The Mathematical Physics of Gravitational Waves and the Polylogarithm and Lambert W Functions

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The long-term vision of my research program is to extend the mathematical physics that underlies gravitational waves and the quantum statistics of D-dimensional Bose and Fermi gases. I propose to realize this vision through two distinct themes summarized below.******Theme 1 ***A key objective in this theme is a theoretical investigation to understand the behaviour of pulsars in the strong gravity of a black hole (BH). Observation runs will begin for second generation detectors, and the Equation of State (EoS) information provided by these detectors will come mainly from tidal interactions during the in-spiral of binary neutron star and BH-compact object (CO) such as neutron star (NS) or White Dwarf (WD) systems, which induce quadrupolar deformations in the NS. Measuring GW signals from NS-NS binaries is a very promising method to learn about the still poorly understood EoS of NS, and one of the primary astrophysical applications of LIGO. We will address the challenging problem of the GW frequency spin-down coefficients of a pulsar, over long observation times required to detect continuous sources of gravitational wave (GW). A renaissance in GW astronomy began after the Laser Interferometer Gravitational-Wave Observatory (LIGO) detected two GW emissions, which both arose from the merger of coalescing binary BH systems. My HQP and I will analyze the weaker GW produced by a pulsar orbiting a BH to gain a more detailed understanding of gravitational physics. In addition, my proposed multi-messenger astronomy study is relevant for LIGO, Virgo (French-Italian GW Observatory), the space-based Laser Interferometer Space Antenna (LISA), the Square Kilometre Array (SKA) and the upcoming LIGO-India. The unique sensitivity of the SKA will locate orbiting pulsars-BH systems, detect single GW sources and provide strong tests of General Relativity. The SKA will also perform a survey of pulsars and this will provide the population statistics of pulsar-BH systems.******Theme 2 ***A key objective in this theme is to use the logarithmic structure of the Polylogarithm and multi-branched Lambert W functions in the understanding and unification of the quantum spin statistics of D dimensional Bose and Fermi gases. We propose to study the fundamental mathematical physics that underlies quantum statistics to unravel characteristic features in problems such as the double polytrope model for Brown Dwarfs (BD), enhancement of the thermoelectric figure of merit in thermoelectric materials (TEM), as well as the zeros of partition functions in the Ising and generalized Ising Models. I will investigate the more exact EoS for astrophysical objects such as BD and NS, and study the superfluid core of a spinning NS. These studies will help to improve the efficiency of TEM and solar cells.******The unifying thread in both of the themes described above is the development of methods in mathematical and theoretical physics to solve challenging physical problems.