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IPM Positions 

Non Resident Researcher (nonresident), School of Mathematics
(2002  2003 ) 

Research Activities 

Algebraic Varieties are the main object of the study in Algebraic Geometry.
These objects, at most points, are locally similar but exhibit exceptional
behavior at a subset of smaller dimension, called singular locus. One of the
main problems in algebraic geometry is seeking for theorems which show the the
existence of a variety which is closely related to a singular variety but it has
a simpler structure on its singular locus. These type of theorems, known as resolution of singularities, are of interest since there are a huge number of theorems in algebraic geometry which depends on existence of desingularization, e.g. local parameterization with power series or solving differential equations near singular locus. Resolution of singularities of algebraic varieties has a long history and goes back to earlier days of axiomatic foundation of Algebraic Geometry, in particular in the works of Max Noether whom did it for algebraic varieties of dimension one (around 1860). In the 1963 Hironaka showed the existence of a resolution of singularities for varieties over field of characteristic zero, of arbitrary dimension n and in 1991, Villamayor, Bierstone and Milman gave a constructive proof for desingularization. At the other hand the importance of resolution of surface singularities is that not only it provides a ground for higher dimensional case but also it is a main step in desingularization of an algebraic variety. There are several methods of resolution of surface singularities: 1Repeat the process { Normalize and then blowup the reduced singular locus }until the singularities become simple. 2Repeat the process { blowup the Jacobian subscheme and Normalize } until the singularities become simple. 3Project to \mathbbP^{2} and do embedded resolution on the branch curve. The monoidal Transformation or blowup, is the central part of all these methods. Based on these methods, some algorithms have been developed. In this project we are planning to investigate on the efficiency of these algorithms and search for a necessary and sufficient condition under which a polynomial number of monoidal transformation yields the smoothed variety. As an another task we analysis the existing constructive proofs and algorithms for resolution of singularities and implement them in computer algebra systems; Macaulay, Singular and COCOA to measure and compare their complexities. To meet this, we need to complete all activities which have been down to setting up a Symbolic Computational Laboratory adjacent to the existing computational laboratory in the IPM. 

Present Research Project at IPM 

Complexity Analysis of Different Algorithms for Resolution of Surface Singularities. 
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