Talks

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.

A key question in holography is how to reconstruct bulk operators in the holographic dual. It is especially interesting to reconstruct operators inside the black hole interior, but also especially difficult to do explicitly. Recently, an explicit form for the bulk-to-boundary `holographic’ map was proposed in JT gravity, by Iliesiu, Levine, Lin, Maxfield, and Mezei, who also proposed and studied an explicit `reconstruction’ map on operators. In this talk, I will discuss various pros and cons of their reconstruction map, and propose an alternative map with perhaps nicer properties.

I will use algebras to revisit and generalize the theory of canonical purifications, explain the general concept of "CRT sewing without gravity," and explain how this technology allows the influential-but-complicated "Araki-Yamagami theorem" from the '80s to be proved with 10x less work using a QI-motivated trick. Based on work with Caminiti and Capeccia.

In holography, when two boundary subsystems have large mutual information, they are connected by their entanglement wedge. However, it remains mysterious whether these subsystems are EPR-like entangled. In this talk, I resolve this problem by finding bulk duals of one-shot distillable entanglement. Namely, I show that in one-shot scenarios: i) there is no distillable entanglement only by local operations at leading order in $G_N$, suggesting the absence of bipartite entanglement in a holographic mixed state, and ii) one-way LOCC-distillable entanglement is related to the entanglement wedge cross section, which is further dual to entanglement of formation. By demonstrating an explicit distillation protocol by holographic measurements, I conclude that a connected wedge does not necessarily imply finite distillable entanglement even when one-way LOCC is allowed. This talk is based on arXiv:2411.03426 [hep-th] and 2502.04437 [quant-ph].

The notion of complementarity, first introduced in the context of the information paradox, posits that the experiences of two observers who cannot communicate need not match in quantum gravity. In the context of the previous talk, I will argue that the recent proposals for explicitly taking observers into account provide a way to describe semiclassical closed universes in holography, at the expense of a resort to complementarity. I will then discuss an analogous setup in the case of an evaporating black hole, and propose a set of general principles for black hole complementarity. This talk is based on an ongoing collaboration with Netta Engelhardt and Daniel Harlow.