Global analysis seminar
Current contact: Gerardo Mendoza
The seminar takes place Wednesdays 11:40 - 1:00 pm in Wachman 527. Click on title for abstract.
Max Reinhold Jahnke, University of São Paulo, Brazil
First, in order to understand the statement of a theorem by Bott we will see a brief exposition of the theory of cohomology of Lie algebras. As an application, we will see how to use it to prove that the study of the Dolbeault cohomology of left-invariant complex structures on semisimple compact Lie groups can be reduced to the study a purely algebraic problem: the study of Dolbeault cohomology of complex structures on semisimple compact Lie algebras. This approach was first used by Pittie.
Max Reinhold Jahnke, University of São Paulo, Brazil
We are going to continue our discussion about Lie algebra cohomology and its relation to classic cohomology theories. We are going to discuss a result by Chevalley and Eilenberg on the De Rham cohomology of compact Lie groups, its relation to Bott's theorem and how to combine them to study certain left-invariant elliptic involutive structures on semisimple compact Lie groups.
Max Reinhold Jahnke, University of São Paulo, Brazil
We continue our discussion of cohomology of left invariant structures on compact Lie groups.
Narek Hovsepyan, Temple University
We are going to discuss some techniques and ideas used by H. Widom${}^1$ for finding the asymptotic behavior of the eigenvalues of certain integral operators, such as $Tf(x) = \int_{-1}^1 k(x-y) f(y) \,\mathrm{d}y$, under the assumptions that $\widehat{k}$ is even, positive and decays at infinity. In particular, we will study the case when $-\ln \widehat{k}(\xi)$ has growth proportional to that of $\xi$, at infinity, in detail, and will touch upon the remaining two cases of the growth (i.e. slower or faster than $\xi$) of $-\ln \widehat{k}(\xi)$.
${}^1$Widom, H., Asymptotic behavior of the eigenvalues of certain integral equations II, Arch. Rat. Mech. Anal. 17 (1964) 215--229.
Luis Fernando Ragognette, Federal University of São Carlos, Brazil
We are going to recall the definition of Gevrey local solvability for a differential complex associated to a locally integrable structure and then we are going to give a necessary condition in terms of an a priori estimate. This kind of estimate was introduced by Hörmander and became a standard technique to study solvability.
Luis Fernando Ragognette, Federal University of São Carlos, Brazil
We are going to recall the definition of Gevrey local solvability for a differential complex associated to a locally integrable structure and then we are going to give a necessary condition in terms of an a priori estimate. This kind of estimate was introduced by Hörmander and became a standard technique to study solvability.
Max Reinhold Jahnke, University of São Paulo, Brazil
First we are going to apply a result by Jacobowitz to prove that every left-invariant elliptic structure on compact Lie groups dominates a left-invariant CR structure and then we will show how this information can be used to study the cohomology of the elliptic structure.
No meeting
Gerardo Mendoza, Temple University
I’ll give an overview of results concerning several aspects of the analysis of elliptic operators on compact stratified manifolds with a single stratum. In the realm of conical singularities I’ll discuss the general asymptotics of the resolvent of an elliptic operator assuming only existence of rays of minimal for the principal symbols, and recent results on the nature of domains of elliptic complexes. In connection with boundary value problems for higher dimensional strata, I’ll describe the bundle of traces (the bundle of Cauchy data), some aspects of the analysis on these, and a result on the domain of the Friedrichs extension (from its minimal domain) of an elliptic semi-bounded second order operator. The results to be presented were obtained in collaboration with Juan Gil or Thomas Krainer or both.
Gerardo Mendoza, Temple University
This is a continuation of the talk of Nov 15. I’ll give an overview of results concerning several aspects of the analysis of elliptic operators on compact stratified manifolds with a single stratum. In the realm of conical singularities I’ll discuss the general asymptotics of the resolvent of an elliptic operator assuming only existence of rays of minimal for the principal symbols, and recent results on the nature of domains of elliptic complexes. In connection with boundary value problems for higher dimensional strata, I’ll describe the bundle of traces (the bundle of Cauchy data), some aspects of the analysis on these, and a result on the domain of the Friedrichs extension (from its minimal domain) of an elliptic semi-bounded second order operator. The results to be presented were obtained in collaboration with Juan Gil or Thomas Krainer or both.
Howard Jacobowitz, Rutgeras University
The local embedding (or realization) problem for a strictly pseudoconvex CR manifold $M^{2n+1}$ is quite challenging and, in fact, remains open for $n=2$. However, the problem is much simpler when the manifold is compact. This was observed by Boutet de Monvel in 1974. The talk will be an exposition of Boutet's paper.