# Global Analysis Seminar

2017

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.

• Wednesday October 18, 2017 at 11:40, Wachman 527

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.

• Wednesday October 11, 2017 at 11:40, Wachman 527

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.

• Wednesday October 4, 2017 at 11:40, Wachman 527
Lie algebra cohomology and left-invariant involutive structures on 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.

• Wednesday September 27, 2017 at 11:40, Wachman 527

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.

• Wednesday September 20, 2017 at 11:40, Wachman 527

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.

2017