Here is a link to my lectures on nonlinear dynamical systems given at York University during the Winter semester of 2017.
These lectures start off with manifold theory, and end with examples in biology, game theory, and general relativity/cosmology.
In the final two lectures of my differential equations class , I discussed how Dynamical Systems theory can be used to understand and describe the dynamics of cosmological solutions to Einstein’s field equations. Videos and lecture notes posted below:
New #cosmology paper: https://arxiv.org/pdf/1609.01310.pdf
Using a dynamical systems approach to provide a unifying framework for the AdS, Minkowski, and de Sitter universes. #physics #mathematics #science
I have a talk today at Perimeter Institute: here are the slides.
I basically showed that even a stochastic multiverse must be generated by precise initial conditions!
Most people when talking about cosmology typically talk about the universe in one context, that is, as a particular solution to the Einstein field equations. Part of my research in mathematical cosmology is to try to determine whether the present-day universe which we observe to be very close to spatially flat and homogeneous, and very close to isotropic could have emerged from a more general geometric state.
What is often not discussed adequately is the fact that not only has our universe emerged from special initial conditions, but the fact that these special initial conditions also must include the geometry of the early universe, and the type of matter in the early universe. Below, I have attached a simulation that shows how the early universe can evolve to different possible states depending on the type of physical matter parametrized by an equation of state parameter . In particular, some examples are:
- : Vacuum energy
- : Radiation
- : Stiff Fluid
Note: Click the image below to access the simulation!
In these simulations, we present phase plots of solutions to the Einstein field equations for spatially homogeneous and isotropic flat, hyperbolic, and closed universe geometries. The different points are:
- dS: de Sitter universe – Inflationary epoch
- M: Milne universe
- F: spatially flat FLRW universe – our present-day universe
- E: Einstein static universe
Note how by changing the value of , the dynamics lead to different possible future states. Dynamical systems people will recognize the problem at hand requires one to determine for which values of is F a saddle or stable node.
I tried to derive a general Einstein field equation for an arbitrary FLRW cosmology. That is, one that can handle any of the possible spatial curvatures: hyperbolic, spherical, or flat. Deriving the equation was easy, solving it was not! It ends up being a nonlinear, second-order ODE, with singularities at a=0, which turns out to be the Big Bang singularity, which obviously is of physical significance. Anyways, here’s a log of my notebook, showing the attempts. More to follow!
My new paper has now been published in Annalen der Physik, which is a great honour, because 100 years ago, Einstein’s General Theory of Relativity was also published in the same journal.
This paper describes a method by which one is able to determine whether a given spatially flat cosmological model produces finite-time singularities, and also gives some examples of interesting cosmological model configurations.
The paper can be accessed by clicking the image below:
The preprint can be accessed here on the arXiv.