In his manuscript "Pursuing stacks", Grothendieck mentions
a certain "schematization problem", which is a far reaching extension of the
rational homotopy theory of Quillen and Sullivan. In this talk, I will
explain the schematization problem, and present several progress , applications and
open questions.
The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. It was initiated by a work of Crandall and Rabinowitz, motivated by the Gelfand problem in combustion theory. The theory experienced a revival in the mid-nineties after new progress made by Brezis and collaborators. I will present these developments and my work in collaboration with Figalli,...
The magnetohydrodynamics (MHD) systems have several important conservative properties, e.g., the magnetic Gauss law and the conservation of energy and (magnetic, cross, hybrid) helicity in the ideal limit. These conserved quantities encode various kinds of intrinsic symmetry of the equations. To achieve physical fidelity and numerical stability, it is desirable to preserve these conditions...
A walk in the quarter plane is a path in the lattice $\mathbb{Z}^2$
with a prescribed set of directions that is confined in the quarter plane. In
the recent years, the enumeration of such walks has attracted the attention
of many authors in combinatorics and probability. The complexity of their
enumeration is encoded in the algebraic nature of their associated generating
series. The main...
We are interested in monitoring patients in remission from cancer. Our aim is to detect their relapses as soon as possible, as well as detect the type of relapse, in order to decide on the appropriate treatment to be given. Available data are some marker level of the rate of cancerous cells in the blood which evolves continuously but is measured at discrete (long) intervals and through noise....
A classical theorem of Liouville asserts that if a map from the sphere to itself is conformal, then it must be a Möbius map: a composition of dilations, rotations, inversions and translations (identifying sphere and euclidean space via stereographic projection). There is a long history of studying stability of this rigidity statement: if a map is nearly conformal, must it be close to a Möbius...
Faced with water scarcity, crop irrigation needs to optimize water inputs over a season, to ensure crop production under both water and nitrogen stress. We study several optimal control problems on a reduced crop model, which allow us to characterize policy structures that can be tested on more complex models or in the field.
In particular, we show the role of non-autonomous and non-smooth...
The Burau representation of braid groups goes back to the thirties, and can be defined by assigning to each generator an explicit matrix. Despite this very easy definition, it is still an open question whether or not the representation is faithful in the 4-strand case (faithfulness in the 3-strand case is an old result, and counterexamples for 5 strands were found by Bigelow in 1999). I'll try...
The substantial development of high-throughput bio-technologies has rendered large-scale multi-omics datasets increasingly available. New challenges have emerged to process and integrate this large volume of information, often obtained from widely heterogeneous sources. Kernel methods have proven successful to handle the analysis of different types of datasets obtained on the same individuals....
Contact phenomena between deformable bodies abound in industry and everyday life. A few simple examples are brake pads in contact with wheels, tires on roads, and pistons with skirts. Because of the importance of contact processes in structural and mechanical systems, considerable effort has been put into their modeling, analysis and numerical simulations and, consequently, the Mathematical...
