In a first part, we present a joint work with Josef Dorfmeister : a Weierstrass type representation for constrained Willmore surfaces in spheres. Using appropriates (moving) frames and a appropriate Lie algebra decomposition of so(1,n+3), we translate the PDE of constrained Willmore surfaces into the Lie algebra setting : namely we rewrite it as a Maurer-Cartan equation of an extended Maurer...
I will present several results and open problems about Koszul-Tate resolutions. Several topics will be addressed : how symmetries "closing" only on a singular subset naturally give a Z-graded Q-manifold. with Koszul-Tate negative part, how to get constructive Koszul-Tate resolutions. Then I will detail several open questions about affine varieties. Joints works with Hancharuk, Kotov, Salnikov, Strobl.
The formation of codimension-one interfaces for multi-well gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The vectorial case in contrast is quite open. This lack of results and insight is to a large extent related to the absence of known monotonicity formula. I will focus on the ...
Gauge fields are the local expression of a principal connection on a principal bundle, and therefore encode the infinitesimal data of a parallel transport between the fibres along curves on the base manifold. One may wonder if there is a field interpretation for parallel transport. Reading principal connections as infinitesimal connections on the associated Atiyah Lie algebroid, this question...
The Atiyah class of a dg manifold $(\mathcal{M},Q)$ is the obstruction to the existence of an affine connection on the graded manifold $\mathcal{M}$ that is compatible with the homological vector field $Q$. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. Using Kontsevich's famous formality theorem, Liao, Xu and I...
Einstein's equation can be obtained as the Euler-Lagrange equation associated to the Hilbert-Einstein Lagrangian, which is essentially the scalar curvature. The curvature tensor, and therefore Einstein's equation, can be formulated and studied on the frame bundle of spacetime. We will introduce a Lagrangian defined on a 10-manifold such that the solutions to the Euler-Lagrange equations equip...
We discuss few very recent results of a work in progress (in collaboration with I. Gaiur and D. Van Straten and with V. Buchstaber and I. Gaiur) about interesting properties of multiplication Bessel kernels, which includes well-known Clausen and Sonin-Gegenbauer formulae of XIX century, special examples of Kontsevich discriminant loci polynomials, raised as addition laws for special...
In a first part, I will report on an ongoing collaboration with Adrien Kassel (CNRS, ENS Lyon) where we elaborate on a classical theorem of G. Kirchhoff (1847). In a second part, I will present recent developments due to Frédéric Hélein on an early work of W.A. Mozart (1762).
I will talk about the problem to prescribe the Jacobian determinant or the volume form, I
will present some old and more recent results on this problem, and mention the link with the
density problem in Sobolev spaces.
In classical differential geometry, geometric transformations have been used to create new curves and surfaces from simple ones: the aim is to solve the underlying defining compatibility equations of curve or surface classes by finding solutions to a simpler system of differential equations arising from the transforms. Classically, the main concern was a local theory. In modern theory,...
There is a long tradition in Differential Geometry of results that deduce topological consequences from pointwise positive curvature hypotheses. In this talk we consider the consequences of pinched curvature of various types and explain how recent developments in Ricci flow have proved to be decisive in establishing results along these lines.
Joint work with ManChun Lee (CUHK)
After a brief review of the classical Positive Mass Theorem and a short introduction to Q-curvature, I will present a theorem on the positive mass for Q-curvature and discuss some applications.
Over 30 years ago Frédéric Hélein proved that all harmonic maps from surfaces into compact Riemannian manifolds are smooth. Despite the existence of several partial results, for $n>2$ the counterpart of this theorem is wide open. In a recent work with two coauthors, Michał Miśkiewicz and Bogdan Petraszczuk, we prove regularity of $n$-harmonic maps into compact Riemannian manifolds and weak...
Hyperbolic systems in one-space dimension appear in various real-life applications (navigable rivers and irrigation channels, heat exchangers, chemical reactors, gas pipes, road traffic, chromatography, ...). This presentation will focus on the stabilization of these systems by means of boundary control. Stabilizing feedback laws will be constructed. This includes explicit feedback laws that...
The first part of the talk gives a review of the loop group method for harmonic maps from Riemann surfaces to (semisimple) symmetric spaces. Applications of this will be illustrated by pointing out the relation between certain types of harmonic maps and certain classes of surfaces. In the second part we will describe in some detail the discussion of Willmore surfaces in spheres along the...
This talk describes a (hypothetical) historical path that lead Batalin and Vilkovisky to the discovery of their (wonderful) approach, starting from the quantization of scalar fields, to the quantization of electrodynamics and non-Abelian gauge fields.
The questions raised here grew from the desire to give an integral representation for members of the dual of $BV$, the Banach space of functions of bounded variation. This potentially has application to the calculus of variations since $BV$ dual contains subgradients of energy functionals to be minimized. The question quickly links to that of identifying the dual of...