Events

A fundamental theorem states that a two-dimensional Riemannian manifold with boundary, equipped with a symmetric tensor field, is locally isometrically embedded in Euclidean space if and only if the symmetric tensor field satisfies the Gauss-Mainardi-Codazzi equations—in which case, the tensor field is the second fundamental form.

When the intrinsic metric is Euclidean, it is a classical result that such tensor fields are Hessians of functions satisfying the Monge-Ampère equation. I shall present a version of this result to arbitrary Riemannian metrics, using a generalized Hodge theory I developed for a broader class of geometric boundary-value problems. I will discuss this theory, its main features, and perhaps give a glimpse of more complicated examples it addresses.

NeNeural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, f:X→Y. Is it possible to identify a pair of non-standard vector spaces for which a conventionally nonlinear function is, in fact, linear? This paper introduces a method that makes such vector spaces explicit by construction. We find that if we sandwich a linear operator between two invertible neural networks, then the corresponding vector spaces are induced by newly defined operations. This framework makes the entire arsenal of linear algebra applicable to nonlinear mappings. We demonstrate this by collapsing diffusion model sampling into a single step, enforcing global idempotency for projective generative models, and enabling modular style transfer. 

Bio:

Assaf is an Assistant Professor at the Technion in the Faculty of Data and Decision Sciences. Previously, he was a Research Scientist at NVIDIA, a Postdoc at UC Berkeley with Alyosha Efros, and a Visiting Scholar at Google DeepMind. I received my PhD from the Weizmann Institute of Science, advised by Michal Irani. I have two Bachelor degrees from Ben-Gurion University in Physics and Electrical-Engineering. Assaf’s research focuses on Deep Neural Networks for computer vision, guided by two core principles: a pursuit of elegant, foundational ideas that offer fundamentally new perspectives, and a focus on dynamic and adaptive learning for real-world scenarios like unannotated data streams and distribution shifts.

The corners problem is a classical problem in additive combinatorics. A corner is a triple of points (x,y), (x+d,y), (x,y+d). It can be viewed as a 2-dimensional analog of a (one-dimensional) 3-term arithmetic progression. An old question of Ajtai and Szemeredi is: how many points can there be in the n x n integer grid without containing a corner? They proved a qualitative bound of o(n^2), but no effective quantitative bounds.

This question has an equivalent description in the language of communication complexity. Given 3 players with inputs x,y,z which are integers in the range 1 to n, what is the most efficient Number-On-Forehead (NOF) deterministic protocol to check if they sum to n. This connection was first observed in the seminal paper of Chandra, Furst and Lipton that introduced the NOF model back in 1983.

In the language of communication complexity, the trivial protocol sends log(n) bits, but there is a better NOF protocol (based on constructions in additive combinatorics) which only sends (log n)^{1/2} bits. However, the best lower bound until our work was double exponentially far off - of the order of log log log n. In this work, we close this gap, and prove a lower bound of (log n)^c for some absolute constant c.

The work is based on combining the high-level approach of Shkredov, who obtained the previous lower bound, which was based on Fourier analysis; with the recent breakthrough of Kelley and Meka on the 3-term arithmetic progression problem, and the ensuing developments. The main message is that "spreadness" based techniques (a notion that I will explain in the talk) give significantly better quantitative bounds compared to classical Fourier analysis.

Joint work with Michael Jaber, Yang P. Liu, Anthony Ostuni and Mehtaab Sawhney

Paper will appear in FOCS 2025
https://arxiv.org/abs/2504.07006