Why I Want to Pursue a Master's Degree in Geometry

There is an undeniable beauty to pure mathematics. Now, after several years of working with deep abstractions, my dream is to go back to the real world and make use of all the theoretical tools that I've acquired, and contribute to further developments in different fields. Last year I attended a week-long event on applied mathematics. It was a life-changing experience for me. For the first time, I could see how beautiful is the role played by mathematics in machine learning, data science, and in many other different areas. I have always been interested in computer science, but since that day, the possibility of working at the intersection of this area and mathematics became my career goal.

Since I was a kid, science has driven my interests. I grew up in a small city in the countryside of Brazil. My parents could not afford a good-quality school for me. At that time, attending college seemed to be almost impossible. During high-school, I split my efforts between part-time jobs, to help my mom pay the bills, and preparation for the college entrance exams. Despite all the difficulties, I was accepted to study mathematics at the State University of Campinas (UNICAMP), one of the most prestigious universities in Latin America. 

At the University, I faced a very challenging environment. My previous education had left many gaps to be filled, making my early years at the University quite difficult. During my undergraduate studies, I took many extracurricular classes, most of them in mathematics but also physics. Also, I took eight post-graduate classes in mathematics, including Analysis, Topology, Abstract Algebra, and Differential Geometry. At the end of my second year, I received a scholarship to spend my summer vacation at IMPA (Institute of Pure and Applied Mathematics) and took classes in linear algebra. Later, I also received two scholarships to develop undergraduate research under the supervision of Professor Dr. Lino Grama. In the first scholarship project, I studied the geometry of the adjoint orbits within the Special Linear group. The results of this project became my first paper, which will be published soon. In the second one, I was introduced to Symplectic Geometry by studying Arnold's conjecture on fixed points of symplectomorphisms. Both of these projects inspired me to pursue a Master's degree in geometry. 

At the beginning of my master's degree, I received a scholarship from FAPESP, to develop a Master's research project together with Prof. Dr. Lino Grama on Higgs Bundles. Since I have completed all the required credits of the Master's degree program during my bachelor's, I could dedicate myself mostly to doctoral level courses and also to my research. At the beginning of the second year, I received the scholarship Research Internships Abroad from FAPESP, and then, in September, I came to the University of Miami to work for six months under the supervision of the Prof. Dr. Katzarkov. Here, I am working mostly on Mirror Symmetry, an active area of research bridging Physics and Mathematics. Since working in this area, I am getting familiar with advanced topics in Algebraic Geometry, Category Theory, and Symplectic Geometry. 

My experience here at the University of Miami has been amazing. I have the chance to get in touch with cutting edge research and also meet renowned mathematicians from all around the world who have shaped and are today shaping the landscape of Mirror Symmetry. I have also met other graduate students from different backgrounds who have enriched my learning experiences here. These reasons helped me decide to pursue my Ph.D. here in the US. Sadly, being involved in my research internship, I did not have enough time to prepare myself properly for the GREs. As a consequence, my grades did not truly reflect my knowledge and dedication. So, I hope these grades do not obscure my achievements presented throughout this application. Berkeley is a natural choice for me. Besides all the remarkable researchers that I will find there, the applied mathematics Ph.D. program is perfectly congruent with my interests. The vast number of courses in computer science offered by the program is exactly what I was looking for to complement my earlier education. Prof. Dr. James Demmel and the BeBOP group in large-scale systems optimization has piqued my interest. 

To illustrate how relevant this area is for our era, let's consider a brief example. Traffic is a problem in most of the big cities around the world. The emergence of the internet and smartphones is giving us the chance to find a solution to this problem through something we have never had in such abundance: data. Self-driving and electrical vehicles will likely drive the next transportation revolution. If we were able to use this data to optimize traffic, in a few decades congestion might not be a problem anymore. However, even a simpler issue - the routing and scheduling problem - is well-known to be NP-hard. A typical routing software, therefore, has to resort to simple optimization heuristics such as greedy route construction or local improvement. This raises the more general question of the computational tradeoffs of large-scale optimization. 

The availability of massive data sets makes it possible to define new real-world applications that are larger than what state-of-the-art optimization algorithms and solvers can handle. As this new science has been showing us wonderful possibilities, we are realizing that our well-established tools are not enough to make them a reality. There is a lot to be discovered about the mathematical and computational nature of these problems. And I, as a mathematician, want to make these discoveries a reality.