LOGOS Seminar

In this talk, we shall discuss the Riemannian and Lorentzian Splitting Theorems and the ideas that go into their proofs. The content of these theorems (roughly) is that a given complete Riemannian manifold or spacetime splits isometrically as a product if it satisfies a certain Ricci curvature condition and contains a complete geodesic that measures distances/time differences accurately everywhere. We shall highlight the similarities, and the differences/complications which arise in the Lorentzian case due to the lack of ellipticity of the Laplace operator.

We prove warped product spacetimes $(0,\infty)\times \bar{M}$ with complete $\bar{M}$, radius bounded below near $t=\infty$ which exhibit past particle horizons at $t=0$ or have infinite radius at $t=0$ are $C^{0,1}$-inextensible using a bound on parallel transport.

We complete the last talk. Then, we prove that Reissner-Nordström-Vaidya spacetimes and Luk-Oh spacetimes are $C^{0,1}$-inextensible using a bound on parallel transport.

We show a straight forward way to determine the explicit form of transformations between inertial frames in special relativity. We only require the transformations to respect the principle of relativity and that they form a group, not that the speed of light is an invariant which is postulated in common derivations.

We will see the PLT (1st, simplified form) for PDE in normal (or Kowalevskian) form. The core result is the Banach fixed point theorem with loss of derivatives in a graded Fréchet space. The theorem can be applied to a class of smooth Cauchy problems which is strictly larger than analytic functions and yields a local existence in "t" and for any "x" in a fixed arbitrary compact set. The simplified form can be of any order in "t", but it cannot contain partial derivatives in the same variable. It applies to at least 23 out of 32 examples of PDE from the book "PDE" by L.C. Evans. It is hence a general existence theorem. If times permits, we present the ideas about how to obtain the 1st general form, without the aforementioned restriction on time derivatives (this would include other 5 examples) and how to enlarge the class of solvable problems to an even larger class of smooth functions (PLT, 2nd form) using Nash-Moser smoothing operators (even if Lewy-Mizohata examples are always in play).
This is a joint work in progress with L. Luperi Baglini (University of Milan) and it's only at the first level of checking (over six), this DIANA seminar being the second one. The real difficulty in this type of attempts, is that one has to psychologically struggle with strong implicit axioms such as: "There is no general theory concerning solutions of PDE", "PDE are deeply different from ODE", "Lewy's counterexample shows that the analogue of the CKT does not hold in the smooth category", "almost always a PDE is not Lipschitz", etc. We will always face these restraining “dogmas” by saying: "Wait... I simply want to understand better whether certain ideas work or not, and if not why".

I will first talk about some elements of statistical mechanics to get a general understanding of what the Boltzmann equation and the Fokker-Planck equation, which is derived from it, are about. Then, I will move to explaining entropy methods for diffusive PDEs and how they can be used to study the long-time behaviour of the Fokker-Planck equation.

The talk will present an example of a causally simple space-time, such that its universal covering is not causally simple as presented in a recent paper by Minguzzi and Costa es Silva "A note on causality conditions on covering spacetimes". Further leading up to this example, we will give an account of some causality properties of space-times related to the topic and a very short overview of the causal ladder.

This talk will cover basics of Gefland-Shilov spaces. We will go through the construction of those spaces as they were originally depicted in "Generalized Functions" by Gelfand and Shilov as spaces of type S.