Differential Geometry and Applications
Organized by Yirmeyahu Kaminski and Serge Lukasiewiecz,
Department of Mathematics, Holon Institute of Technology
Differential geometry in all its aspects, which include differential topology, Riemannian manifolds and symplectic geometry, is by itself a very central and important mathematical subject. It also has numerous applications in various domains like physics, statistics (information geometry), data science, computer vision (shape spaces), control theory and others.
Therefore our seminar which is aimed at covering all these aspects is entitled "Differential Geometry and Applications".
Frédéric Hélein: Dynamical mechanisms for Kaluza-Klein theories, Thursday November 24, 2022 at 3pm (Israel time, GTM + 2).
I will present a theory based on a variational principle leading to solutions of Einstein-Maxwell or, more generally, Einstein-Yang-Mills systems in the spirit of Kaluza-Klein theories. In these theories, no fibration is assumed: the fields are defined on a spacetime Y of dimension 4+r without a priori structure, where r is the dimension of the structure group. If the latter is compact and simply connected, the classical solutions allow the construction of a manifold X of dimension 4 which can be interpreted as the physical space-time, so that Y acquires the structure of a principal bundle over X, and provide solutions of the Einstein-Yang-Mills systems. I will essentially detail the (degenerate but simpler) case leading to the Einstein-Maxwell system.
Jake Solomon: The cylindrical transform, Thursday December 22, 2022 at 2 pm (Israel Time, GTM+2)
The space of positive Lagrangian submanifolds admits a Riemannian metric of non-positive curvature. Understanding the geodesics of the space of positive Lagrangian submanifolds would shed light on questions ranging from the uniqueness and existence of volume minimizing Lagrangian submanifolds to Arnold's nearby Lagrangian conjecture. The geodesic equation is a nonlinear degenerate elliptic PDE. I will describe work with A. Yuval on the cylindrical transform, which converts the geodesic equation to a family of non-degenerate elliptic boundary value problems. As a result, we obtain large families of geodesics of positive Lagrangians of arbitrary dimension. The talk will be aimed at a broad audience.
Marina Ville: Minimal surfaces in R4 , Thursday January 26, 2023 at 2 pm (Israel Time, GTM+2)
Minimal surfaces minimize the area w.r.t. deformations with small support. In R3 basic exemples are given by the plane, the helicod and catenoid; and the topic is fairly well understood with many more examples and existence or non existence theorems. Much less is known about R4: the algebraic curves in C2 are all minimal surfaces in R4 but we do not have many explicit non holomorphic examples of minimal surfaces, especially embedded ones. I will describe the search for such examples and some tools we use from topology and analysis. The talk will be as self-contained as possible.
Eric Leitchnam: Deninger’s programme: a motivation for developing new mathematics, Thursday March 23, 2023 at 2 pm (Israel Time, GTM+2)
Deninger started in the 90’s to develop an infinite dimensional cohomological formalism in order to explain the expected conjectures for the arithmetic zeta functions. He conjectured that these (infinite dimensional) cohomology groups should be constructed as leafwise cohomology groups on suitable foliated spaces. These foliated spaces are not known to exist and maybe they do not exist at all. But it turns out that, having in mind this motivation, one can develop interesting new mathematics in differential geometry (i.e. outside number theory). We shall review in some details the history and the basic ideas underlying the origin of Deninger’s programme and at the end describe briefly new results, motivated by it, in particular by AlvarezLopez-Kordyukov-L in differential geometry.
Jean Levine: Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions, Thursday April 27, 2023 at 2 pm (Israel Time, GTM+2)
In this talk, we consider a general finite dimensional control system, i.e. a smooth $n$-dimensional manifold and a familiy of vector fields on this manifold, depending on $m < n$ control inputs.
Different control systems being susceptible of producing the same integral curves in suitable local coordinates and with suitable controls, various equivalence relations have been introduced in the past: equivalence by diffeomorphism and feedback or the coarser one, called Lie-Bäcklund equivalence. The equivalence class of linear controllable systems, called “static feedback linearizable” for the former equivalence relation, and “differentially flat” for the latter, are of practical importance for the design of motion planning or trajectory tracking by feedback.
The class of static feedback linearizable systems has been fully characterized, in Lie algebraic terms, in 1983 by B. Jakubczyk and W. Respondek. The class of differentially flat systems has been thoroughly studied by M. Fliess, Ph. Martin, P. Rouchon and the author since 1991, and, with the viewpoint of “dynamic feedback linearization”, by B. Charlet, R.Marino and the author in the late 1980’s, but a computationally tractable characterization still remains to be found.
Here, we introduce the notion of “differential flatness by pure prolongation”: loosely speaking, a system admits this property if, and only if, there exists a pure prolongation of finite order such that the prolonged system is feedback linearizable.
We obtain Lie-algebraic necessary and sufficient conditions for a general nonlinear multi-input system to satisfy this property.
These conditions are comprised of the involutivity and relative invariance of a pair of filtrations of distributions of vector fields. An algorithm computing the minimal prolongation lengths of the input channels that achieve the system linearization, yielding the associated flat outputs, is deduced. Examples that show the efficiency and computational tractability of the approach will be presented.
Leonid Polterovich: Topological Persistance in Geometry and Analysis, Thursday May 18, 2023 at 2 pm (Israel Time, GTM+2)
Persistence modules and barcodes is an emerging field of algebraic
topology which originated in data analysis. I will discuss its applications to
function theory and to symplectic geometry.
Joint with Lev Buhovsky, Jordan Payette, Iosif Polterovich, Egor Shelukhin, and
Vukasin Stojisavljevic.