Canonical algebraic and geometric structures that play fundamental role in description of various systems in physics are studied in the language of differential geometry and super-geometry. Among them are Poisson and Jacobi structures (e.g. symplectic and contact), Lie and Courant algebroids, Dirac structures, generalized (e.g. complex) geometries, Nijenhuis tensors and the corresponding contractions, principles of variational calculus – all this with applications to Theoretical Mechanics, especially to frame-independent description of mechanical systems, foundations of Quantum Mechanics and the geometry of quantum states, quantum information and description of entanglement.
Main activity are applications of methods of statistics and the theory of stochastic processes to the problem of detection of gravitational waves in the noise of the detector. In particuar in the paper [1] optimal statistic to search for modulated periodic gravitational waves from rotating neutron stars is derived. This optimal statistic is now commonly used in the gravitational wave data analysis in particular in the Einstein@Home project.
In paper [2] it is shown that the response of a detector to the superposition of tens of millions of gravitational wave signals from binary white dwarf systems is a cyclo-stationary random process. Using these methods real data from resonant bar detectors EXPLORER and NAUTILUS are analyzed. Also several problems of the Lorentzian geometry in the large, in particular properties of Cauchy horizons have been studied.
The main areas of research are geometrical structures of nonlinear field theories, in particular of the Einstein theory of gravitation and Yang-Mills gauge theories, and statistical methods of detection and estimation of signals of gravitational and electromagnetic origin.
In particular we carried out research on the long-standing hypothesis of Roger Penrose - the cosmic censorship hypothesis, which asserts that the final state of gravitational collapse of a star of sufficiently large mass is always a black hole. To approach this problem we have used methods of geometry and differential topology. We work on theoretical and practical aspects of analysis of data from gravitational-wave detectors. Ground-based detectors of gravitational waves are currently working in Germany, France, Italy, Japan, and the USA. NASA and the European Space Agency are planning to put a gravitational wave detector (LISA project) in an orbit around the Sun. Detection of gravitational waves will be a final confirmation of Einstein’s theory of gravity and will open a new window on the Universe. Our main interest is detection of very weak, quasi-periodic signals in large parameter spaces. Using our theoretical methods and algorithms we currently analyse data from Italian NAUTILUS resonant bar detector. Analysis is performed on a large network of computers. We study theoretical methods and we develop data analysis tools to analyse the gravitation wave signal originating from superposition of many signals from binary systems in our Galaxy. This is the dominant gravitational wave signal that will be present in the data of a space-borne detector LISA.
We have taken part in organization of several conferences in Banach International Mathematical Center:
The team has strong collabor
New approach to logarithmic mappings as invertible selectors of multifunctions is proposed together with some applications of logarithmic and antilogarithmic mappings to linear and nonlinear equations in algebras of matrices. Also problems of algebraic analysis of algebraic structures with right-invertible operators.
Problems concerning strongly paraconvex functions on an open convex subset in a Banach space X are studied, e.g. their differentiability [1], as well as an extension of the notion ofstrongly paraconvex (uniformly approximate convex) functions on differentiable manifolds and Φ-convexity in metric spaces [2].
Wilson systems play an essential role in the construction of orthonormal bases from the Gabor tight frames with certain redundancy. The research shows that there is a strong connection between the Wilson system construction and the automorphisms of the underlying system of operators, especially as they are unitarily implemented.
The wavelet analysis of time series is investigated, showing the so called Long-Range Dependence. This behaviour is observed, e.g. in packet load in web traffic or in economical processes. To this group of stochastic processes belong, in particular, 1/f noises.
Explicit constructions of symplectic manifolds with prescribed properties are performed. The most important problem in this area is the existence problem for symplectic structures on closed manifolds. It is generally believed that closed symplectic manifolds do not possess any special topological properties, however, the main problem in this area (Thurston's conjecture) still is not solved. Several questions of the same type were aked about some other homotopic properties. For example, it was asked if there are relations between hard Lefschetz property, formality and Betti numbers of closed symplectic manifolds.
Integrability, and Poisson pensils are studied in the context of bihamiltonian systems.