My Fulbright grant began in 2014 when I left for Princeton to study a PhD in Mathematics. So far my experience has been great. My first year was mostly theoretical: I had to take a preliminary exam, which consisted of an oral exam with three subjects, students chose these at the beginning of the year. My three subjects were focused on Partial Differential Equations, Numerical Analysis and Probability, and Statistics. The exam was extremely tough which I expected as I had chosen the best professors possible for each part, something that I could have only done at Princeton. During the first year, I also had the opportunity to participate in several projects with professors from a a variety of areas, from astrophysics and neuroscience to fluid dynamics. This is an extraordinary experience that was possible since I was studying in the US, the Spanish system is very different.

In the summer of 2016, I started working with Professor Sergiu Klainerman. This professor is one of the most outstanding researchers in Partial Differential Equations, a field that studies how different complicated systems in nature work. He is particularly an expert in Mathematical General Relativity, the field which studies Einstein’s Equation Theory as an attempt to establish rigorous and fundamental results about how our universe works. The project was related to my interests: Developing numerical algorithms to simulate the formation of a Black Hole. However, I soon (well, after 3 months) realized that very little is understood about how numerical algorithms work in general, thus I had to go deeper into the theory of Numerical Analysis.

This realization led me to begin working in November with Professor Charles L. Fefferman, my current advisor. Professor Fefferman, a Fields medalist and also a great person, works in theoretical results for interpolation and reconstruction of functions and shapes, among many other things (fluids, quantum mechanics, geometry…). I have started working on several projects with him. This work we have begun could further our understanding of what a computer (which is finite in its nature) can predict about smooth functions (which are essentially infinite). Furthermore, some of these projects have immediate applications in the fields of optimization, computer vision and the study of graphene. All of these problems are extremely complicated, but I am certain that, together with my advisor, we will be able to achieve in-depth and successful results and hopefully gain a better understanding into the relationship between what we can compute and what already exists.

In closing, I am very fortunate to be able to study a PhD at Princeton University. In addition, carrying this out with a Fulbright-Telefonica scholarship has given me the freedom in my research that I wouldn´t have had otherwise. I cannot express how grateful I am for this opportunity. I believe that the best way to express my gratitude is to make the most of it, which is what I have been doing and what I plan to do for the rest of my time here.

**Bernat Guillen Pegueroles**

**2014-16 Telefónica/Fulbright **

**Princeton University **