his first memoir a characterization of locally most economic
packings is given, but a corresponding result for coverings has been missing.
In this paper we bridge the two classical memoirs.
By looking at the covering problem from a different perspective, we discover
the missing analogue. Instead of trying to find lattices giving economical
coverings we consider lattices giving, at least locally, very uneconomical
ones. We classify local covering maxima up to dimension 6 and prove their
existence in all dimensions beyond.
New phenomena arise: Many highly symmetric lattices turn out to give
uneconomical coverings; the covering density function is not a topological
Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in
Annales de l'Institut Fourie
Homological stability for classical groups
We prove a slope 1 stability range for the homology of the symplectic,
orthogonal and unitary groups with respect to the hyperbolic form, over any
fields other than , improving the known range by a factor 2 in the case of
finite fields. Our result more generally applies to the automorphism groups of
vector spaces equipped with a possibly degenerate form (in the sense of Bak,
Tits and Wall). For finite fields of odd characteristic, and more generally
fields in which -1 is a sum of two squares, we deduce a stability range for the
orthogonal groups with respect to the Euclidean form, and a corresponding
result for the unitary groups.
In addition, we include an exposition of Quillen's unpublished slope 1
stability argument for the general linear groups over fields other than ,
and use it to recover also the improved range of
Galatius-Kupers-Randal-Williams in the case of finite fields, at the
characteristic.Comment: v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams
improved stability range for general linear groups over finite field
On the integrability of symplectic Monge-Amp\'ere equations
Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij})
the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a
linear relation among all possible minors of U. Particular examples include the
equation det U=1 governing improper affine spheres and the so-called heavenly
equation, u_{13}u_{24}-u_{23}u_{14}=1, describing self-dual Ricci-flat
4-manifolds. In this paper we classify integrable symplectic Monge-Ampere
equations in four dimensions (for n=3 the integrability of such equations is
known to be equivalent to their linearisability). This problem can be
reformulated geometrically as the classification of 'maximally singular'
hyperplane sections of the Plucker embedding of the Lagrangian Grassmannian. We
formulate a conjecture that any integrable equation of the form F(u_{ij})=0 in
more than three dimensions is necessarily of the symplectic Monge-Ampere type.Comment: 20 pages; added more details of proof
Generalising the Hardy-Littlewood Method for Primes
The Hardy-Littlewood method is a well-known technique in analytic number
theory. Among its spectacular applications are Vinogradov's 1937 result that
every sufficiently large odd number is a sum of three primes, and a related
result of Chowla and Van der Corput giving an asymptotic for the number of
3-term progressions of primes, all less than N. This article surveys recent
developments of the author and T. Tao, in which the Hardy-Littlewood method has
been generalised to obtain, for example, an asymptotic for the number of 4-term
arithmetic progressions of primes less than N.Comment: 26 pages, submitted to Proceedings of ICM 200
A brief introduction to Enriques surfaces
This is a brief introduction to the theory of Enriques surfaces over
arbitrary algebraically closed fields. Some new results about automorphism
groups of Enriques surfaces are also included.Comment: Minor corrections, to appear in "Development of Moduli Theory---Kyoto
2013
Neutrino Interactions in Hot and Dense Matter
We study the charged and neutral current weak interaction rates relevant for
the determination of neutrino opacities in dense matter found in supernovae and
neutron stars. We establish an efficient formalism for calculating differential
cross sections and mean free paths for interacting, asymmetric nuclear matter
at arbitrary degeneracy. The formalism is valid for both charged and neutral
current reactions. Strong interaction corrections are incorporated through the
in-medium single particle energies at the relevant density and temperature. The
effects of strong interactions on the weak interaction rates are investigated
using both potential and effective field-theoretical models of matter. We
investigate the relative importance of charged and neutral currents for
different astrophysical situations, and also examine the influence of
strangeness-bearing hyperons. Our findings show that the mean free paths are
significantly altered by the effects of strong interactions and the
multi-component nature of dense matter. The opacities are then discussed in the
context of the evolution of the core of a protoneutron star.Comment: 41 pages, 25 figure
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Invisibility and Inverse Problems
This survey of recent developments in cloaking and transformation optics is
an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual
Meeting of the American Mathematical Society.Comment: 68 pages, 12 figures. To appear in the Bulletin of the AM
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