A joint online scientific meeting between the American University of Beirut, Lebanese University and University of Potsdam. This meeting, organized in the framework of the Research Group Linkage sponsored by the Humboldt Foundation, brings together mathematicians from Lebanon and elsewhere to present their research and exchange ideas around spectral, geometric and complex analysis, highlighting how geometry, spectral and complex analysis fruitfully intertwine.
Abstract: The harmonic map equation is a second order semilinear elliptic PDE for maps between Riemannian manifolds. Many results on questions of existence and qualitative behavior of solutions have been obtained in the literature. After providing a pedagogical introduction to the field of harmonic maps we will introduce the notion of k-harmonic maps which are characterized by a semilinear elliptic PDE of order 2k. We will discuss various recent results on these maps including unique continuation properties and the question under which conditions k-harmonic maps must be harmonic. Finally, we will focus on a second higher order version of harmonic maps initially proposed by Eells and Sampson in 1964 and present several recent results on the latter.
This is joint work with Stefano Montaldo, Cezar Oniciuc and Andrea Ratto.
Abstract: In the study of the geometry of domains and real submanifolds of complex spaces, it is important to identify and explore suitable biholomorphic invariants. Some of these can be given as invariant metrics defined on the interior of the domain (such as the Kobayashi metric) while others can be derived from the geometry of the boundary, coming in the form of tensors or, more indirectly, of families of curves related to these tensors (an important example of the latter are the Chern-Moser chains). In some situations, important insights can be gained by comparing invariants of different types. In joint work with F. Bertrand and B. Lamel, we consider the relation between the extremal discs for the Kobayashi metric and other invariant families of curves defined on strongly pseudoconvex hypersurfaces, with particular regard to the characterization of local sphericity.
Break : 11h-14h30/ 12h-15h30
Abstract: The study of stationary probability probability measures on homogeneous spaces have been the subject of many research works in the last decade, especially after the seminal works of Benoist--Quint and then Eskin--Linderstrauss. The goal of the talk is present new results on stationary probability measures on projective spaces with some applications to algebraic homogeneous dynamics. This is a joint work with C. Sert.
15h20-16h / 16h20-17h Fida El Chami (UL): Eigenvalue estimates of differential operators on manifolds with boundary
Abstract: We study the spectrum of some differential operators that appear naturally on manifolds with boundary which are closely related to the Laplacian. We establish comparison results for the smallest eigenvalue of some of these operators and give estimates that depend on geometric quantities.
Break: 16h10-16h20/ 17h10-17h20
Abstract: Motivated by the locality principle in physics, we consider locality in terms of binary relations and partial operations in algebraic structures. With applications to regularizations and renormalizations in mind, special attention is given to locality for meromorphic germs with linear poles. Classes of such germs are shown to be polynomial algebras in the locality context, leading to a transitive action of a Galois type group on the generalized evaluators defined on these germs. This is a joint work with Sylvie Paycha and Bin Zhang.
Abstract: In 1887, G. Koenigs gave a full description of the point spectrum of composition operators on Hol(D) associated with a symbol f with a fixed point in D. We will describe the spectrum of such operators for every symbols and then we will give the spectral properties for the weighted composition operators associated with a symbol with an interior fixed point.