Periods in action, PhD seminar of Ghent University.
Calculs symboliques et numériques avec les équations de Picard-Fuchs, Institut Fourier, Grenoble.
Periods in action, Combinatorics and Arithmetic for Physics at IHES.
SIAM Activity Group on Algebraic Geometry Early Career Prize, received in Atlanta.
Finding one root of a polynomial system: Smale’s 17th problem, plenary talk at FoCM 2017 in Barcelona.
“Computing the homology of basic semialgebraic sets in weak exponential time”, with P. Bürgisser and F. Cucker.
Periods in action, plenary talk at MEGA 2017 in Nice.
2017
How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ bound.
The new algorithm relies on numerical continuation along rigid continuation paths. The central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems. This leads to a better average condition number and allows for bigger steps. We show that on the average, we can compute one approximate root of a random Gaussian polynomial system of $n$ equations of degree at most $D$ in $n+1$ homogeneous variables with $O(n^5 D^2)$ continuation steps. This is a decisive improvement over previous bounds that prove no better than $\sqrt{2}^{\min(n, D)}$ continuation steps on the average.