In this talk, we will discuss a systematic approach to gravitational waves generated by localized sources in de Sitter spacetime, starting from quadrupolar radiation and moving toward a general multipolar framework. We will first review the geometry of de Sitter spacetime and explain how it modifies the standard flat-space description of radiation. Next, we will discuss de Sitter Teukolsky waves and demonstrate their relation, through explicit gauge transformations, to other quadrupolar solutions in the literature. We will explain how a consistent truncation of the source multipole expansion can be performed while preserving stress-tensor conservation and the generalized harmonic gauge conditions, alongside its asymptotic Bondi description. Going beyond the quadrupole, we will describe the construction of the source-generated metric through octupolar order, including its electric and magnetic ($\ell=3$) sectors. Finally, we will present recent work extending this construction to arbitrary multipole order using an exact map to an auxiliary flat-space problem. The resulting framework provides a systematic starting point for studying gravitational radiation and its observables at arbitrary multipole order in an accelerating background.

Neutrino Astronomy started with the observation of Solar neutrinos and then the Supernova neutrinos. Establishing that neutrinos are massive through the observation of neutrino oscillations led to neutrino physics becoming an integral part of astrophysics. The kilometer-cubed size detector IceCube has taken the astrophysical role of neutrinos  further by detecting extremely high energy neutrinos from extra galactic sources. This has also led to the concept of Multi-Messenger astronomy. In this talk, I will describe some of the highlights of IceCube results and how we hope to learn more about the universe from these results.

A classical theorem of Kronecker states that the Mahler measure, $M(P) = 1$ for any monic, irreducible polynomial with integer coefficients if and only if $P$ is a $n$-th cyclotomic polynomial for some natural number $n$. If a monic irreducible polynomial $P(x)\in \mathbb{Z}[x]$ with $P(0) e 0$ which is not a cyclotomic polynomial, then $M(P) > 1$. In 1933, D. H. Lehmer posed the following problem. Does there exist a positive constant $c$ such that $M(P) \geq 1+c$ for all monic, polynomial $P$ with $P(0) e 0$ and integer coefficients? Recently, we proved the following results. Let $\mathcal{P}$ be the collection of all monic, irreducible non-cyclotomic polynomials $P$ of degree $d\geq 1$ with integer coefficients and $P(0) e 0$. For a given prime $p$, we partition $\mathcal{P} = \bigcup_{R\geq 0} \mathcal{P}_{p,R}$. Then for a fixed $R\geq 0$, we prove Lehmer's Problem for all primes $p$. Also, for any fixed prime $p$ and $d$ large enough, we proved a quantitative uniform lower bound for the Lehmer's Problem by varying $R\leq d/\log d$. We also note that Lehmer's problem is equivalent to prove a uniform lower bound by varying $R\leq d-1$ and for any fixed prime $p$. In this talk, we discuss the proof of a case when $R = 1$. This is a joint work with Piyush Kumar Jha.