Theta Operators on v-adic Modular Forms and v-adic Families of Goss Polynomials and Eisenstein Series

The first part of the dissertation is mainly from my first paper joint with my advisor Papanikolas and the second part will be our second paper. In 1973, Serre introduced p-adic modular forms for a fixed prime p, which are defined to be p-adic limits of Fourier expansions of holomorphic modular forms on SL2(ℤ) with rational coefficients. He also established fundamental results about families of p-adic modular forms by developing the theories of differential operators and Hecke operators acting on p-adic spaces of modular forms. In particular, he showed that the weight 2 Eisenstein series E2 is also p-adic. If we let ϑ := 1 /2πi d/dz be Ramanujan’s theta operator acting on holomorphic complex forms, then letting q(z) = e^2πiz, we have ϑ = q d/dq; ϑ(q^n) = nq^n. Although ϑ does not preserve spaces of complex modular forms, Serre proved the induced operation ϑ : ℚ⨂ℤp[[q]] → ℚ⨂ℤp[[q]] does take p-adic modular forms to p-adic modular forms and preserves p-integrality. Moreover, the Bernoulli numbers Bm and the Eisenstein series Em have p-adic limits as m goes to a p-adic limit. To extend the theory to function fields, we investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Similar to the classical case, the false Eisenstein series E is a v-adic modular form, though it is not a Drinfeld modular form. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality, which can be proved using Goss polynomials. Furthermore, we can show that the Bernoulli-Carlitz numbers BCmj have a v-adic limit if mj have the form aq^dj + b with a, b non-negative. Using the same method, we can also prove that the Goss polynomials have v-adic limits after multiplication by a Carlitz factorial. Because of this, we can also prove the limit of ПmjΘ^mj exists. Therefore, since the Eisenstein series En can be expressed as the sum of Bernoulli-Carlitz numbers and Goss polynomials, we can derive that Emj also have a v-adic limit in K⨂AAv[[u]]. Notice for the Eisenstein series in function fields, the result we get is different from the classical number fields. In the classical case, Serre proved that if mj has a limit m in the p-adic topology and mj goes to infinity in the Euclidean norm, then the classical Eisenstein series Emj has a p-adic limit only depending onm. However, for example in function fields, even if the two series aq^dj + b and (q - 1)q^2dj + aq^dj + b satisfy the previous two condition and their corresponding Eisenstein series are non-zero, they do not have the same v-adic limit

Kawasaki dynamics in continuum: micro- and mesoscopic descriptions

The dynamics of an infinite system of point particles in $\mathbb{R}^d$, which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval $[0,T)$, the evolution of states $\mu_0 \mapsto \mu_t$ is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution $k_0 \mapsto k_t$, $t\in [0,T)$, in a scale of Banach spaces; (b) proving that each $k_t$ is a correlation function for a unique measure $\mu_t$. The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles' density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution $\varrho_t$, $t\in [0,+\infty)$.Comment: revised versio

Wavefront modelling and sensing for advanced gravitational wave detectors

The Advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) directly detected gravitational waves for the first time on the 14th of September 2015. In 2017 the detection of gravitational waves from a binary neutron star merger was subsequently followed up by observations by optical and radio astronomers — the first time an astrophysical event was observed by two completely separate astrophysical signals. This marked the beginning of multi-messenger astronomy. Since then 90 astrophysical events have been observed using gravitational waves. To increase the rate of event detection the sensitivity of gravitational wave detectors must be improved. Current state of the art gravitational wave detectors are optical interferometers in the dual recycled Fabry-Perot Michelson (DRFPMI) configuration with quantum squeezed light injected to reduce vacuum noise. Future plans to improve the sensitivity further rely on increasing the circulating laser power and improving the efficiency of quantum squeezing. Squeezing efficiency is drastically reduced by optical losses in the interferometer of which mode mismatch is a large component. Higher laser power introduces larger thermal distortions in the interferometer, which increase mode mismatch. This thesis covers topics relevant to optical modelling of coupled cavity interferometers such as the DRFPMI with a focus on mode mismatch. Novel applications in aLIGO commissioning based on existing mode mismatch sensing techniques using the output mode cleaner (OMC) are presented. A new mode mismatch sensing technique based on transverse higher order mode sidebands is demonstrated on an optical tabletop and its applications to mode mismatch sensing in aLIGO is discussed. A new optical modelling framework based on linear canonical transformations and signal flow graph theory is also presented.Thesis (Ph.D.) -- University of Adelaide, School of Physical Sciences, 202