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# Upcoming Seminars

Universal learning is considered from an information theoretic point of view following the universal prediction approach pursued in the 90's by F&Merhav. Interestingly, the extension to learning is not straight-forwar

Understanding statistical properties of zeros of Laplacian eigenfunctions is a program which is attracting much attention from mathematicians and physicists. We will discuss this program in the setting of "quantum graphs", self-adjoint differential operators acting on functions living on a metric graph. Numerical studies of quantum graphs motivated a conjecture that the distribution of nodal surplus (a suitably rescaled number of zeros of the n-th eigenfunction) has a universal form: it approaches Gaussian as the number of cycles grows. The first step towards proving this conjecture is a result established for graphs which are composed of cycles separated by bridges. For such graphs we use the nodal-magnetic theorem of the speaker, Colin de Verdiere and Weyand to prove that the distribution of the nodal surplus is binomial with parameters p=1/2 and n equal to the number of cycles. Based on joint work with Lior Alon and Ram Band.

In this talk I will describe some recent work on distributed property testing in the networks with bounded bandwidth (the CONGEST model): we have a network of computing nodes communicating over some initially-unknown network graph, where every communication link can carry a bounded number of bits per round. Some simple-looking problems, such as checking if the network contains a 4-cycle, are known to be very hard in this model, and this motivates us to consider property testing instead of exact solutions.

I will describe distributed property testing algorithms for two problems: subgraph-freeness, where we wish to determine whether the network graph contains some fixed constant-sized subgraph H; and uniformity testing, where every node of the network draws samples from an unknown distribution, and our goal is to determine whether the distribution is uniform or far from uniform. I will also discuss lower bounds.

We consider the geo-localization task - finding the pose (position & orientation) of a camera in a large 3D scene from a single image. We aim toexperimentally explore the role of geometry in geo-localization in a convolutional neural network (CNN) solution. We do so by ignoring the often available texture of the scene. We therefore deliberately avoid using texture or rich geometric details and use images projected from a simple 3D model of a city, which we term lean images. Lean images contain solely information that relates to the geometry of the area viewed (edges, faces, or relative depth). We find that the network is capable of estimating the camera pose from the lean images, and it does so not by memorization but by some measure of geometric learning of the geographical area. The main contributions of this work are: (i) providing insight into the role of geometry in the CNN learning process; and (ii) demonstrating the power of CNNs for recovering camera pose using lean images.

This is a joint work with Moti Kadosh & Ariel Shamir

Periods of automorphic forms have an important place in the theory of automorphic representations. They are often related to (special values of) L-functions and have applications to arithmetic geometry and analytic number theory. For an automorphic form on a group G, a period is its integral over a subgroup of G. If the automorphic form is not cuspidal such integrals are usually divergent. It is nonetheless possible in many cases to extend the definition of the period to almost all automorphic forms which has direct applications to the study of the given period. In this talk I will describe a general procedure of defining such periods in the case when the subgroup is reductive.

I will also discuss the joint work with A. Pollack and C. Wan that applies this to the study of certain periods and their relations to special values of L-functions confirming predictions of Sakellaridis and Venkatesh.