Basic zeta functions and some applications in physics

It is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the Casimir effect and Bose-Einstein condensation. A brief introduction into these areas is given in the respective sections. We will consider exclusively spectral zeta functions, that is zeta functions arising from the eigenvalue spectrum of suitable differential operators. There is a set of technical tools that are at the very heart of understanding analytical properties of essentially every spectral zeta function. Those tools are introduced using the well-studied examples of the Hurwitz, Epstein and Barnes zeta function. It is explained how these different examples of zeta functions can all be thought of as being generated by the same mechanism, namely they all result from eigenvalues of suitable (partial) differential operators. It is this relation with partial differential operators that provides the motivation for analyzing the zeta functions considered in these lectures. Motivations come for example from the questions "Can one hear the shape of a drum?" and "What does the Casimir effect know about a boundary?". Finally "What does a Bose gas know about its container?"Comment: To appear in "A Window into Zeta and Modular Physics", Mathematical Sciences Research Institute Publications, Vol. 57, 2010, Cambridge University Pres

Embedding into Banach spaces with finite dimensional decompositions

This paper deals with the following types of problems: Assume a Banach space $X$ has some property (P). Can it be embedded into some Banach space $Z$ with a finite dimensional decomposition having property (P), or more generally, having a property related to (P)? Secondly, given a class of Banach spaces, does there exist a Banach space in this class, or in a closely related one, which is universal for this class?Comment: 26 page

The instability of naked singularities in the gravitational collapse of a scalar field

One of the fundamental unanswered questions in the general theory of relativity is whether ``naked'' singularities, that is singular events which are visible from infinity, may form with positive probability in the process of gravitational collapse. The conjecture that the answer to this question is in the negative has been called ``cosmic censorship.'' The present paper, which is a continuation previous work, addresses this question in the context of the spherical gravitational collapse of a scalar field.Comment: 35 pages, published version, abstract added in migratio

An extension of the Artin-Mazur theorem

Let M be a compact manifold. We call a mapping f in C^r(M,M) an Artin-Mazur mapping if the number of isolated periodic points of f^n grows at most exponentially in n. Artin and Mazur posed the following problem: What can be said about the set of Artin-Mazur mappings with only transversal periodic orbits? Recall that a periodic orbit of period n is called transversal if the linearization df^n at this point has for an eigenvalue no nth roots of unity. Notice that a hyperbolic periodic point is always transversal, but not vice versa. We consider not the whole space C^r(M,M) of mappings of M into itself, but only its open subset Diff^r(M). The main result of this paper is the following theorem: Let 1 <= r < \infty. Then the set of Artin-Mazur diffeomorphisms with only hyperbolic periodic orbits is dense in the space Diff^r(M).Comment: 13 pages, published version, abstract added in migratio

Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology

Classical definitions of locally complete intersection (l.c.i.) homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms phi : R \longrightarrow S of commutative noetherian rings. It is defined in terms of the structure of phi in a formal neighborhood of each point of Spec S. We characterize the l.c.i. property by different conditions on the vanishing of the Andr\'e-Quillen homology of the R-algebra S. One of these descriptions establishes a very general form of a conjecture of Quillen that was open even for homomorphisms of finite type: If S has a finite resolution by flat R-modules and the cotangent complex \cot SR is quasi-isomorphic to a bounded complex of flat S-modules, then phi is l.c.i. The proof uses a mixture of methods from commutative algebra, differential graded homological algebra, and homotopy theory. The l.c.i. property is shown to be stable under a variety of operations, including composition, decomposition, flat base change, localization, and completion. The present framework allows for the results to be stated in proper generality; many of them are new even with classical assumptions. For instance, the stability of l.c.i. homomorphisms under decomposition settles an open case in Fulton's treatment of orientations of morphisms of schemes.Comment: 33 pages, published versio

Effect of long-range Coulomb interaction on shot-noise suppression in ballistic transport

We present a microscopic analysis of shot-noise suppression due to long-range Coulomb interaction in semiconductor devices under ballistic transport conditions. An ensemble Monte Carlo simulator self-consistently coupled with a Poisson solver is used for the calculations. A wide range of injection-rate densities leading to different degrees of suppression is investigated. A sharp tendency of noise suppression at increasing injection densities is found to scale with a dimensionless Debye length related to the importance of space-charge effects in the structure.Comment: RevTex, 4 pages, 4 figures, minor correction

Quantum Magnetic Deflagration in Mn12 Acetate

We report controlled ignition of magnetization reversal avalanches by surface acoustic waves in a single crystal of Mn12 acetate. Our data show that the speed of the avalanche exhibits maxima on the magnetic field at the tunneling resonances of Mn12. Combined with the evidence of magnetic deflagration in Mn12 acetate (Suzuki et al., cond-mat/0506569) this suggests a novel physical phenomenon: deflagration assisted by quantum tunneling.Comment: 4 figure

Any flat bundle on a punctured disc has an oper structure

We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.Comment: 12 page

Obstructions to the existence of fold maps

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be interpreted as primary and secondary obstructions to the elimination of certain singularities. We also discuss the relationship between the existence problem of fold maps and that of vector fields of stabilized tangent bundles.Comment: 17 pages; the final version of the preprint will be published in the Journal of the LM