Weekly Bulletin

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.

A scheme X is a splinter if for any finite surjective morphism f: Y \to X the pullback map O_X \to f_* O_Y splits as O_X-modules. By the direct summand conjecture, now a theorem due to André, every regular Noetherian ring is a splinter. Whilst for affine schemes the splinter property can be viewed as a local measure of singularity, the splinter property imposes strong constraints on the global geometry of proper schemes over a field of positive characteristic. For instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology. I will report on joint work with Charles Vial concerning further obstructions on the global geometry of proper splinters in positive characteristic.

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.

More information to be announced closer to date.

'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).'