Spectral gaps,additive energy,and a fractal uncertainty principle

We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension \({\delta}\) of the limit set close to \({{n-1\over 2}}\). The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors–David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.     

Convexity bounds for BSDE solutions, with applications to indifference valuation

We consider backward stochastic differential equations (BSDEs) with a particular quadratic generator and study the behaviour of their solutions when the probability measure is changed, the filtration is shrunk, or the underlying probability space is transformed. Our main results are upper bounds for the solutions of the original BSDEs in terms of solutions to other BSDEs which are easier to solve. We illustrate our results by applying them to exponential utility indifference valuation in a multidimensional It? process setting.     

Random geometry, quantum gravity and the Kähler potential

We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a background-independent measure on the space of metrics can be easily constructed from first principles. Our framework suggests the relevance of a new gravitational effective action and we show that it occurs when coupling the massive scalar field to two-dimensional gravity. This yields new types of quantum gravity models generalizing the standard Liouville case.     

Valuations on Manifolds and Integral Geometry

We construct new operations of pull-back and push-forward on valuations on manifolds with respect to submersions and immersions. A general Radon-type transform on valuations is introduced using these operations and the product on valuations. It is shown that the classical Radon transform on smooth functions, and the well-known Radon transform on constructible functions, with respect to the Euler characteristic, are special cases of this new Radon transform. An inversion formula for the Radon transform on valuations has been proven in a specific case of real projective spaces. Relations of these operations to yet another classical type of integral geometry, Crofton and kinematic formulas, are indicated.     

Synthesis and Structure of the Bis- and Tris-Polyhedral Hybrid Carboranoclathrochelates with Functionalizing Biorelevant Substituents—The Derivatives of Propargylamine Iron(II) Clathrochelates with Terminal Triple C≡C Bond(s)

A synthetic strategy for obtaining structurally flexible hybrid iron(II) carboranoclatrochelates functionalized with biorelevant groups, based on a combination of a 1,3-dipolar cycloaddition reaction with nucleophilic substitution of an appropriate chloroclathrochelate precursor, was developed. In its first stage, a stepwise substitution of the dichloroclathrochelate precursor with amine N-nucleophiles of different natures in various solvents was performed. One of its two chlorine atoms with morpholine or diethylamine in dichloromethane gave reactive monohalogenoclathrochelate complexes functionalized with abiorelevant substituents. Further nucleophilic substitution of their remaining chlorine atoms with propargylamine in DMF led to morpholine- and diethylamine-functionalized monopropargylamine cage complexes, the molecules of which contain the single terminal C≡C bond. Their “click” 1,3-cycloaddition reactions in toluene with ortho-carborane-(1)-methylazide catalyzed by copper(II) acetate gave spacer-containing di- and tritopic iron(II) carboranoclatrochelates formed by a covalent linking between their different polyhedral(cage) fragments. The obtained complexes were characterized using elemental analysis, MALDI-TOF mass, UV-Vis, ¹H, ¹H{¹¹B}, ¹¹B, ¹¹B{¹H}, ¹⁹F{¹H} and ¹³C{¹H}-NMR spectra, and by a single crystal synchrotron X-ray diffraction experiment for the diethylamine-functionalized iron(II) carboranoclathrochelate. Its encapsulated iron(II) ion is situated almost in the center of the FeN₆-coordination polyhedron possessing a geometry intermediate between a trigonal prism and a trigonal antiprism with a distortion angle φ of approximately 28°. Conformation of this hybrid molecule is strongly affected by its intramolecular dihydrogen bonding: a flexibility of the carborane-terminated ribbed substituent allowed the formation of numerous C–H…H–B intramolecular interactions. The H(C) atom of this carborane core also forms the intermolecular C–H…F–B interaction with an adjacent carboranoclathrochelate molecule. The N–H…N intermolecular interaction between the diethylamine group of one hybrid molecule and the heterocyclic five-membered 1H-[1,2,3]-triazolyl fragment of the second molecule of this type caused formation of H-bonded carboranoclathrochelate dimers in the X-rayed crystal.     

Discrete Kontorovich–Lebedev transforms

The Ramanujan Journal - Discrete analogs of the classical Kontorovich–Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald...     

On the measure of intersection of cylinders

In this paper we study an extremal problem on the minimum of the measure of intersection of cylinders satisfying certain geometric conditions. The main result generalizes previous results of Khatri, Sidak and Gluskin to the wider class of all rotation invariant measures. Partially supported by a Minkowski Center grant.     

Pluripotential theory on quaternionic manifolds

On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Ampère operator is defined. It is shown that it satisfies a version of the theorems of A. D. Alexandrov and Chern–Levine–Nirenberg. For more special classes of manifolds analogous results were previously obtained in Alesker (2003) [1] for the flat quaternionic space Hⁿ${\mathbb{H}}^n$ and in Alesker and Verbitsky (2006) [5] for hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we also prove a new multiplicativity property.