Letter of motivation: suggestions and criticism.

Hi everyone!
I am trying to apply for a joint Master degree programme in Mathematics (named ALGANT, an acronym for ALgebra Geometry And Number Theory, http://www.algant.eu/ ). As a mandatory requirement for the application, I am supposed to write

a personal letter of motivation explaining the student's interest in the ALGANT Master Course in particular and mathematics in general.

In what follows, I put my first draft. Since it is my first time ever that I try to write such a cover letter and I think there are several possibilities of improvement, I am completely open to any kind of suggestion or criticism both from the side of language/style and that of contents. Just some information to help you reading. Firstly, I will attach to the letter a detailed curriculum vitae and a transcript of all my academic records, so I decided not to mention in the draft anything that can be found looking at that papers, a part from what could be useful for my argument. Secondly, throughout the text there are some technical words, about the name of different branches of Mathematics involved and of mathematical objects (for example, at the end of the second paragraph "limits", "adjunction", "ring", "sheaf", "site" etc.). Hence, if something is not so clear, do not exhitate to ask me explanations about it!

Dear Sir or Madam,

I hereby wish to apply to the ALGANT Master Programme and specifically move to the University of Leiden for the next academic year. My professional goal and main interest as a growing mathematician is that of becoming a researcher in the mathematical field of Category Theory, without explicitly excluding, at the moment, any possible topic of study and application within such an extensive and substantial subject.

I discovered Category Theory at the end of my second year as an undergraduate student at the University of Trento (Italy), by virtue of some profitable talks with a friend and colleague of mine and of enlightening visits to an on-line forum about Mathematics, namely http://www.forumscienzematematiche.it. I therefore decided to buy the book “Categories for the Working Mathematician” by Saunders Mac Lane. As soon as I started adventuring myself through the delightful pages of that book, it was immediately clear to me what I would have liked to try and study in the forthcoming future. Hence, I devoted the draft of my Bachelor degree thesis to the purpose of enjoying myself in getting a solid background in the most basic concepts of Category Theory, as (co)limits and adjunctions, aiming at the concrete objective of proving a charming result in Sheaf Theory, that is, the existence of a canonical isomorphism between the category of sheaves of rings on a site and the category of ring objects in the category of sheaves (of sets) on a site.

In my opinion, the best merit which ought to be acknowledged to Category Theory is that of having provided Mathematics with new and unifying language and tools which enable a relational and interdisciplinary approach to the study of different mathematical entities. However, what I like the most about such a subject is its high level of abstraction and its prowess in properly formalizing a wide range of concepts within the most appropriate context. Actually, it is precisely within these features that I can explain my interest for Category Theory, as I deem one of my best skill is exactly the one of adopting a completely rigorous and conceptual method of studying and learning Mathematics. I tend to require myself as much effort as sustainable to pursue such a goal, since my whole academic experience have definitively brought me to believe that clarity (of concepts) demands rigour. This is also what makes me believe that I might be a suitable and rather good candidate for the ALGANT extremely onerous programme.

As an aspirant categorist and quite an open-minded person , it is my major concern and necessity to strive to grant me the widest mathematical education and knowledge, especially about those branches of mathematics where a categorical approach can be extremely effective. Undoubtedly, (abstract) Algebra and Geometry (mainly Algebraic Geometry and Topology as well as Differential Geometry) are definitely among those areas and indeed they heavily match my own mathematical attitude and interests too. Concerning this and just to mention the most recent didactic experience, during the fall semester of this academic year at the University of Padova, I have followed, among the others, classes in Commutative Algebra and Ring and I have been able to appreciate categorical frameworks, ideas and instruments at work. I have also attended the course named Topology 2, where I have been amused by the significant use and meld of methods from Homological Algebra (in the most appropriate environment of abelian categories) as well as from Sheaf Theory in the study of topological spaces. I have followed a class in (Algebraic) Number Theory too and I have been fascinated by some of its ideas and the underlying interplay between Commutative Algebra and Galois Theory that is needed and used in its studying. In the next spring semester I am taking courses in Module Theory, Algebraic Geometry and Complex Analysis in order to keep on completing my mathematical instruction.

Actually, the main reason for my application to the ALGANT Programme can certainly be found in the extremely qualifying education and professional perspectives provided by the consortium, precisely within the sectors of Algebra and Geometry with which I hope and need to become as much familiar as possible. This is why I would feel really fortunate and honoured to have the exciting opportunity in becoming one of your students. On the other hand, I chose the University of Leiden as my first mobility option for the next academic year, due to its worldwide, cosmopolitan bearings, in addition to its highly rated instruction and courses, about which I have been told by one of its former students . Among these classes, I noticed the national one in Category Theory and Topos Theory that I am eager for being allowed to attend. Furthermore, my attention was drawn by the apparent interest of Dutch mathematical community (or at least part of it) in the use of categorical methods within Geometry and Logic (e.g. in the study of Topos Theory) and in Algebraic Topology, which was the historical birth nest of Category Theory. I had some basic, introductory notions about this subject (such as homotopy, foundamental group, covering theory and (singular) homology) by virtue of my undergraduate course in Geometry 3 and of some personal studies I did following the book “Algebraic Topology” by Allen Hatcher. I absolutely enjoy such a branch of Mathematics for its ability to concretely realize one of the goal, in my humble opinion, of the geometrical approach, that is spotting algebraic, hence computable, (topological) invariants to classify spaces.

I am sincerely confident of my decision, also due to some meaningful speaking with some ALGANT students in Padova, some of whom were at the University of Leiden during the previous academic year and convinced me even more about the importance of try to apply to ALGANT. Giving my past carrier of commitment to success and excellence, shown by my academic records, as well as my fascination for the whole (pure) Mathematics, I truly believe I will be able to bring a positive, ambitious and competitive attitude to your programme. Furthermore, I suppose that my disposition to cooperate will help me to proficiently share my knowledge with my colleagues and spread a collaborative mentality among them.

Thank you for considering my application. I keenly look forward for your acceptance.

Yours faithfully,
Marco Vergura.