Abstract:
We obtain an asymptotic formula for the counting function of the discrete spectrum for Hankel-type pseudo-differential operators with discontinuous symbols.

Abstract:
The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a `degenerate spin network', where the rotation group SO(4) is replaced by the Euclidean group of isometries of R^3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones.

Abstract:
We formulate a "correct" version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory "homological symbols algebra". As a byproduct we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.

Abstract:
The main result in this paper is a one term Szego type asymptotic formula with a sharp remainder estimate for a class of integral operators of the pseudodifferential type with symbols which are allowed to be non-smooth or discontinuous in both position and momentum. The simplest example of such symbol is the product of the characteristic functions of two compact sets, one in real space and the other in momentum space. The results of this paper are used in a study of the violation of the area entropy law for free fermions in [18]. This work also provides evidence towards a conjecture due to Harold Widom.

Abstract:
We investigate the asymptotics of the determinant of N by N Hankel matrices generated by Fisher-Hartwig symbols defined on the real line, as N becomes large. Such objects are natural analogues of Toeplitz determinants generated by Fisher-Hartwig symbols, and arise in random matrix theory in the investigation of certain expectations involving random characteristic polynomials. The reduced density matrices of certain one-dimensional systems of impenetrable bosons can also be expressed in terms of Hankel determinants of this form. We focus on the specific cases of scaled Hermite and Laguerre weights. We compute the asymptotics using a duality formula expressing the N by N Hankel determinant as a 2|q|-fold integral, where q is a fixed vector, which is valid when each component of q is natural.We thus verify, for such q, a recent conjecture of Forrester and Frankel derived using a log-gas argument.

Abstract:
The authors analyze the asymptotics of eigenvalues of Toeplitz matrices with certain continuous and discontinuous symbols. In particular, the authors prove a conjecture of Levitin and Shargorodsky on the near-periodicity of Toeplitz eigenvalues.

Abstract:
Recent interest in the Kashaev-Murakami-Murakami hyperbolic volume conjecture has made it seem important to be able to understand the asymptotic behaviour of certain special functions arising from representation theory -- for example, of the quantum 6j-symbols for SU(2). In 1998 I worked out the asymptotic behaviour of the classical 6j-symbols, proving a formula involving the geometry of a Euclidean tetrahedron which was conjectured by Ponzano and Regge in 1968. In this note I will try to explain the methods and philosophy behind this calculation, and speculate on how similar techniques might be useful in studying the quantum case.

Abstract:
It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.

Abstract:
We study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate. We generalize an asymptotic expansion by Widom [22], which was known for symbols supported on an arc. We highlight applications of our results in the Circular Unitary Ensemble and in the study of Fredholm determinants associated to the sine kernel.