I am applying to the mathematics PhD program at the University of California, Berkeley. I am an undergraduate mathematics major, and I have taken many advanced and graduate level courses; I have done uniformly well in all of them, and nearly always at the top of the class. Overall I have maintained a subject GPA of 7.81/8 and am the top mathematics student of my batch. Throughout my studies I have tried to explore as many areas of mathematics as possible to find what I enjoy most; based on what I have learned, it is clear to me that my interests lie in analysis, particularly in probability. My aim is to eventually be in a position where I understand both of these areas and their interplay deeply, and to be able to contribute to and share …show more content…
Niranjan Balachandran. I read about the basics of graph theory and extremal set theory, and then studied the proof of the complete intersection theorem (a generalization of the Erd\H{o}s-Ko-Rado theorem). I also worked on a conjecture in extremal graph theory about the minimum possible size of the largest cycle in $k$-cyclable graphs (that it equals $2k$). While we did not prove it, we did make headway in a few specific cases. (Project write-ups are available at \href{http://milindhegde.github.io}{milindhegde.github.io}.) This project was my first exposure to mathematics beyond the undergraduate level. I found it challenging to complete the details of the proof of the complete intersection theorem and to come up with my own arguments for the conjecture. Ultimately, however, I found the fruits of the effort enjoyable and I learned a great deal. I gained many new skills, including reading mathematical papers, considering problems which may not have solutions, and finding ways to present arguments clearly in writing. It was my first glimpse of what research may be …show more content…
Sandeep Varma. This is a selective program for which only 20 undergraduates across India are chosen. I learned about the original motivation for ideals and the recovery of unique prime factorization in Dedekind domains, which was very satisfying. I also studied the finiteness of the class number and Dirichlet's unit theorem, and gave a short talk on the sign of the Gauss sum. I have also given several other talks as part of courses; details are in my