ICM 2026 Invited Section Lecture - Thomas Rothvoss

The Subspace Flatness Conjecture The covering radius $\mu(\Lambda,K)$ of a convex body $K \subseteq \setR^n$ with respect to a full rank lattice $\Lambda \subseteq \setR^n$ is the minimum scaling needed so that translates of $K$ placed at every lattice point cover the whole $\setR^n$. A question going back to work of Kannan and Lov\'asz (1988) is how well the covering radius is described solely by the volume of $K$ and the density of the lattice, after projecting both into a suitable subspace. We report on significant progress on this question due to Regev and Stephens-Davidowitz (2016) who resolve this problem for ellipsoids and Reis and Rothvoss (2023) who cover the general case. We also survey applications of these results that give near-optimal bounds on the flatness constant as well as a $(\log n)^{O(n)}$-time algorithm for solving $n$-variable integer programs, following work of Dadush (2012). This is based on joint work with Victor Reis. Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices