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Let F be a function field and G a connected split reductive group over F. We define a "strange" operator between different spaces of automorphic functions on G(A)/G(F), and show that this operator is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. We discuss how to define this operator over a number field by relating it to pseudo-Eisenstein series and inversion of the standard intertwining operator. This operator is also connected to Deligne-Lusztig duality and cohomological duality of representations over a local field.

The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model.

Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

Generative Adversarial Networks (GANs) is a recent algorithmic framework that has won considerable attention. In a nutshell, GANs receive as input an IID sample and outputs synthetic data that should resemble data from the true underlying distribution. For example, consider an algorithm that receives as input some tunes from a specific music genre (e.g. jazz, rock, pop) and then outputs a new, original, tune from that genre.

From a theoretical perspective, the distinction between algorithms that genuinely generate original new examples vs. algorithms that perform naive manipulations (or even merely memorization) of the input sample is an elusive distinction. This makes the theoretical analysis of GANs algorithms challenging.

In this work we introduce two mathematical frameworks for the task of generating synthetic data. The first model we consider is inspired by GANs, and the learning algorithm has only an indirect access to the target distribution via a discriminator. The second model, called DP-Foolability, exploits the notion of differential privacy as a criterion for "non-memorization".

We characterize learnability in each of these models as well as discuss the interrelations. As an application we prove that privately PAC learnable classes are DP-foolable. As we will discuss, this can be seen as an analogue of the equivalence between uniform convergence and learnability in classical PAC learning.

Joint work with Olivier Bousquet and Shay Moran.

https://arxiv.org/pdf/1902.03468.pdf

Modern robotic and vision systems are often equipped with a direct 3D data acquisition device, e.g. a LiDAR or RGBD camera, which provides a rich 3D point cloud representation of the surroundings. Point clouds have been used successfully for localization and mapping tasks, but their use in semantic understanding has not been fully explored. Recent advances in deep learning methods for images along with the growing availability of 3D point cloud data have fostered the development of new 3D deep learning methods that use point clouds for semantic understanding. However, their unstructured and unordered nature make them an unnatural input to deep learning methods. In this work we propose solutions to three semantic understanding and geometric processing tasks: point cloud classification, segmentation, and normal estimation. We first propose a new global representation for point clouds called the 3D Modified Fisher Vector (3DmFV). The representation is structured and independent of order and sample size. As such, it can be used with 3DmFV-Net, a newly designed 3D CNN architecture for classification. The representation introduces a conceptual change for processing point clouds by using a global and structured spatial distribution. We demonstrate the classification performance on the ModelNet40 CAD dataset and the Sydney outdoor dataset obtained by LiDAR. We then extend the architecture to solve a part segmentation task by performing per point classification. The results here are demonstrated on the ShapeNet dataset. We use the proposed representation to solve a fundamental and practical geometric processing problem of normal estimation using a new 3D CNN (Nesti-Net). To that end, we propose a local multi-scale representation called Multi Scale Point Statistics (MuPS) and show that using structured spatial distributions is also as effective for local multi-scale analysis as for global analysis. We further show that multi-scale data integrates well with a Mixture of Experts (MoE) architecture. The MoE enables the use of semi-supervised scale prediction to determine the appropriate scale for a given local geometry. We evaluate our method on the PCPNet dataset. For all methods we achieved state-of-the-art performance without using an end-to-end learning approach.

The theory of locally compact quantum groups grew out of the need to extend Pontryagin's duality for locally compact abelian groups to a wider class of objects, as well as from a modern "quantum" point of view suggesting the replacement of some algebras of functions on a group by non-commutative objects, namely operator algebras. In this talk, which will be split into two parts, we will show how several fundamental notions from probability and geometric group theory fit in this framework.

The first part will be an introduction to locally compact quantum groups. We will present the rationale and the definitions, give examples, and explain how the theory is related to other branches of math. If time permits, we will also touch upon more specific notions related to the second part.

In the second part we will discuss convolution semigroups of states, as well as generating functionals, on locally compact quantum groups. One type of examples comes from probability: the family of distributions of a L\'evy process form a convolution semigroup, which in turn admits a natural generating functional. Another type of examples comes from (locally compact) group theory, involving semigroups of positive-definite functions and conditionally negative-definite functions, which provide

important information about the group's geometry. We will explain how these notions are related and how all this extends to the quantum world; derive geometric characterizations of two approximation properties of locally compact quantum groups; see how generating functionals may be (re)constructed and study their domains; and indicate how our results can be used to study cocycles.

Based on joint work with Adam Skalski.

No background in operator algebras will be assumed.

Inverse problems appear in many applications, such as image deblurring, inpainting and super-resolution. The common approach to address them is to design a specific algorithm (or recently - a deep neural network) for each problem. The Plug-and-Play (P&P) framework, which has been recently introduced, allows solving general inverse problems by leveraging the impressive capabilities of existing denoising algorithms. While this fresh strategy has found many applications, a burdensome parameter tuning is often required in order to obtain high-quality results. In this work, we propose an alternative method for solving inverse problems using off-the-shelf denoisers, which requires less parameter tuning (can be also translated into less pre-trained denoising neural networks). First, we transform a typical cost function, composed of fidelity and prior terms, into a closely related, novel optimization problem. Then, we propose an efficient minimization scheme with a plug-and-play property, i.e., the prior term is handled solely by a denoising operation. Finally, we present an automatic tuning mechanism to set the method's parameters. We provide a theoretical analysis of the method, and empirically demonstrate its impressive results for image inpainting, deblurring and super-resolution. For the latter, we also present an image-adaptive learning approach that further improves the results.

The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this talk, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the MA-completeness of the problem of deciding whether the groundenergy of a uniform stoquastic local Hamiltonian is zero or inverse polynomial. We show that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at least some constant, is in NP. Thus, if there exists a gap-amplification procedure for uniform stoquastic Local Hamiltonians (in analogy to the gap amplification procedure for constraint satisfaction problems in the original PCP theorem), then MA = NP (and vice versa). Furthermore, if this gap amplification procedure exhibits some additional (natural) properties, then P = RP. This is a joint work with Dorit Aharonov.

A graph is automatically also a metric space, but is there anything interesting to say about such metric spaces? Many fascinating and concrete questions are encapsulated in the more general (and vague) question "to what extent can a finite graph emulate the properties of a infinite regular tree ?". We will see how this leads us to investigate expansion in graphs and questions about the large scale aspects of graph metrics including girth and diameter and the relations between the two. If time permits (probably not) I may also say a little about the local geometry of graphs.

This talk is based on many collaborations which I have had over the years, among my more recent collaborators are Michael Chapman, Yuval Peled and Yonatan Bilu. The talk requires no particular previous background and should be accessible to general mathematical audience.