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09:45 to 10:00 |
Prof. Rajesh Gopakumar (ICTS-TIFR, Bengaluru, India) |
Welcome remarks |
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10:00 to 11:15 |
Ram Murty (Queen's University, Kingston, Canada) |
Ramanujan, Modular Forms and Beyond 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!
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11:45 to 13:00 |
Nick Andersen (Brigham Young University, Provo, USA) |
Introduction to harmonic Maass forms 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.
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14:30 to 15:30 |
Michael Griffin (Vanderbilt University, USA) |
Maas operators in integral weight 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.
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15:35 to 16:25 |
Ranveer Kumar Singh (Rutgers University, New Jersey, USA) |
Generalization of Monster denominator identity to higher level using harmonic Maass forms 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.
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17:00 to 18:00 |
Ken Ono (University of Virginia, Charlottesville, USA) |
Eichler-Selberg Relations for Traces of Singular Moduli 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.
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