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In the first part of this talk, I will provide a brief introduction to classical summation formulas and their significance in number theory. We will review the foundational contributions of Bochner, Koshliakov, and the seminal work of Chandrasekharan and Narasimhan on summation formulas for a broad class of arithmetical functions.

In the second part, I will present recent developments involving new summation formulas in the theory of harmonic Maass forms. As an application of our summation formula, I will discuss the asymptotic behavior of the Riesz means of the Hurwitz class numbers.\\

This talk is based on recent joint work with Olivia Beckwith, Nikolaos Diamantis, Larry Rolen, and Kalani Thalagoda.

We review the basic properties of hecke operators, Atkin-Lehner involutions and trace operators, and show how these operators act on harmonic Maass forms. We also discuss certain p-adic results that follow from this theory.

We give motivating examples of harmonic Maass forms of half-integral weight, including Poincare series and Zagier's Eisenstein series. We also discuss the sesquiharmonic lift of Zagier's Eisenstein series by Duke, Imamoglu, and Toth.

We give explicit examples of harmonic Maass forms of integral weight, including non-holomorphic Eisenstein series and Weierstraass-mock modular forms. We also discuss Borcherds' obstruction theorem (a consequence of Serre-duality).

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.