Amir Hashemi
An introduction to computational algebraic geometry.

In this talk we give an introduction  to computational commutative algebra and algebraic geometry.  In particular, we will present the theories of  Grobner bases and involutive bases. Grobner bases are a powerful computational tool which were introduced by Buchberger in his PhD thesis. In the first part of this talk, we will address the construction and discuss then some well-known applications of Grobner bases.  It should be noted that a Grobner basis does not inherit, in general, all the algebraic and combinatorial properties of the ideal it generates. Therefore, in the second part of this talk we deal with involutive bases as a special kind of Grobner bases, with additional properties. These bases have their origin in the Riquier-Janet Theory of partial differential equations and were introduced by Gerdt, Blinkov and Zharkov. We will review the theory of involutive bases and present some of their  applications  in commutative algebra and algebraic geometry.