Integrated Calendar

The concept of "quasi-periodic" sets, functions, and measures is

prevalent in diverse mathematical fields such as Mathematical Physics,

Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for discrete periodic sets, saying that the Fourier transform of the counting measure of a discrete periodic set is also a counting measure of a discrete periodic set. Fourier Quasicrystals (FQ) generalize this notion of periodicity: a counting measure of a discrete set is called a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition.   

 

Recently Kurasov and Sarnak provided a method for construction of one-dimensional counting measures which are FQ (motivated by quantum graphs) using the torus zero sets of multivariate Lee-Yang polynomials. In this talk, I will show that the Kurasov-Sarnak construction generates all FQ counting measures in 1D.

 

A discrete set on the real line is fully described by the gaps between consecutive points. A discrete periodic set has finitely many gaps. We show that a non-periodic FQ has uncountably many gaps, with a well-defined gap distribution. This distribution is given explicitly in terms of an ergodic dynamical system induced from irrational flow on the  torus.

 

The talk is aimed at a broad audience, no prior knowledge in the field is assumed.

 

Based on joint works with Alex Cohen and Cynthia Vinzant.

We study the persistence event ${f(x)>0 for all x in D}$ for stationary Gaussian fields on d dimensional Euclidean space, with a spectral singularity at the origin of order $alpha$, i.e. such that their spectral density at the origin behaves like $|lambda|^{-alpha}$. Our main result is that, under certain mild regularity conditions, the probability of persistence on a large ball $D=B(T)$ decays at a universal log-asymptotic rate of $m (d-alpha)(log T) capa(B(T))$, where $capa(B(T))$ is the capacity of $B(T)$ with respect to the field, and $m$ is the mass of the absolutely continuous component of its spectrum. This generalises a result of Bolthausen, Deuschel and Zeitouni for the Gaussian free field (GFF) on $^d$, $d ge 3$, to a wide class of Gaussian fields with spectral singularity.  Using this estimate, we establish an `entropic repulsion’ phenomenon,  showing that the field conditioned to persist is tending in distribution to an independent copy of the unconditioned process, shifted by a constant level of $sqrt{m (d-alpha)(log T)}$.

 

This work is the first to introduce the notions of capacity and equilibrium potential to the spectral theory of persistence. In the course of the talk I will explain these notions, provide an overview of the proof and present the role of these notions in improving the precision of persistence probability estimates. I will also try to put this work in the context of current study of rare events of stationary Gaussian processes. No prior knowledge of the subject is assumed.

 

Joint work with Naomi Feldheim from Bar-Ilan and Stephen Muirhead from Melbourne.

: G-protein coupled receptors (GPCRs) play a paramount role in diverse brain functions. Twenty years ago, GPCR activity was shown to be regulated by membrane potential in vitro, but whether the voltage dependence of GPCRs contributes to neuronal coding and behavioral output under physiological conditions in vivo has never been demonstrated. We show in two different processes that muscarinic GPCR mediated neuromodulation in vivo is voltage dependent. First, we show that muscarinic type A receptors (mAChR-A) mediated neuronal potentiation is voltage dependent. This potentiation voltage dependency is abolished in mutant flies expressing a voltage independent receptor. Most important, muscarinic receptor voltage independence caused a strong behavioral effect of increased odor habituation. Second, we show that muscarinic type B receptors (mAChR-B) voltage dependency is required for both efficient and accurate learning and memory. Normally, to prevent non-specific olfactory learning and memory, mAChR-B activity suppress both signals that are required for plasticity. Behavior experiments demonstrate that mAChR-B knockdown impairs olfactory learning by inducing undesired changes to the valence of an odor that was not associated with the reinforcer. On the other hand, mAChR-B voltage dependence prevents mAChR-B to interfere with plasticity in neurons that are required for the learning and memory process. Indeed, generating flies with a voltage independent mAChR-B resulted in impaired learning. Thus, we provide the very first demonstrations of physiological roles for the voltage dependency of GPCRs by demonstrating crucial involvement of GPCR voltage dependence in neuronal plasticity and behavior. As such, our findings create a paradigm shift in our thinking on GPCR recruitment and activity. Together, we suggest that GPCR voltage dependency plays a role in many diverse neuronal functions including learning and memory and may serve as a target for novel drug development.
Light refreshments before the seminar.

Large arrays of trapped neutral atoms have emerged over the past few years as a promising platform for quantum information processing, combining inherent scalability with high-fidelity control and site-resolved readout. In this talk, I will discuss ongoing work with arrays of Alkaline-earth atoms. These divalent atoms offer unique properties stemming largely from their long-lived metastable states, which form the basis of the optical atomic clock. I will describe the design of a universal quantum processor based on clock qubits and its application in quantum metrology, and I will address the challenge of generating and benchmarking highly entangled states in an analog quantum simulator. First, we realize scalable local control of individual clock qubits, which we utilize to extend the Ramsey interrogation time beyond the coherence time of a single atom [1]. To realize a universal quantum processor, we demonstrate record high-fidelity two-qubit entangling gates mediated by Rydberg interactions, which we combine with dynamical reconfiguration to entangle clock probes in GHZ states and perform Ancilla-based detection [2]. We then use the narrow clock transition to measure and remove thermal excitations of atoms in tweezers (a technique known as erasure conversion) and generate hyperentangled states of motion and spin [3]. In the second part of the talk, I will describe a different approach for generating large scale entangled states in an analog quantum simulator configuration [4], including error mitigation [5] and benchmarking of a 60-atom simulator [6]. Together, these show the great promise and the large variety of experiments accessed with this emerging platform. [1] A. Shaw*, R. Finkelstein*, R. Tsai, P. Scholl, T. Yoon, J. Choi, M. Endres, Multi-ensemble metrology by programming local rotations with atom movements, arxiv:2303.16885, Nature Physics in press (2023). [2] R. Finkelstein, R. Tsai, A. Shaw, X. Sun, M. Endres, A universal quantum processor for entanglement enhanced optical tweezer clocks, in preparation. [3] P. Scholl*, A. Shaw*, R. Finkelstein*, R. Tsai, J. Choi, M. Endres, Erasure cooling, control, and hyper-entanglement of motion in optical tweezers, arXiv:2311.15580 (2023). [4] J. Choi, A. Shaw, I. Madjarov, X. Xie, R. Finkelstein, J. Covey, J. Cotler, D. Mark, H.Y. Huang, A. Kale, H. Pichler, F. Brandão, S. Choi, and M. Endres, Preparing random states and benchmarking with many-body quantum chaos, Nature 617 (2023) [5] P. Scholl, A. Shaw, R. Tsai, R. Finkelstein, J. Choi, M. Endres, Erasure conversion in a high-fidelity Rydberg quantum simulator, Nature 622 (2023). [6] A. Shaw, Z. Chen, J. Choi, D.K. Mark, P. Scholl, R. Finkelstein, A. Elben, S. Choi, M. Endres, Benchmarking highly entangled states on a 60-atom analog quantum simulator, arXiv:2308.07914 (2023).