**
This is a set of on-line lectures dealing with classical General Relativity and Differential Geometry.
As of now the lectures are classified into topics. If you are interested in a topic which is not listed, please contact me
at alfonso AT math.uminho.pt and I will check if I can add a lecture about
the corresponding topic. Of course, I accept questions about the lectures posted here.
**

**
Penrose diagrams and global properties of the space-time**

In these lectures I introduce the notion of conformal boundary and conformal compactification. I explain in detail the conformal compactification of Minkowski, de Sitter and anti de Sitter solutions. I also introduce the notion of Penrose diagram and construct the Penrose diagram for a number of exact solutions of the Einstein field equations. To follow the lectures of this topic you are assumed to be familiar with the representation of the space-time as a Lorentzian manifold. You also need to know that observers in the space-time are represented by (inextendible) causal curves.

**Conformal compactification of the Minkoski solution**

- The Minkowski solution in standard Euclidean coordinates.
- The conformal embedding of the Minkowski space-time into the Einstein static universe.
- Conformal diagram of Minkowski space-time.
- The conformal boundary of Minkowski space-time and its different regions.
- The global causal properties of the Minkowski space-time.

**Conformal compactification of de Sitter and anti-de Sitter.**

- Maximally symmetric spaces in pseudo-Riemannian geometry.
- Maximally symmetric solutions of the Einstein equations: Minkowski, de Sitter and anti-de Sitter solutions.
- Embedding of de Sitter solution in five dimensional flat space-time.
- The conformal embedding of de Sitter solution into the Einstein static universe. Construction of the conformal boundary and study of the global properties of the de Sitter solution.
- Embedding of anti de Sitter solution in five dimensional flat space-time.
- The conformal embedding of anti-de Sitter solution into the Einstein static universe. Construction of the conformal boundary and study of the global properties of the anti-de Sitter solution.

**Penrose diagram of the Schwarzschild solution.**

- Spherically symmetric space-times in dimension four.
- The notion of Penrose diagram of a spherically symmetric space-time.
- Example: Penrose diagram of the Minkowski space-time.
- The Schwarzschild solution in Schwarzschild coordinates.
- Penrose diagram of the Schwarzschild solution and construction of the Kruskal extension.
- Global causal properties of the Schwarzschild solution.

**Penrose diagram of the Reissner Nordström solution.**

- The Reissner-Nordström solution and its classification:
*m>q*,*m=q*and*m< q*. - Penrose diagram of the case
*m>q*: maximal extension and global causal properties. - Penrose diagram of the case
*m=q*: maximal extension and global causal properties. - Penrose diagram of the case
*m< q*: maximal extension and global causal properties.

In this lecture I explain in detail the ΛCDM model and its implications to the global causal structure of the universe. I also compute the Penrose diagram of the universe using the material introduced in the lectures about Penrose diagrams. The lecture can be understood as a mini-course in Cosmology. Familiarity with elementary General Relativity is assumed.

Lecture contents:- The cosmological principle and the Weyl postulate. Scales of the known Universe.
- Construction of the FLRW space-time and its classification: open, flat and closed universes.
- The cosmological redshift and blueshift. Relation to the expansion and contraction of the universe.
- Matter-energy content of the universe. Review of geometrized units.
- The FLRW cosmological models and their dynamics. Cosmological equations.
- Cold dark matter hypothesis and its implications for the dynamics of the universe. Definitions
of Ω
_{Λ}, Ω_{M}, Ω_{R}and Ω_{k}. - The ΛCDM model.
- Construction of the Penrose diagram of the Universe and discussion of its global properties.

**
A brief course on probability & statistics **

Brief course taught at the undergraduate level. The course material can be downloaded here (in Spanish).

Course contents:- Introduction to probability. Sample space and definition of a probability space.
- Conditional probability. Bayes theorem.
- Random variable and probability distributions. Continuous and discrete probability distributions.
- The binomial distribution.
- The Poisson distribution.
- The hypergeometric distribution.
- The normal distribution
- The central limit theorem.
- Notion of sample statistic.
- Estimators.
- Confidence intervals.

**
A course about the xAct system for tensor analysis **

Course being taught at the graduate level. The course slides can be found here.

Course contents:- Package xTensor: coordinate-free tensor analysis.
- Package xCoba: tensor analysis in coordinates.
- Package xTerior: exterior calculus and its applications.
- Package Spinors: Penrose's spinor analysis in General Relativity.

On-line presentation about xTerior: