# Notes

Here I will list the reading groups or seminars in which I will be participating/organising. I will also occasionally upload some notes taken at each of the seminars. My notes can (and will usually) be a bit sloppy, and I declare myself responsible for any mistakes in them.

**Quantum theory for mathematicians:** here I will post (handwritten) notes from our reading group (co-organized with Gastón Burull). The discussions are lead by myself and the notes taken by Gastón.

Description | Notes |
---|---|

In preparation… |

**Notes on dynamical systems and ergodic theory:** here I will post some notes on topics related to my research that I have written.

Description | Notes |
---|---|

The ergodic theorem | here |

Statistical laws in Dynamical Systems | here |

**Notes on probability theory and statistics:** here I will post notes on a variety of topics related to probability theory which are of my interest in one way or another. They have an expository character while keeping the formality.

Description | Notes |
---|---|

Is overfitting… good? | here |

PCA and supervised learning | here |

Martingales 0 | here |

Extreme value theory III | here |

Empirical error | here |

Extreme value theory II | here |

Confidence intervals | here |

The law of Anomalous numbers | here |

Understanding bias and variance | here |

Extreme value theory I | here |

Large deviations | here |

Central limit Theorem | here |

Quantitative Borel-Cantelli lemma | here |

Law of large numbers, part 1 | here |

Borel-Cantelli lemma | here |

Weak law of large numbers | here |

**Problems in data science:** here I will post a number of problems and solutions, related to the discipline of data science. This includes statistics, probability, algorithms, among others.

- Problem 5: Unbiased and consistent esimators
- Problem 4: producing normal vectors
- Problem 3: rearrange of lists
- Problem 2: expectation of minimum of uniform rv
- Problem 1: random unfair coin

**Statistical properties in hyperbolic dynamics:** course from the Houston Summer School on Dynamical Systems 2019 by the Department of Mathematics of the University of Houston. For more information, see here.

Date | Speaker | Description | Notes |
---|---|---|---|

03/06 | Matt Nicol | Limit laws in Dynamical systems | here |

04/06 | Matt Nicol | Martingale approximation | here |

05/06 | Andrew Török | Quasi-compactness | here |

06/06 | Will Ott | Coupling | here |

**Fractal Weyl laws. This reading group was held on the first term of year 2016-2017:**

Date | Speaker | Description | Notes |
---|---|---|---|

17/10 | Jimmy Tseng | Symbolic spaces, Hausdorff dimension, Thermodynamic formalism | here |

24/10 | Sebastian Muller | Non-fractal Weyl laws | here |

31/10 | Felipe Pérez | Thermodynamic formalism | here |

07/11 | Felipe Pérez | Iterated functions systems, transfer operator | here |

14/11 | Thomas Jordan | Julia sets, determinant formulas | here |

21/11 | Felipe Pérez | Zeta functions, zeroes counting | here |

28/11 | Sebastian Muller | Non-fractal Weyl laws | Lost |

**M.I. Gordin, On the central limit for stationary processes:** this is a *very* rough translation of the paper by Gordin, where he introduced the martingale approximation method to derive statistical results in dynamical systems. Russian original/English translation.

**Bowen’s formula and transfer operators:** here. In these notes I present a sketch of the proof of the Bowen’s formula for the Hausdorff dimension of the repeller of an IFS, as well as introducing notions of thermodynamic formalism.