# Maths

I spent 10 years of my life (2007-2017) first studying and then teaching and doing research in mathematics in the Mathematics Department of Universidad Autónoma de Madrid. Nowadays I only do any research in mathematics very rarely, but I’m quite fond of this period of my life. Here you can find some of my work from those years.

###### PhD thesis

K-spectral sets, operator theory and related function theory, February 2017

###### MSc thesis

This includes a self-contained introduction to the Sz.-Nagy–Foias theory of dilations and Ando’s theorem about the dilation of pairs of commuting contractions. I am particularly happy about the way that the analytic model for $$C_{\cdot 0}$$ contractions is written, and I think it provides a good alternative exposition to the Sz.-Nagy–Foias book. Perhaps this can be useful to postgraduate students interested in these topics.

MSc thesis defense slides (in Spanish).

###### Teaching Statement

Clearly, my favourite activity during my PhD studies was teaching. I enjoyed a lot helping the students and finding unconventional ways of explaining things and getting the students engaged. Life has now taken me through other paths and I don’t teach maths any longer (except for the odd exceptional private class). Now I’m well busy with other things that are perhaps more intelectually challenging, but probably not so emotionally rewarding as teaching. Certainly I miss teaching, but who knows what will happen in the future.

As an example of my style of teaching (and for some anecdotes), you can read my Teaching Statement, which I wrote in Spring 2017.

(If any of my former students finds themselves reading this, please drop me a line. I would love to know how are you doing.)

###### Miscellaneous works

The Hodge decomposition theorem, January 2013. A self-contained proof of the Hodge decomposition theorem. This was done for an MSc course in differential geometry given by Gabino González.

Interpolación en espacios de Hardy, December 2011 (in Spanish). A self-contained proof of the Shapiro-Shields theorem about weighted interpolation in $$H^p(\mathbb{D})$$, with an application to the study of eigenvectors of the model operator $$T = P_KM_z|K$$, where $$K = H^2 \ominus BH^2$$. This was done for an undergraduate course in complex analysis given by Ana Vargas.

###### Trivia

Erdös number: 3 (co-authored with M. Dritschel, who co-authored with J.M. Anderson, who co-authored with Erdös)

Mathematics genealogy