SOP sample for Mathematics

This application is very important to me because completion of a PhD degree in Mathematics will be the best chance for a unifying career development with my labor of love - mathematics. I am confident that when I become a serious and mature researcher my desire to make a contribution to mathematics and to our understanding of this world will remain my main driving force.

I don't remember when I became keen on mathematics. At first it was only entrainment as I found it exciting to solve intricate problems. Later it became something more than a mere hobby. I particularly enjoyed the ineffable feeling of triumph when you realize that the problem is solved; that you have got the idea. I think it is the profundity of this feeling that made mathematics my chief enthusiasm. I have taken part in Olympiads and Conferences. The most valuable contribution of these competitions was a possibility to meet the same enthusiasts in mathematics. I was particularly impressed when I was invited to participate in an International Summer School and Conference in the ancient town of Pereslavl-Zalessky. There I had an opportunity not only to work on appealing research problems but also to interact closely with working mathematicians such as X and Y. It is difficult to describe the feelings that overwhelmed me but they did incite me to further progress.

Being a high school student I was doing a course on inequalities at Kiev State University where I conducted my first research work. It was essentially proof of Karamat inequality that utilized properties of convex functions and Murhead inequality. It was unforgettable, how the main idea of my central proof dawned upon me. That evening I went to sleep the happiest boy in the world. Although, relatively simple, I did something really innovative, something that no one had ever done before with elementary methods. That was my first moderate contribution to mathematics.

Later, while being a freshman at the Moscow Institute of Physics and Technology (MIPT), I refined the proof and presented it at the 52nd MIPT Scientific Conference where it was honored the first prize.

In my fourth year when students of our University get involved in research activity, Professor X suggested to be my supervisor. Under his guidance I wrote my thesis ”Integration of Multivalued Mappings“ and defended it with Honors. Essentially it consisted in a study of necessary and sufficient conditions under which there exists Riemann integral of multivalued maps. My task was to find some classes of sets of attainability and to investigate relation between Lebesgue and Riemann integral for Multivalued Maps. In this work I analyzed properties of spaces of compacts with Housdorff metric and properties of support functions of convex compacts and applied these theoretical findings to several problems of theoretical mechanics. During this work I acquired a broad range of research experience and background necessary for further research in convex analysis.

My current research work is devoted to differentiating of multivalued mappings and differential inclusions. The main task is to become familiar with ideas and approaches introduced in the Sci. D. thesis of Professor X and to improve the results that are obtained in it. One of the most challenging tasks in the project is to obtain Pontryagin maximum principle in Hamiltonian form from Lagrange form (in terms of tangent cones).

After graduation I plan to continue my scientific career in mathematics. Differential games, convex analysis and Optimization theory are of particular interest to me. I have the strongest incentive to advance as far as I can in this captivating science and feel confident that application to the University of Chicago is the best possible step to accomplish it. I would regard my admission to your University not only as a great honor but also as a great responsibility and an obligation to work hard.

Ivan Ivanov