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.

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).'