Date:	Tue, October 6, 2009
Time:	14:00
Place:	Research I Seminar Room

Abstract: One can form from any "function" $f$ on $\mathbb C^n$ something called module with chosen element $(M_f, m_f)= ((\mathbb C[\bar x, \partial_{\bar x}])f, f)$. It happened so that the pair $(M_f, m_f)$ determine in a strong sense expressions of next types: $(p(x)/q(x))^s, s\in \mathbb C$, $exp(p(x)/q(x))$, $ln x$ and they products and sums. We introduce something called Gelfand-Kirillov dimension of a pair $(M,m)$ which is more or less "order of a differential growth" of a pair $(M, m)$. We prove that this "order of a growth" can not be less than $n$ and greater than $2n$. Pair $(M,m)$ called holonomic if Gelfand-Kirillov is precisely $n$. A function $f$ called holonomic if pair $(M_f, m_f)$ is holonomic. All mentioned functions are holonomic. Holonomic pair $(M_f, m_f)$ determine practically uniquely "function" f. It rise a question: what else can determine such pairs? For example it can determine a $\delta$ "function" and all its derivatives. From the other hand it can determine something called a bundle with a flat connection and it is more or less the same with a formal sum of functions. Introducing a support of a holonomic module we can state the following theorem: Theorem. Any simple holonomic pair $(M, m)$ is determined by its support and bundle with a flat connection on it. Than any holonomic module determine only direct sum of products of holonomic functions with $\delta$ functions of some closed sets. At the end I introduce something called holonomic modules with regular singularities(RS) (more or less it is a set functions which have no singularities of type exp(1/x)) and something called perverse sheaves. Theorem. A system (or category) of (RS)holonomic modules on variety (manifold) $X$ is equivalent to a system (or category) of perverse sheaves on $X$.