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The well-known Zig-Zag product and related graph operators, like derandomized squaring, are fundamentally combinatorial in nature. Classical bounds on their behavior often rely on a mix of combinatorics and linear algebra. However, these traditional bounds are not tight and frequently fail to align with experimental results. In this talk, we will present a more refined analysis that utilizes the full spectrum of the graph, rather than relying solely on its spectral expansion. This approach produces results that both match experimental observations and, in a sense, are proved to be optimal. Our technique is analytic, diverging from classical methods: for the upper bound, we apply finite free probability, while for the lower bound, we draw on results from analytic combinatorics.
Based on joint works with Itay Cohen, Gal Maor and Yuval Peled. No prior knowledge is required.

I will show that every acylindrically hyperbolic group Γ admits a weakly mixing probability measure preserving action which is faithful but not essentially free. In other words, Γ admits a weakly mixing nontrivial faithful IRS. I will also show that every hyperbolic group admits a characteristic random subgroup with the same properties. Both results rely on works of Sun and Kechris–Quorning. 

This is a joint work with Anton Hase.

Let $\mathcal{R} : L^\infty(\mathbb{S}^{n-1}) \rightarrow L^\infty(\mathbb{S}^{n-1})$ denote the spherical Radon transform, defined as $\mathcal{R}(f)(\theta) = \int_{\mathbb{S}^{n-1} \cap \theta^{\perp}} f(u) d\sigma(u)$. A long-standing question in non-linear harmonic analysis due to Lutwak, Gardner, and Fish--Nazarov--Ryabogin--Zvavitch, is to characterize those non-negative $\rho \in L^\infty(\mathbb{S}^{n-1})$ so that $\mathcal{R}(\rho^{n-1}) = c \rho$ when $n\geq 3$. We show that this holds iff $\rho$ is constant, and moreover, $\mathcal{R}(\mathcal{R}(\rho^{n-1})^{n-1}) = c \rho$ iff $\rho$ is either identically zero or is the reciprocal of some Euclidean norm. Our proof recasts the problem in a geometric language using the intersection body operator $I$,  introduced by Lutwak following the work of Busemann, which plays a central role in the dual Brunn-Minkowski theory. We show that for any star-body $K$ in $\mathbb{R}^n$ when $n \geq 3$, $I^2 K = c K$ iff $K$ is a centered ellipsoid, and hence $I K = c K$ iff $K$ is a centered Euclidean ball. To this end, we interpret the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of $I K$, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when $n\geq 3$.

Joint work with Shahar Shabelman and Amir Yehudayoff.

I will explain the conjectured duality between double-scaled SYKat infinite temperature and the static patch of a particularversion of JT-de Sitter space. The conjectured duality raises thequestion of what particle physics is like in the bulk of the staticpatch.The answer: I will give it in the lecture.

Horizontal gene transfer (HGT) is a central mechanism in bacterial evolution, allowing organisms to exchange genetic material outside of traditional reproduction. This process is a key driver of antibiotic resistance and the emergence of virulence traits.Unlike vertical inheritance, HGT leads to non-tree-like evolutionary relationships, motivating a network-based view of microbial evolution.In this talk I will present a minimal model for HGT and show how it captures distinctive statistical features of bacterial genomes. By combining the model with the genomic data, I infer general properties of the underlying HGT networks. FOR THE LATEST UPDATES AND CONTENT ON SOFT MATTER AND BIOLOGICAL PHYSICS AT THE WEIZMANN, VISIT OUR WEBSITE: https://www.biosoftweizmann.com/ 

2D materials offer vast variability, with nearly unlimited combinations of composition, properties, and structures. This versatility can be further extended through layer stacking and twisting, enabling unique electronic and mechanical behaviours. The diversity in chemical composition necessitates various approaches for their crystal growth and chemical modifications.This discussion will cover the synthesis and crystal growth methods for different classes of 2D materials, including chalcogenides, halides, chalcogen-halides, and beyond. The impact of experimental conditions on their structural and functional properties will also be explored.Exfoliation techniques, particularly those involving intercalation, provide a pathway for obtaining large-area monolayer flakes and bulk intercalated compounds with tailored properties. The effects of these methods on material characteristics will be examined. Additionally, chemical exfoliation methods for materials with layered structures held together by covalent bonds will be presented.Finally, the applications of 2D materials across multiple fields will be discussed, including electronics, energy storage, catalysis, and beyond. This overview aims to highlight the transformative potential of 2D materials from fundamental synthesis to practical technological implementations.