CAMGSD Seminars

Vishnu Jejjala, 2019/05/20, 15h, Deep Learning the Hyperbolic Volume of a Knot

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, Vol($K$). A less studied question is whether Vol($K$) can be recovered directly from the original Jones polynomial ($N=1$). In this report we use a deep neural network to approximate Vol($K$) from the Jones polynomial.

Juan Antonio Valiente Kroon, 2019/02/13, 11h, Construction of anti de Sitter-like spacetimes using the metric conformal field equations

In this talk I with describe how to make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function.

Bruno Oliveira, 2019/02/26, 15h, Big jet-bundles on resolution of orbifold surfaces of general type.

The presence of symmetric and more generally $k$-jet differentials on surfaces $X$ of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from $\mathbb{C}$ to $X$). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio $c_1^2/c_2$ a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or $k$-jet differentials.

Pedro Girão, 2019/02/22, 11h 30m, Solutions of the wave equation bounded at the Big Bang

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(\Sigma)$ (where $\Sigma=\mathbb{R}^n$, $\mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $\phi$ of the wave equation converging to $A$ in $H^1(\Sigma)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma)$.

Jarrod Williams, 2019/03/21, 14h 30m, The Friedrich-Butscher method for the construction of initial data in General Relativity

The construction of initial data for the Cauchy problem in General Relativity is an interesting problem from both the mathematical and physical points of view. As such, there have been numerous methods studied in the literature the "Conformal Method" of Lichnerowicz-Choquet-Bruhat-York and the "gluing" method of Corvino-Schoen being perhaps the best-explored. In this talk I will describe an alternative, perturbative, approach proposed by A. Butscher and H.

Anne Franzen, 2019/01/30, 11h, Flat FLRW and Kasner Big Bang singularities analyzed on the level of scalar waves

We consider the wave equation, $\square_g\psi=0$, in fixed flat Friedmann-Lemaitre-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface $\{t=0\}$.

Carlos Herdeiro, 2019/04/10, 15h, Light ring stability in ultra-compact objects

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities.

Federico Sau, 2019/02/12, 15h, Self-duality for conservative interacting particle systems

In this talk, we will sketch some recent developments about the notion of duality for conservative interacting particle systems. In particular, we will show the simplification that arises in presence of self-duality when considering hydrodynamic limits in a dynamic disorder (joint work with F. Redig and E. Saada). We will find all particle systems which admit a special form of self-duality (joint work with F. Redig) and, in conclusion, we will use the spectral point of view of this notion to address some open questions.