A program for branching problems in the representation theory of real reductive groups

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages:
(A) abstract features of the restriction;
(B) branching laws (irreducible decompositions of the restriction); and
(C) construction of symmetry breaking operators on geometric models.
We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are {\it{a priori}} known to be "nice" in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A.
The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.

Decorated super-Teichmueller space

La suite spectrale de Hodge-Tate

The Hodge-Tate spectral sequence for a proper smooth variety over a p-adic field provides a framework for us to revisit Faltings' approach to p-adic Hodge theory and to fill in many details. The spectral sequence is obtained from the Cartan-Leray spectral sequence for the canonical projection from the Faltings topos to the étale topos of an integral model of the variety. Its abutment is computed by Faltings' main comparison theorem from which derive all comparison theorems between p-adic étale cohomology and other p-adic cohomologies, and its initial term is related to the sheaf of differential forms by a construction reminiscent of the Cartier isomorphism.

On the modular structure of the genus-one Type II superstring low energy expansion

The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions
up to order $D^{10} \cR^4$ are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.

Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus

The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure modulus of a torus. In the case of the four-graviton amplitude, each of these modular functions is a multiple sum associated with a Feynman diagram for a free massless scalar field on the torus. The lines in each diagram join pairs of vertex insertion points and the number of lines defines its weight $w$, which corresponds to its order in the low energy expansion. Previous results concerning the low energy expansion of the genus-one four-graviton amplitude led to a number of conjectured relations between modular functions of a given $w$, but different numbers of loops $\le w-1$. In this paper we shall prove the simplest of these conjectured relations, namely the one that arises at weight $w=4$ and expresses the three-loop modular function $D_4$ in terms of modular functions with one and two loops. As a byproduct, we prove three intriguing new holomorphic modular identities.

More Graviton Physics

The interactions of gravitons with spin-1 matter are calculated in
parallel with the well known photon case. It is shown that graviton
scattering amplitudes can be factorized into a product of familiar
electromagnetic forms, and cross sections for various reactions
are straightforwardly evaluated using helicity methods. Universality relations are
identified. Extrapolation to zero mass yields scattering amplitudes for
photon-graviton and graviton-graviton scattering.

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.

Energetics and phasing of nonprecessing spinning coalescing black hole binaries

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.

Fourth post-Newtonian effective one-body dynamics

Analytic determination of high-order post-Newtonian self-force contributions to gravitational spin precession

Detweiler's gauge-invariant redshift variable: analytic determination of the nine and nine-and-a-half post-Newtonian self-force contributions

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.