160 research outputs found
Lattice packings with gap defects are not completely saturated
We show that a honeycomb circle packing in with a linear gap defect
cannot be completely saturated, no matter how narrow the gap is. The result is
motivated by an open problem of G. Fejes T\'oth, G. Kuperberg, and W.
Kuperberg, which asks whether of a honeycomb circle packing with a linear shift
defect is completely saturated. We also show that an fcc sphere packing in
with a planar gap defect is also not completely saturated
Four symmetry classes of plane partitions under one roof
In previous paper, the author applied the permanent-determinant method of
Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to
obtain a determinant or a Pfaffian that enumerates each of the ten symmetry
classes of plane partitions. After a cosmetic generalization of the Kasteleyn
method, we identify the matrices in the four determinantal cases (plain plane
partitions, cyclically symmetric plane partitions, transpose-complement plane
partitions, and the intersection of the last two types) in the representation
theory of sl(2,C). The result is a unified proof of the four enumerations
Non-involutory Hopf algebras and 3-manifold invariants
We present a definition of an invariant #(M,H), defined for every
finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for
every closed, framed 3-manifold M. When H is a quantized universal enveloping
algebra, #(M,H) is closely related to well-known quantum link invariants such
as the HOMFLY polynomial, but it is not a topological quantum field theory.Comment: 36 page
Circumscribing constant-width bodies with polytopes
Makeev conjectured that every constant-width body is inscribed in the dual
difference body of a regular simplex. We prove that homologically, there are an
odd number of such circumscribing bodies in dimension 3, and therefore
geometrically there is at least one. We show that the homological answer is
zero in higher dimensions, a result which is inconclusive for the geometric
question. We also give a partial generalization involving affine
circumscription of strictly convex bodies.Comment: 6 pages. This version has minor changes suggested by the referee.
Note that Makeev, and independently Hausel, Makai, and Szucs, also obtained
the main resul
A volume-preserving counterexample to the Seifert conjecture
We prove that every 3-manifold possesses a , volume-preserving flow with
no fixed points and no closed trajectories. The main construction is a
volume-preserving version of the Schweitzer plug. We also prove that every
3-manifold possesses a volume-preserving, flow with discrete closed
trajectories and no fixed points (as well as a PL flow with the same geometry),
which is needed for the first result. The proof uses a Dehn-twisted Wilson-type
plug which also preserves volume
A tracial quantum central limit theorem
We prove a central limit theorem for non-commutative random variables in a
von Neumann algebra with a tracial state: Any non-commutative polynomial of
averages of i.i.d. samples converges to a classical limit. The proof is based
on a central limit theorem for ordered joint distributions together with a
commutator estimate related to the Baker-Campbell-Hausdorff expansion. The
result can be considered a generalization of Johansson's theorem on the
limiting distribution of the shape of a random word in a fixed alphabet as its
length goes to infinity [math.CO/9906120,math.PR/9909104].Comment: 7 page
Random words, quantum statistics, central limits, random matrices
Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved
[math.CO/9906120] that the expected shape \lambda of the semi-standard tableau
produced by a random word in k letters is asymptotically the spectrum of a
random traceless k by k GUE matrix. In this article we give two arguments for
this fact. In the first argument, we realize the random matrix itself as a
quantum random variable on the space of random words, if this space is viewed
as a quantum state space. In the second argument, we show that the distribution
of \lambda is asymptotically given by the usual local limit theorem, but the
resulting Gaussian is disguised by an extra polynomial weight and by reflecting
walls. Both arguments more generally apply to an arbitrary finite-dimensional
representation V of an arbitrary simple Lie algebra g. In the original
question, V is the defining representation of g = su(k).Comment: 11 pages. Minor changes suggested by the refere
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