I will discuss Dye's Theorem and the uniqueness of hyperfinite equivalence relations following mostly [Kerr-Li] Section 4.9.

For more details and the exact reference, click here.  

Suppose $u_j$ is an orthonormal basis of eigenfunctions of the Laplacian on a compact Riemannian manifold, and consider the probability measures $|u_j|^2 dvol$. The quantum ergodicity theorem of Shnirelman, Zelditch and Colin de Verdiere, states that if the geodesic flow on the manifold is ergodic then there is a density one subsequence of these measures that equidistributes. What about the remaining eigenfunctions? The quantum unique ergodicity conjecture of Rudnick and Sarnak states that for hyperbolic manifolds (or more generally for negatively curved manifolds), any sequence of measures as above equidistributes.  In 2006 Lindenstrauss proved that this is true for Hecke—Maass forms on congruence surfaces. These are eigenfunctions of both the Laplacian and the Hecke operators, which are discrete averaging operators coming from the arithmetic structure of the manifold. 

We will discuss our extensions of Lindenstrauss’ results to the three and (partially) four dimensional cases. One of the challenges that arise in the higher dimensional case is to rule out concentration of measure on totally geodesic submanifolds. We will focus on this issue and our new methods for showing non-concentration. 

Based on joint works with Alexandre de Faveri and Lior Silberman.

How likely is $G(n;p)$ to have a less-than-typical number of triangles? This is a foundational question in non-linear large deviation theory. When $p \ll 1/\sqrt{n}$ or $p \gg 1/\sqrt{n}$ the answer is fairly well-understood, with Janson's inequality applying in the former case and regularity- or container-based methods applying in the latter. We study the regime $p=c/\sqrt{n}$, with $c$ fixed, with the large deviation event having at most $\eta$ times the expected number of triangles, for a fixed $\eta \in [0,1)$.

We prove explicit formula for the log-asymptotics of the event in question, for a wide range of pairs $(c,\eta)$. In particular, we show that for sufficiently small $\eta$ (including the triangle-free case $\eta=0$) there is a phase transition as $c$ increases, in the sense of a non-analytic point in the rate function. On the other hand, if $\eta > 1/2$, then there is no phase transition.

As corollaries, we obtain analogous results for the $G(n;m)$ model, when $m = Cn^{3/2}$. In contrast to the $G(n;p)$ case, we show that a phase transition occurs as $C$ increases for \textit{all} $\eta$.

Finally, we show that the probability of $G(n;m)$ being triangle free, where $m=Cn^{3/2}$ for a sufficiently small constant $C$, conforms to a Poisson heuristic.

Joint with Matthew Jenssen, Will Perkins, and Aditya Potukuchi. Based on arXiv:2410.22951 and arXiv:2411.18563.

The rapid growth of generative models allows an ever-increasing variety of capabilities. Yet, these models may also produce undesired content such as unsafe images, private information, or copyrighted material.

In this talk, I will discuss practical methods to prevent undesired generations. First, I will show how the challenge of avoiding undesired generations manifested itself in a simple Capture-the-Flag LLM setting, where even our top defense strategy was breached. Next, I will demonstrate a similar vulnerability in state-of-the-art concept erasure methods for Text-to-Image models. Finally, I will describe the notion of ‘Unconditional Concept Erasure’ aiming to mitigate such vulnerabilities. I will show that Task Vectors can achieve Unconditional Concept Erasure, and discuss the challenge of applying Task Vectors in practice.

Bio: Niv is a postdoctoral researcher at New York University hosted by Prof. Chinmay Hegde. He received a BSc in mathematics with physics as part of the Technion Excellence Program. He received his PhD in computer science from the Hebrew University of Jerusalem, advised by Prof. Yedid Hoshen. Niv was awarded the Israeli data science scholarship for outstanding postdoctoral fellows (VATAT). He is interested in anomaly detection, model personalization, and AI safety for Vision & Language models.