Narrow Results

In this article, unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables sharing small identical sets for moving targets are given.

In this talk various spin effects in hard exclusive electroproduction of mesons are briefly reviewed. The data are discussed in the light of recent theoretical calculations within the frame work of the handbag approach. This talk has been presented at the Conference in Honor of Prof. Anatoly Efremov's 75th Birthday held at Trento, July, 2009.

The aim of this paper is to explore the possibility of discovering a fermionic field with mass dimension one, the Elko field, in the Large Hadron Collider. Due to its mass dimension, an Elko can only interact either with Standard Model spinors and gauge fields at one-loop order or at tree level through a quartic interaction with the Higgs field. In this Higgs portal scenario, the Elko is a viable candidate to a dark matter constituent which has been shown to be compatible with relic abundance measurements from WMAP and direct dark matter searches. We propose a search strategy for this dark matter candidate in the channel at the LHC. We show the LHC potential to discover the Elko considering a triple Higgs–Elkos coupling as small as ~0.5 after 1 ab^-1 of integrated luminosity. Some phenomenological consequences of this new particle and its collider signatures are also discussed.

On the basis of the one-band t–t′-Hubbard model a self-consistent renormalization theory of magnetic short-range order (SRO) in the paramagnetic phase is presented combining the four-field slave-boson functional-integral scheme with the cluster variational method. Contrary to previous SRO approaches the SRO is incorporated at the saddle-point and pair-approximation levels. A detailed numerical evaluation of the theory is performed at zero temperature, where both the hole- and electron-doped cases as well as band-structure effects are studied. The ground-state phase diagram shows the suppression of magnetic long-range order in favour of a paramagnetic phase with antiferromagnetic SRO in a wide doping region. In this phase the uniform static spin susceptibility increases upon doping up to the transition to the Pauli paraphase. Comparing the theory with experiments on high-T_c cuprates a good agreement is found.

Layered models are models in which the coupling constants depend in an arbitrary way on one spatial coordinate, usually the distance from a free surface or boundary. Here the theory of the boundary critical behaviour of two-dimensional layered Ising models, including the Hilhorst–van Leeuwen model and models for aperiodic systems, is reviewed, with a particular attention to exact results for the critical behaviour and the boundary order parameter.

We study an unconventional cosmology, in which we investigate the consequences that antigravity would pose to cosmology. We present the main characteristics of the Dirac-Milne Universe, a cosmological model where antimatter has a negative active gravitational mass. In this non-standard Universe, separate domains of matter and antimatter coexist at our epoch without annihilation, separated by a gravitationally induced depletion zone. We show that this cosmology does not require a priori the Dark Matter and Dark Energy components of the standard model of cosmology. Additionally, inflation becomes an unnecessary ingredient. Investigating this model, we show that the classical cosmological tests such as primordial nucleosynthesis, Type Ia supernovæ and Cosmic Microwave Background are surprisingly concordant.

An interesting two-dimensional model theory has been proposed that allows one to consider black-hole evaporation in the semiclassical approximation. The semiclassical equations will give a singularity where the dilaton held reaches a certain critical value. This singularity will be hidden behind a horizon. As the evaporation proceeds, the dilaton held on the horizon will approach the critical value but the temperature and rate or emission will remain finite. These results indicate either that there is a naked singularity, or (more likely) that the semiclassical approximation breaks down.

We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin's homomorphism and of a certain analog in characteristic p.

In June of 1940, shortly after the fall of Prance in which Doeblin died, I received a package of reprints sent from France by Fréchet. It was probably among those reprints that I first saw Doeblin's paper [1] cited. I was then a student at Kunming, China and had read Fréchet's book [2]. Kolmogorov's paper [3] which laid the foundation of the theory of Markov chains with denumerable states was not accessible to me, although the journal containing it might have been under the caves of Yunnanfu to escape Japanese bombing, as I reminisced at the Paul Lévy memorial in 1987¹. I read it later after learning Russian, and translated it in an exercise book which I still have. Kolmogorov mentioned Doeblin's independent work but did not state the two problems treated in [1]. I think Feller first showed that paper to me. Later Mrs. Doeblin sent me several of her son's reprints, among which was this one, with a cover but without the year of publication nor the number of the volume, testifying to the wartime conditions in France. What I remembered distinctly is that I asked a visitor, M.L., to report its content to a small audience, including Doob and Snell. Afterwards we had a good chuckle over it because the oral presentation did not sound any different from the rather obscure original script. The following year I was living in New York City and giving a course on limit theorems for sums of independent random variables at the Department of Mathematical Statistics of Columbia University, and the Korean war broke out. I must have tired of those sums and been looking for something else to think about. It was in this way that I returned to the first problem in [1]…

To honor and to please our friend Krzysztof P. Wojciechowski I will review the milestones of his mathematical work. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderón projector and the topology of space of elliptic boundary problems. The theme of the 90s is the eta invariant. The paper with Douglas was fundamental for establishing spectral invariants for manifolds with boundary and for the investigation of the behavior of spectral invariants under analytic surgery. This was so influential that many different proofs of the gluing formula for the eta-invariant were published. Finally turning to the new millennium we will look at the zeta-determinant. Compared to eta this is a much more rigid spectral invariant which is technically challenging.

We review the gluing formulae of the spectral invariants - the ζ-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator. In particular, we explain the crucial role of the Cauchy data spaces in these gluing formulae.

The Atiyah-Singer index formula states that the analytic index of an elliptic pseudo-differential operator equals the topological index of the K-theoretic class of its asymptotic symbol. In the embedding proof of the index formula, it is shown that both indices obey certain axioms and which guarantee that they are the same. For the most part, the verification that the topological index obeys the axioms is fairly direct, once one has the relevant background in K-theory. However, the verification is not as straightforward for the analytic index. This is particularly true for the multiplication axiom which is used to show how the index behaves under embedding. Complications arise when the normal bundle of the embedding is nontrivial. Our aim is to clarify the verification of the multiplication axiom for the analytic index by defining the participating elliptic pseudo-differential operators in terms of global symbols invariantly defined on the cotangent bundles of the relevant base spaces.

In this survey we describe situations when analytic invariants of an isolated complex surface singularity can be computed by topological methods.

We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators, appear, in particular, in gauge field theories, string theory as well as models of non-commutative gravity theories, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays the role of a “non-commutative” metric. It is well known that there is a small-time asymptotic expansion of the trace of the corresponding heat kernel in half-integer powers of time. We initiate the development of a systematic approach for the explicit calculation of these coefficients, construct the corresponding parametrix of the heat equation and compute explicitly the first two heat trace coefficients.

For a compact manifold with boundary, M, there are well known local boundary conditions that make the de Rham operator A = d+δ elliptic, namely the absolute and relative boundary conditions. We study the eta invariants of such elliptic boundary value problems under the metric deformation

Since the heat kernel of the sub-Laplacian on Heisenberg group was constructed in an explicit integral form by A. Hulanicki, we have several ways to construct the heat kernel for the sub-Laplacian and the Laplacian on 2-step nilpotent Lie groups. In this note we explain a method effectively employed by Beals-Gaveau-Greiner, the so called complex Hamilton-Jacobi theory, and illustrate the construction of the heat kernel for general 2-step cases. We discuss the solution of the generalized Hamilton-Jacobi equation and a quantity similar to the van Vleck determinant and their roles in the integral expression of the heat kernel. We expect this method will work also for 3-step cases to construct the heat kernel together with the theory of elliptic functions. So as an example, we consider the solution of the generalized Hamilton-Jacobi equation for the lowest dimensional 3-step nilpotent Lie group (Engel group). Then we discuss a hierarchy of heat kernels for the three dimensional Heisenberg group and Heisenberg manifolds as a simple example.

In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^−s) at the possible poles. In particular, they show a formula for the zero'th coefficient at s = 0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-Guillemin noncommutative residue density of an associated operator. The purpose of this note is to provide a proof of that formula relying entirely on resolvent techniques (for the sake of possible generalizations to situations where powers are not an easy tool). — We also give some corrections to transition formulas used in our earlier works.

We extend the definition, in the extended cohomology framework, of the L²- analytic torsion for covering spaces due to Braverman-Carey-Farber-Mathai to the case of manifolds with boundary, and prove an associated anomaly formula. Our main result may be viewed as a common generalization of the anomaly formula for Ray-Singer analytic torsion for manifolds with boundary proved by Brüning-Ma, as well as the anomaly formula for L²-analytic torsions for closed manifolds proved by Zhang. It generalizes also an earlier result of Lück-Schick, without the assumptions on the unitary representations as well as the technical “determinant class condition”.

Using Laurent expansions of canonical traces of holomorphic families of classical pseudodifferential operators, we define functionals on the space of Riemannian metrics and investigate their conformal properties, thereby giving a unified description of several conformal invariants and anomalies.

The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow first analysed in the classical setting by Getzler and extended to Breuer-Fredholm operators by some of the current authors. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (non-compact) covering spaces of a compact spin manifold using the adiabatic method. This answers a question of Mathai connecting Atiyah's L²-index theorem to our analytic spectral flow. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on R^N using some ideas of Shubin. This provides us with an opportunity to describe the deepest part of the theory namely the semifinite local index theorem in noncommutative geometry. This theorem, which gives a formula for spectral flow was recently proved by some of the present authors. It provides a far-reaching generalisation of the original 1995 result of Connes and Moscovici.