This project aim at studying the growth of finitely presented algebras. These are quotient algebras of the free algebra on the words of a finite set X, such that the ideal related to this quotient possesses a finite number of generators. We intend to define a notion of growth to these algebras and study their asymptotic behaviour. The study will be divided in some parts. First, the student will acquire some basic knowledge in modern algebra, such as commutative algebra. Then, he will advance gradually in the main theme of the research. For that, he will learn some fundamental tools for the development of the theory, such as the Gröbner bases, the Hilbert series, and the Ufnarovsky Graphs. In the mean time, he will pass by the concept of Gelfand-Kirillov dimension. Finally, he will get to study the finitely presented algebras. In this work, some particular cases are going to be studied, such as the case where our algebras are commutative, when we will learn, for example, the Hilbert-Serre Theorem. Another particular case that will be considered in this work is the case of monomial algebras, whose Gröbner bases are easily calculated. Beyond that, we will see some applications, such as the calculation of the Gröbner bases of some given algebras, some results like the Poicaré-Birkoff-Theorem, the Kuroush problem, among others. Therefore, one can notice that the importance of this work is justified because it will provide the student with a solid base in algebra. Some of the themes of this work are very close to modern themes of research, and therefore this project might be extended to a future project at graduate level. At last, in the final months, we will try to generalize some of the results learnt for the case of non finitely presented algebras.
News published in Agência FAPESP Newsletter about the scholarship: