DIAMANT
Day 1 (April 6) @ 13:30–15:00
Juan Peypouquet (University of Groningen)
A fast convergent first-order method bearing second-order information
We propose a model for a class of first-order methods as an inertial system with Hessian-driven damping. The model combines several features of the Levenberg-Marquardt algorithm and Nesterov’s acceleration scheme for first-order algorithms. We obtain a second-order system (in time and space), which can be interpreted as a first-order one by an appropriate transformation. The resulting method is more stable than classical accelerated methods, and just as fast.
Lola Thompson (Utrecht University)
Summing $\mu(n)$: an even faster elementary algorithm
We present a new elementary algorithm for computing $M(x) = \sum_{n\le x} \mu(n),$where $\mu(n)$ is the Möbius function. Our algorithm takes $$ \mbox{time}\quad O_\epsilon\left(x^\frac{3}{5}(\log x)^{\frac{3}{5}+\epsilon}\right)\quad \mbox{and space}\quad O\left(x^\frac{3}{10}(\log x)^\frac{13}{10}\right),$$which improves on existing combinatorial algorithms. While there is an analytic algorithm due to Lagarias-Odlyzko with computations based on the integrals of $\zeta(s)$ that only takes time $O(x^{1/2+\epsilon}),$our algorithm has the advantage of being easier to implement. The new approach roughly amounts to analyzing the difference between a model that we obtain via Diophantine approximation and reality, and showing that it has a simple description in terms of congruence classes and segments. This simple description allows us to compute the difference quickly by means of a table lookup. This talk is based on joint work with Harald Andrés Helfgott.
Cecilia Sagado (Rijskuniversiteit Groningen)
Large Mordell-Weil rank jumps and the Hilbert Property
We discuss recent progress on the variation of the Mordell-Weil rank in families of elliptic curves. We show that if the underlying surface admits a conic bundle structure then the subset of fibres for which the Mordell-Weil rank is strictly larger than the generic rank is not thin, as subset of the base of the fibration. We obtain larger rank jumps (+2 and +3) also on non-thin subsets of the line under extra hypotheses on the elliptic fibration. The results presented are based on joint work with D. Loughran (Bath) and on work in progress with R. Dias (UFRJ).
