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Typical time-efficient encoding and decoding algorithms for error correcting codes use linear space. We construct asymptotically good codes that can be deterministically encoded in almost linear time and sub-linear space, as well as asymptotically good codes that can be deterministically decoded in this complexity. The encodable codes are based on condenser graphs. The decodable codes are based on locally correctable codes and a new efficient derandomization method. We believe that the new derandomization method is of independent interest.

The talk is based on joint works with Joshua Cook (University of Texas at Austin).

In this talk we will introduce a notion slightly weaker than the coboundary expansion notion of Linial–Meshulam called the cosystolic expansion,  which is strong enough to implies Gromov’s topological property for high dimensional complexes. We will then give a criterion for proving this notion assuming our complex has good enough local structure. Examples of complexes with such good local structure are quotient of affine buildings (e.g. the LSV complexes). This allow us to present a family of bounded degree complexes with the topological overlapping property.

This is based in a joint work with Tali Kaufman.

While stochastic optimization methods drive continual improvements in machine learning, choosing the optimization parameters—particularly the learning rate (LR)—remains challenging. In this talk, I will describe our work on eliminating LR tuning from stochastic gradient descent (SGD) under convexity assumptions. Our starting point is a novel post-hoc empirical certificate for the SGD step size choice, which yields strong parameter-free guarantees via a bisection procedure. This certificate also inspires a tuning-free dynamic SGD step size formula, which we call Distance over Gradients (DoG). For smooth stochastic objectives, we combine DoG with UniXGrad (Kavis et al., 2019) to obtain the first accelerated parameter-free method. Finally, we develop a “price of adaptivity” framework that allows us to evaluate the inherent cost of not knowing problem parameters in advance. In several settings, our lower bounds nearly match existing upper bounds, establishing there is no parameter-free lunch. 

Joint work with Maor Ivgi, Itai Kreisler, and Oliver Hinder.

A cooperative resonance effect in a stochastic nonlinear dynamical system subjected to external weak periodic forcing, called stochastic resonance (SR), has been extensively studied for the past forty years. Here I discuss the experimentally unexpected observation of SR above an elastic non-modal instability of an inertia-less channel flow of polymer solution (much more complicated than stochastic dynamical flow) due to finite-size white noise perturbations. This flow is shown to be linearly stable similar to Newtonian parallel shear flow. First, I briefly describe viscoelastic flow with curved streamlines, where linear elastic normal mode instability at the critical Weissenberg number, Wic, has been observed and characterized, and the elastic instability mechanism has been explained and experimentally validated. Furthermore, at Wi>>Wic, “elastic turbulence” (ET), a chaotic flow arising via secondary instability, is experimentally discovered, characterized and theoretically explained, while elastic instability in straight channel flow is found from the direct transition from laminar to chaotic flow in the transition flow regime is found. At the secondary instability, ET is observed, and further on the next transition to the unexpected drag reduction flow regime takes place, accompanied by elastic waves previously discovered and characterized earlier. Moreover, we propose and experimentally validate a mechanism of amplification of the wall normal fluctuating vortices by the elastic waves. The elastic waves play the key role in the energy transfer from the main flow to the wall-normal fluctuating vortices. Finally, we report on recently discovered SRs only in a limited subrange of weak elastic waves just above Wic, their characterization, and their role in the transition to a chaotic flow.

Expander is a family of constant degree graphs with spectral gaps larger than some positive constant. From a Metric Geometry perspective, the spectral gap implies their non-embeddability in Hilbert space in any meaningful way. It is therefore customary in Metric Geometry to generalize them to expanders that do not embed well in a given metric space X, called X-expanders. Non-Hilbertian Expanders were first studied by Linial-London-Rabinovich, Matousek, and Gromov. In a breakthrough result, V. Lafforgue proved the existence of a super(-reflexive)-expander. It is an open question whether every (classical) expander is also a super-expander.

In this talk I will discuss the existence of expanders with respect to random regular graphs. It gives the first example of a metric space X for which some expanders are X-expanders and some are not X-expanders. The construction is based on the zigzag expanders of Reingold-Vadhan-Wigderson and uses concepts and techniques from expander graph theory, random graph theory, probability, local theory of Banach spaces, and Alexandrov spaces.

The talk is based on joint papers with A. Naor and a forthcoming joint paper with A. Eskenazis and A. Naor.

Our work answers a nearly 60-year quest to derive the turbulent spectrum of weakly interacting internal gravity waves from first principles. The classical wave-turbulence approach didn’t work, as the underlying equation, both in 2D and 3D, is an anisotropic, non-canonical Hamiltonian equation.
A key consequence of the non-canonical Hamiltonian is the conservation of a sign-indefinite quadratic invariant alongside the sign-definite quadratic energy. In 2D, this allows us to derive a much simpler kinetic equation. We leverage this simplification into the derivation of solutions of the kinetic equation, one of which is the turbulent spectrum of weakly interacting 2D internal gravity waves. Our spectrum exactly matches the phenomenological oceanic Garrett-Munk spectrum in the limit of large vertical wave numbers and zero rotation.
This talk is based on recent joint works with Oliver Bühler and Jalal Shatah
arXiv:2311.04183 (to appear soon in PRL).
arXiv:2406.06010.
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The study of human memory is a rich field with a history that spans over a century, traditionally investigated through the prism of psychology. Drawing inspiration from this vast pool of findings, we approached the subject with a more physics-oriented mindset based on first principles. For this reason, we combined mathematical modelling of established ideas from the literature of psychology with large-scale experimentation. In particular, we created a model based on the concept of retroactive interference that states that newly encoded items hinder the retention of older ones in memory. We show that this simple mechanism is sufficient to describe a variety of experimental data of recognition memory with different categories of verbal and pictorial stimuli. The model has a single free parameter and can be solved analytically. We then focus on recall and recognition memory of stories. This transition from discrete random lists to coherent continuous stimuli such as stories introduces a new challenge when it comes to the quantification and the analysis of the results. To address this, we have developed a pipeline that employs large language models and showed that it performs comparably to human evaluators. Using this tool we were able to show that recall scales linearly with recognition and story size for the range we examined. Finally, we discovered that when stories are presented in a scrambled manner, even though recall performance drops, subjects seem to reconstruct the material in their recall in alignment to the unscrambled version.

The aim of the talk is to show that simple groups over local fields have a spectral gap absorption principle. That is, that if a representation that doesn't have almost invariant vectors is tensored with another representation, then the tensored representation still doesn't have almost invariant vectors. This property was conjectured by Uri Bader and Roman Sauer in their paper about unitary cohomology, and was proved there in some of the cases. We prove the general result. Such tensor products appear naturally when one works with restriction and induction of representations, and it is useful to know that the spectral gap is preserved.

 

I will (try to) give a survey of the rich and beautiful theory of representations of semisimple groups, and show how to use the celebrated Langlands classification theorem, as well as some more modern results, in order to prove the theorem.

In recent years the cell nucleus emerged as a dynamic mechanosensor capable of sensing and transducing mechanical signals into cellular responses to facilitate homeostasis and adaptation to changing environmental conditions. The constantly beating heart has a remarkable ability to adapt its structure and contractility in response to changes in mechanical load. I am introducing unique, live, and dynamic imaging approaches to investigate how nuclei in the mature heart can provide such mechano-protection and mechano-regulation of the genome. I will present a novel assay to couple cytoskeletal to nuclear strain transfer in the beating cardiomyocyte, and its further application to decipher mechanisms of nuclear damage in dilated cardiomyopathy caused by mutations in the LMNA gene (LMNA-DCM). This work pinpoints localized microtubule-dependent forces, but surprisingly not actomyosin contractility, as drivers of nuclear damage in LMNA-DCM, highlighting new therapeutic avenues. I will further discuss the role of mechanical signaling in spatial organization of the genome within the nucleus, to regulate transcriptionally active and repressed hubs, and downstream gene expression.