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We discuss the definitions and basic properties of harmonic Maass forms having arbitrary weight, level, and multiplier system. These properties include: (1) the Fourier expansion, (2) decomposition into holomorphic & non-holomorphic parts, (3) the Maass raising and lowering operators, and the xi operator, and (4) mock modular forms and shadows.

This is an informal survey of the theories of modular forms, quasi-modular forms, and mock modular forms. These topics owe much to Srinivasa Ramanujan whose mystic vision of mathematics shaped 20th century number theory and now continues to shape the 21st century going beyond number theory and into combinatorics, representation theory and even mathematical physics!

The Eichler–Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz–Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function \( j_0(\tau) = 1 \). More generally, we consider the singular moduli for the Hecke system of modular functions. For each \( \nu \geq 0 \) and \( m \geq 1 \), we obtain an Eichler–Selberg relation. For \( \nu = 0 \) and \( m \in \{1, 2\} \), these relations are Kaneko’s celebrated singular moduli formulas for the coefficients of \( j(\tau) \). For each \( \nu \geq 1 \) and \( m \geq 1 \), we obtain a new Eichler–Selberg trace formula for the Hecke action on the space of weight \( 2\nu + 2 \) cusp forms, where the traces of \( j_m(\tau) \) singular moduli replace Hurwitz–Kronecker class numbers.

The Monster denominator identity is an infinite product representation of j(z)-j(\tau), where j is the Klein’s j-function invariant under the action of SL(2,Z). I will describe a generalization of the Monster denominator formula to higher level using harmonic Maass forms.

We discuss the Maass operators in integer weight. We consider Bol's identity, along with its image and relation to the xi operator, as well as the Petersson inner product and the Bruinier-Funke pairing.

In this lecture, I will give an introduction to the physics of ultra-cold atoms and Quantum gases near absolute zero temperature. I will also describe the experimental methods for their realisation in the Laboratory. Ultra-cold Rydberg atoms and Quantum gases in optical traps are highly controllable systems which offer a versatile platform for Quantum Technology applications such as Quantum computing, Quantum sensing and Quantum Simulation of Many-body physics. I will give an overview of Quantum Science and Technologies with ultra-cold Rydberg atoms and Quantum gas mixtures in the final part of this lecture.

Pierre de Gennes received the Nobel prize in Physics in 1991 for ‘discovering that methods developed for studying order phenomena in simple systems can be generalised to more complex forms of matter, in particular to liquid crystals and polymers’. Liquid crystals, polymers, foams, emulsions and suspensions are common examples of a class of matter called ‘soft materials’ – materials characterised by structural complexity and mechanical flexibility. In this talk, we will together try to decipher the intriguing flow and deformation properties of some very common soft materials that we encounter everyday - materials categorised  as colloidal suspensions (clay, smoke, fog, ink and milk), emulsions (mayonnaise, lotions and creams), liquid foams, pastes (tomato ketchup and toothpaste), granular media (a bag of rice or sand) and polymers. 

I will give a brief discussion of my journey as a scientist over the years, including how I got into the question of gender in physics. I will then end my lecture with a popular science level discussion of my current work.

The talk will go over some crucial developments in astronomy over the last hundred years, such as the classification of stars, the distance scale of the universe, the abundance and production of the chemical elements, compact objects like neutron stars and black holes, and many more. Women astronomers had a crucial role in all of these examples

In the two sessions we will explore what it means to approach physics through the lens of computation. I’ll introduce some of the ways in which computation has shaped physics in terms of modelling, measurement and also how we learn the subject. No prior programming knowledge is expected. We’ll begin from first principles, using just our hands, some paper, and a few simple rules to build up a system. 

In the second session, we’ll move to a visual programming environment (Tinkercad and Seelab). You’ll use a laptop (preferable) or smartphone to try out basic computational constructs like loops, conditionals, and sensor-based responses. The goal is to begin thinking computationally in a  physics context, and to see how we can even start doing simple experiments by interfacing with real hardware.