LOGOS Seminar

In this weeks seminar, I will talk about the notion of strong differentiation (defined by Peano), which generalizes the notion of Fréchet differentiable functions and thus also the classical notion of differentiability of functions in IR^n. As the name suggests, strong differentiability implies (Fréchet -) differentiability. The interesting aspect of strong differentiation is that it enables us to "improve" (- meaning simpler formulations, or weaker assumptions or more elegant proofs) some of the key-theorems in Analysis like the Inverse function theorem, the Implicit function theorem, or theorems about the equality of mixed partial derivatives under certain conditions. Mathematicians who have encountered this rather unknown notion of strong differentiation emphazize its usefulness, some of them even suggest to teach strong differentiation instead of classical differentiation at university courses! I aim to present Peano's notion of strong differentiation in a very digestible way by starting (of course) with the basic definitions and results, then returning to functions on IR to get a better intuition of this notion, and then present some of the above mentioned key-theorems using strong differentiation and compare the results to their related classical versions.

I will talk about a recent review paper by Kontou and Sanders about Energy conditions in general relativity and quantum field theory. I will give a very brief introduction to relativity and then introduce the various energy conditions discussed by Kontou and Sanders and their application to singularity theorems.

I will introduce the relevant notions of causality theory and prove that non-compact spacetimes with compact causal diamonds and dimension at least are automatically globally hyperbolic, as well as a similar statement for causal simplicity.

First, I will motivate the topic by considering the -spaces as concrete examples. Then I will formulate and proof a theorem by Cohen (1959) which, applied to $L^1$ yields the factorization $L^1 = L^1 * L^1$. Finally, I will discuss a further result which gives rise to a factorization of $L^p$ into $L^1*L^p$

After introducing the basics of the theory of strongly continuous semigroups on real Banach spaces, I will prove in detail the Theorem of Hille-Yosida for $\omega$-contractive semigroups. In short, this theorem classifies when a time evolution problem of the form $u_t=Au, u(0)=u_0$, where $u$ is Banach-space-valued, can be solved by constructing its solution from a semigroup generated by $A$ . Finally, I will sketch how to apply this result to simple classes of second-order hyperbolic and parabolic partial differential equations.

I will motivate the setting in which the splitting problem in Riemannian Geometry arises, then I will go on to prove the Riemannian Splitting Theorem in full detail. At the end, I will state the Lorentzian Splitting Theorem and outline (briefly) the main ideas that go into the proof. In particular, I will give an outlook on how the techniques used in the Riemannian case can be adapted to the Lorentzian setting.

Since the ring of Robinson-Colombeau is non-Archimedean, a classical series $\sum_{n=0}^{+\infty}a_n$ of generalized numbers is convergent if and only if $a_n$ converges to $0$ in the sharp topology. Therefore, this property does not permit us to generalize several classical results. Introducing the notion of hyper series, we solve this problem by recovering classical examples of analytic functions as well as several classical results.