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.
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.
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.
Max Reinhold Jahnke, University of São Paulo, Brazil
We continue our discussion of cohomology of left invariant structures on compact Lie groups.
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
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.