Given an Abelian group G, a Boolean-valued function f: G -> {-1,+1}, is said to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the domain G. In a seminal paper, Gopalan et al. proved "Granularity" for Fourier coefficients of Boolean valued functions over Z_2^n, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over Z_2^n which are approximately Fourier-sparse. In this work, we obtain structural results for approximately Fourier-sparse Boolean valued functions over Abelian groups G of the form,G:= Z_{p_1}^{n_1} \times ... \times Z_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m^{2}s)^ceiling(phi(m)/2), on the absolute value of the smallest non-zero Fourier coefficient of an s-sparse function, where m=p_1 ... p_t, and phi(m)=(p_1-1) ... (p_t-1). We carefully apply probabilistic techniques from Gopalan et al., to obtain our structural results, and use some non-trivial results from algebraic number theory to get the lower bound. We construct a family of at most s-sparse Boolean functions over Z_p^n, where p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is 1/omega(n). The "Granularity" result of Gopalan et al. implies that the absolute values of non-zero Fourier coefficients of any s-sparse Boolean valued function over Z_2^n are 1/O(s). So, our result shows that one cannot expect such a lower bound for general Abelian groups. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient testing algorithm for Fourier-sparse Boolean functions, that requires poly((ms)^phi(m),1/epsilon)-many queries. Further, we prove an Omega(sqrt{s}) lower bound on the query complexity of any adaptive sparsity testing algorithm. Joint work with: Sourav Chakraborty, Pranjal Dutta, Ariji Ghosh, Swagato Sanyal

Fast radio bursts (FRBs) are energetic millisecond duration pulses, located at cosmological distances, whose physical origin is still unresolved. Detection of a Galactic FRB in April 2020 suggested that some FRBs can originate from magnetars. Characterizing FRB source population will optimize future search strategies and provide valuable insights regarding progenitor models. In this talk, I will first describe a generalized framework utilized to constrain properties of FRB source and host galaxy directly from multi-band radio observations. Next, I will discuss the mechanism for production of coherent radio bursts that are likely accompanied by persistent radio emission originating from magnetar wind nebula. Such late-time emission has been detected for three localized FRBs, and provide direct constraints on the NS age and magnetic field. Lastly, I will discuss how cosmological FRBs can be used as potential probes to investigate He reionization history and can help reveal energetic processes in the early Universe.

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Many insect species live in societies that parallel, if not better, human societies in the sophistication and complexity of their organization, communication, division of labour and even their caste system. How do these insects achieve such social complexity? How has natural selection favoured social life over solitary life? This is especially paradoxical because most individuals in the insect societies function as sterile slaves and help one or a small number of queens to reproduce. In my quest of the origins of sociality, I have focussed on Ropalidia marginata, a primitively social wasp that appears to be on the brink of sociality. With warm temperatures and favourable conditions throughout the year, tropical southern India provides a perpetual stage for these wasps to play out the drama of war and peace. Perhaps the most significant message from my studies is that conflict and cooperation are intimately linked and maintaining a fine balance between the two is what social life is all about.

https://indico.imsc.res.in/event/5/