Fractal Tube Formulas and a Minkowski Measurability Criterion for Compact Subsets of Euclidean Spaces

We establish fractal tube formulas valid for a large class of compact subsets (and more generally, relative fractal drums, RFDs) in Euclidean spaces of any dimension. These formulas express the volume of the tubular neighborhoods of the fractal under consideration in terms of the residues of the associated fractal zeta functions and their poles (i.e., the complex dimensions). Under suitable assumptions, we also show that a compact subset of R^N is Minkowski measurable if and only if its only complex dimension of maximum real part is the Minkowski (or box) dimension D of the fractal (or RFD), and D is simple. These results extend to arbitrary dimensions N greater than 1 the corresponding ones obtained by Lapidus and van Frankenhuijsen for fractal strings (i.e., when N =1). We illustrate them by means of several examples.

Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums

Distance and Tube Zeta Functions of Fractals and Arbitrary Compact Sets

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Deformation approach to quantisation of field models

Associativity-preserving deformation quantisation via Kontsevich's summation over weighted graphs is lifted from the algebras of functions on finite-dimensional Poisson manifolds to the algebras of local functionals within the variational Poisson geometry of gauge fields over the space-time.

The Sound of Fractal Strings and the Riemann Hypothesis

Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis

An asymmetric criterion for the Riemann hypothesis is provided, expressed in terms of the invertibility of the spectral operator. This criterion is asymmetric, in the sense that it is valid for all fractal dimensions c in (0,1/2) if and only if the Riemann hypothesis is true, but that (unconditionally) it fails to be true for any value of c in (1/2, 1), a fact which is closely connected to the universality of the Riemann zeta function.

Deformations of complex structures on Riemann surfaces and integrable structures of Whitham type hierarchies

We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space
which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which appear to be the same as the ones that emerged in the
integrability theory of Whitham type hierarchies. Driven by this observation, we develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application
we prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.

On the large-scale geometry of the $L^p$-metric on the symplectomorphism group of the two-sphere

We prove that the vector space $R^d$ of any finite dimension $d$ with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms of the two-sphere endowed with the $L^p$-metric for $p>2$. Along the way we show that the $L^p$-metric on this group is unbounded for $p>2$ by elementary methods.

The Equivalence Principle in a Quantum World

Lectures on Regular and Irregular Holonomic D-modules

This is a survey paper based on lectures given by the authors at Ihes, February/March 2015.
In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part,
we present the enhanced version of the first part, treating along the same lines the irregular holonomic case.

Differential symmetry breaking operators I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations.
We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential
equations.
In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators
in the holomorphic setting.
In this case symmetry breaking operators are characterized by differential equations of second order via the F-method.

Differential symmetry breaking operators. II. Rankin--Cohen operators for symmetric pairs

Rankin--Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of SL(2,R). We address a general problem to find explicit formulae, for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs.
For this purpose we use a new method (F-method) developed in the first part of the series and based on the algebraic Fourier transform for generalized Verma modules.
The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order.
We discover explicit formulae, of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin--Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.

Global uniqueness of small representations

We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.