Events

Large language models (LLMs) have taken the field of AI by storm. But how large do they really need to be? I'll discuss the phi series of models from Microsoft, which exhibit many of the striking emergent properties of LLMs despite having merely a few billion parameters.

More information: https://zucmap.ethz.ch/call_made

In a seminal paper, Kannan and Lov\’asz (1988) considered a quantity $\mu_{KL}(\Lambda,K)$ which denotes the best volume-based lower bound on the \emph{covering radius} $\mu(\Lambda,K)$ of a convex body $K$ with respect to a lattice $\Lambda$. Kannan and Lov\’asz proved that $\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log n)$ factor suffices, which would match the lower bound from the work of Kannan and Lov\’asz. We settle this conjecture up to a constant in the exponent by proving that $\mu(\Lambda,K) \leq O(\log^{3}(n)) \cdot \mu_{KL} (\Lambda,K)$. Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a $(\log n)^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal \emph{flatness constant} of $O(n \log^{3}(n))$. This is joint work with Victor Reis.

The aim of this talk is to investigate what survives of the standard estimates valid for operators with constant coefficients in the case of variable coefficients. The general strategy is based on quantifying how far the (inverse) operator with variable coefficients is from an (inverse) operator with constant coefficients, and obtain the desired estimates by perturbation. Whereas this is classically done at the level of the coefficients themselves (Meyers’ estimates, Schauder theory e.g.), in this talk I will use closeness in the sense of homogenization. As an illustration, I will discuss large-scale Meyers’ estimates, large-scale Lipschitz estimates, and conclude with large-scale dispersive estimates.

''The moduli space of curves was first studied by Riemann. I will explain what it is, how to compactify it, and how to attempt to compute the cohomology of its compactifications. Ideas from a broad range of mathematics are necessary, including low dimensional topology, algebraic geometry, and number theory.

I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain O in R^m. These minimizers can be thought of as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of O. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.

A day of talks on Random Matrix Theory, in an informal environment encouraging discussion. The planned scheduled is: 10:15-11:00 Afonso Bandeira 11:15-12:00 Antti Knowles 14:15-15:00 Ashkan Nikeghbali 15:15-16:00 Dominik Schroder 16:15-17:00 Benjamin Schlein

In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and Eaton lenses. The three of them are motivated by physics. The wind-tree model was instoduced by Paul and Tatyana Ehrenfest to study a gaz: a particule is moving in a plane where obstacles are periodically placed, on which the particule bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens. In the beginning of the 2000's, physicists have conceived metamaterials with negative index of refraction. Tilling billiards' trajectories consist of light rays moving in a arrangement of metamaterials with opposite indew of refraction. After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle. I show that these trajectories are traped in a strip, for almost every parameter. This behavior is similar to the one of Eaton lenses.

The Cremona group of rank n is the group of birational transformations of the projective n-space. Cremona groups have been studied for around 150 years and attracted a lot of attention in the last decades due to their rich group theoretical and dynamical properties. In this talk, I will first give a short introduction to Cremona groups (no prior knowledge of algebraic geometry assumed). Then I will present natural constructions of CAT(0) cube complexes on which Cremona groups act by isometries, and explain how we can deduce old and new results from this action. This is joint work with Anne Lonjou.

A graphic sequence is a non-increasing sequence of natural numbers that can occur as the degree sequence of a graph. We show that the number of graphic sequences of length n grows like cn^{-3/4}4^n for some constant c, answering a question by Royle. The foundation of our proof consists of a few reformulations that turn our problem into a question about the lazy simple symmetric random walk bridge. To be precise, we calculate the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge never goes negative. Our reformulation also yields a new, efficient algorithm for exact enumeration of graphic sequences, with which we are able to calculate many more exact values than previously known. This talk is based on joint work with Paul Balister, Carla Groenland, Tom Johnston and Alex Scott. In the last part of the talk, I will also touch upon a work with Brett Kolesnik, in which we use random walk techniques to show that the possible (non-increasing) score sequences of a round-robin tournament with n players grows like c'n^{-5/2}4^n for some constant c', answering a question by Erdös and Moser. We also obtain a Brownian scaling limit of a uniformly random score sequence.

'We prove the instantaneous cusp formation from a single corner of the vortex patch solutions. This positively settles the conjecture given by Cohen-Danchin in Multiscale approximation of vortex patches, SIAM J. Appl. Math. 60 (2000), no. 2, 477-502. This is a joint work with Tarek Elgindi (Duke University).'