``Ars combinatoria'' chez Gian-Carlo Rota ou le triomphe du symbolisme

Gian-Carlos Rota est l'inventeur d'une nouvelle science : la combinatoire algébrique qui n'était jusque-là qu'un ensemble de questions disparates, dont la solution, parfois ingénieuse, ne laissait entrevoir la méthode. Il s'agit ici de décrire la stratégie mise en oeuvre sous sa forme mathématique dont les enjeux philosophiques ne sont pas négligeables, en poursuivant le geste de Rota, pour sortir la combinatoire de son ghetto.

Generalized conformal Hamiltonian dynamics and the pattern formation equations

We demonstrate the
significance of the Jacobi last multiplier in Hamiltonian theory by
explicitly constructing the Hamiltonians of certain well known
first-order systems of differential equations arising in the activator-inhibitor (AI) systems.
We investigate the generalized Hamiltonian dynamics of the AI systems of Turing pattern formation problems,
and demonstrate that various subsystems of AI, depending on the choices of parameters,
are described either by conformal or contact Hamiltonian dynamics or both. Both these dynamics
are subclasses of another dynamics, known as Jacobi mechanics.
Furthermore we show that for non Turing pattern formation, like the Gray-Scott model,
may actually be described by generalized conformal Hamiltonian dynamics using two Hamiltonians.
Finally, we construct a locally defined dissipative Hamiltonian generating function \cite{HHG}
of the original system. This generating function coincides with the ``free energy'' of the associated system if
it is a pure conformal class. Examples of pattern formation equation are presented to illustrate the
method.

Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy

The hierarchy of equations belonging to two different but related integrable systems,
the Nonlinear Schrödinger and its derivative variant, DNLS are subjected to two distinct
deformation procedures, viz. quasi-integrable deformation (QID) that generally do not
preserve the integrability, only asymptotically integrable, and non-holonomic deformation
(N HD) that does. QID is carried out generically for the NLS hierarchy while for the
DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of
the family. No QI anomaly is observed at the level of EOMs which suggests that at that
level the QID may be identified as some integrable deformation. NHD is applied to the
NLS hierarchy generally as well as with the specific focus on the NLS equation itself and
the coupled KdV type NLS equation. For the DNLS hierarchy, the Kaup-Newell(KN)
and Chen-Lee-Liu (CLL) equations are deformed non-holonomically and subsequently,
different aspects of the results are discussed.

Noncommutative Catalan Numbers

The goal of this paper is to introduce and study non-commutative Catalan numbers C_n which belong to the free Laurent polynomial algebra in n generators. Our non-commutative numbers admit interesting (commutative and non-commutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another to solving non-commutative quadratic equations. We also establish total positivity of the corresponding non-commutative Hankel matrices and introduce accompanying nonc-mmutative binomial coefficients.

Theory of Morphogenesis

A model of morphogenesis is proposed based upon seven explicit postulates. The mathematical import and biological significance of the postulates are explored and discussed.

Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription

Properties of soliton surfaces associated with integrable $\mathbb{C}P^{N-1}$ sigma models

We investigate certain properties of $\\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable $\\mathbb{C}P^{N-1}$ sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined on the two-dimensional Riemann sphere with finite action functional. Several new properties of the projectors mapping onto one-dimensional subspaces as well as their relations with three mutually different immersion formulas, namely, the generalized Weierstrass, Sym-Tafel and Fokas-Gel\'fand have been discussed in detail. Explicit connections among these three surfaces are also established by purely analytical descriptions and, it is demonstrated that the three immersion formulas actually correspond to the single surface parametrized by some specific conditions.

q-deformed quadrature operator and optical tomogram

In this paper, we define the homodyne q-deformed quadrature operator and find its eigenstates in terms of the deformed Fock states. We find the quadrature representation of q-deformed Fock states in the process. Furthermore, we calculate the explicit analytical expression for the optical tomogram of the q-deformed coherent states.

SKEW PRODUCT SMALE ENDOMORPHISMS OVER COUNTABLE SHIFTS OF FINITE TYPE

We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous
potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic
formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets.
We describe then the thermodynamic formalism for Smale skew products over
countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We
then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi (EMR) maps, and we settle the question of the exact dimensionality of such measures. In
particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we

M-theoretic Lichnerowicz formula and supersymmetry

A suitable generalisation of the Lichnerowicz formula can relate the squares of supersymmetric
operators to the effective action, the Bianchi identities for fluxes, and some equations of
motion. Recently, such formulae have also been shown to underlie the (generalised) geometry
of supersymmetric theories. In this paper, we derive an M-theoretic Lichnerowicz formula that
describes eleven-dimensional supergravity together with its higher-derivative couplings. The
first corrections to the action appear at eight-derivative level, and the construction yields two
different supersymmetric invariants, each with a free coefficient. We discuss the restriction of
our construction to seven-dimensional internal spaces, and implications for compactifications
on manifolds of G 2 holonomy. Inclusion of fluxes and computation of contributions with
higher than eight derivatives are also discussed.

Modification of Schrodinger-Newton equation due to braneworld models with minimal length

We study the correction of the energy spectrum of a gravitational quantum well due to the combined effect of the braneworld model with infinite extra dimensions and generalized uncertainty principle. The correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed by the Schrodinger-Newton equation based on a minimal length framework. The two fold correction in the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory.

Quantum Supersymmetric Cosmological Billiards and their Hidden Kac-Moody Structure

Software modules and computer-assisted proof schemes in the Kontsevich deformation quantization

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the non-commutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter ħ. Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, modulo o(ħ^4), for the newly built ⋆-product expansion.

Galois equivariance of critical values of $L$-functions for unitary groups

The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic $L$-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of $\GL_{n}\times\GL_{1}$. We show that our results are compatible with Deligne's conjecture.

Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra

We continue the study undertaken in [DV] of the exceptional Jordan algebra $J = J_3^8$ as (part of) the finite-dimensional quantum algebra in an almost classical space-time approach to particle physics. Along with reviewing known properties of $J$ and of the associated exceptional Lie groups we argue that the symmetry of the model can be deduced from the Borel-Siebenthal theory of maximal subgroups of simple compact Lie groups.

Examples of pre-CY structures, associated operads and cohomologies

On the four-loop static contribution to the gravitational interaction potential of two point masses