Undergraduate (LMAC) students have to complete two small projects that go by the names

• Seminario e Monografia
• Projecto de Matematica

Here is a list of topics suitable for such projects as well as for the Master's dissertation. There is some overlap but of course the dissertation would require a much more detailed study of the topic. Depending on the background of the student, the small projects are also suitable for students in Gulbenkian's program "Novos Talentos em Matematica".

I have given very summary descriptions below. Please feel free to contact me for further details. Also, I have electronic copies of almost all the references below so if you want to have a look at any of them (and can't be bothered to find them in the library) just send me email.

Projects

• Mordell's Theorem (small project) : The aim of this project is to understand the proof of Mordell's theorem to the effect that the group of rational points on an elliptic curve is finitely generated. It involves learning the basic theory of elliptic curves (Riemann surfaces of genus 1). The pre-requisites are only basic complex analysis. The reference is "Elliptic curves" by A. Knapp, Princeton Univ. Press (1992) (just the first 3 chapters).
• Representations of symmetric groups (small project) : The aim of this project is to understand the classification of the irreducible representations of the symmetric groups via Young diagrams. It involves learning the basics of representations of finite groups (which is not usually covered in the undergraduate algebra courses). If there is time the relationship with the representation theory of the general linear groups can also be studied. The basic reference is "Representation Theory: a first course" by W. Fulton, Springer (1991) (just the first 4 sections of Part I).
• Classification of quadratic forms (small project) : The aim of this project is to understand the Hasse-Minkowski theorem classifying quadratic forms over the rationals. The prerequisites are the basic algebra courses. The project involves learning to work with the p-adic numbers and some elementary arithmetic. Depending on the time available we could also study the classification of unimodular forms over the integers or Arf's classification of quadratic forms over F_2 which have interesting topological applications. The basic reference is "A course in arithmetic" by J.P. Serre, Springer (1973) (just the first 4 chapters).
• Topology of hyperplane arrangements (small project or dissertation) : The aim of this project is to learn about the topology of the complement of a finite union of complex hyperplanes (or possibly even subspaces) in C^n. This is the subject of much current research. The aim for a small project might be to understand the computation of the fundamental group or the cohomology ring (a Theorem of Arnold!) of (the complement of) an arrangement of complex hyperplanes (for the latter the student should be familiar with differential forms). The only pre-requisite is the introductory topology class (which covers the fundamental group and covering spaces) and linear algebra. There is a lot of interesting combinatorics involved and we could concentrate on that . A comprehensive reference is "Arrangements of hyperplanes" by P. Orlik and H. Terao, Springer (1992).
• 2d Topological quantum field theories (small project or dissertation) : The aim of this project is understand the algebraic classification of 2 dimensional topological quantum field theories in terms of Frobenius algebras. Despite the title, no physics background is required (I have no understanding of the Physics myself). The necessary background in math is just some basic algebra and basic acquaintance with manifolds. The study of TQFTs is a very active area of topology and this classification theorem is a beautiful mixture of topology and algebra. The reference is "Frobenius algebras and 2 dimensional Topological Quantum Field Theories"by Joachim Kock, London Mathematical Society Student Texts (2003).
• Quaternionic analysis (small project and maybe dissertation) : Complex analysis generalizes to the quaternions (to a certain extent)! This was discovered by Fueter in the 1930s. For example, the Cauchy theorem holds for quaternionic holomorphic functions leading to Taylor and Laurent series expansions. The pre-requisites for understanding this are just multivariable calculus! (although the use of differential forms streamlines things somewhat). Reference for this are "Quaternionic Analysis" by A. Sudbery, Math. Proc. Camb. Phil. Soc. 85 (1979), pp 199-225, and "The quaternion calculus" by C. A. Deavours, Amer. Math. Monthly 80 (1973) pp. 995-1008. More recently an algebraic formulation of this theory has been developed by Dominic Joyce. Understanding this and studying some examples could be a suitable topic for a master's dissertation. A reference is A theory of quaternionic algebra by Dominic Joyce.
• String Topology (dissertation) : String Topology is the study of the topology of the space of maps from a circle to a manifold (called the free loop space). This is an infinite dimensional manifold and in the late nineties Chas and Sullivan discovered the existence of very interesting algebraic structure in the homology of the free loop space. This is an area of current interest in algebraic topology. The aim of this project would be to understand the construction of the Chas-Sullivan product on homology and the computations of some examples such as spheres. An overview is given by Notes on string topology by R. Cohen and A. Voronov. The algebraic topology course is a pre-requisite and differential topology and homotopy theory would also be useful.
• Differential graded categories (dissertation to be supervised jointly with Goncalo Tabuada (Universidade Nova de Lisboa)) : Differential graded categories are categories enriched over chain complexes (this means that there is a chain complex of morphisms between two objects). They are of much current interest in algebraic geometry, topology and representation theory. The aim of this project would be to understand the basic homotopy theory of differential graded categories and to try to do some explicit computations in this homotopy theory (which is due to Goncalo Tabuada). This would involve learning some abstract homotopy theory (Quillen model categories). Pre-requisites are the introductory course on algebraic topology (mostly for the homological algebra and basic category theory though) and the basic algebra sequence. References are Homotopy theories and model categories by W. Dwyer and J. Spalinski, in Handbook of Algebraic Topology, Elsevier, 1995, and Une structure de categorie de modeles de Quillen sur la categorie des dg-categories by Goncalo Tabuada.