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Math Lounge before the exam.

Departmental PhD Thesis Exam

Wednesday, April 21, 2010, 1:10 p.m.,
in BA 6183, 40 St. George Street

PhD Candidate:  Mikhail Mazin

PhD Advisor:  Askold Khovanskii

Thesis Title:  Geometric Theory of Parshin Residues.

Thesis Abstract:
In the early 70's Parshin introduced his notion of the
multidimensional residues of meromorphic top-forms on
algebraic varieties. Parshin's theory is a generalization
of the classical one-dimensional residue theory. The main
difference between the Parshin's definition and the
one-dimensional case is that in higher dimensions one computes the
residue not at a point but at a complete flag of irreducible
subvarieties X=X_n\supset ... \supset X_0,  dim(X_k)=k.
Parshin, Beilinson, and Lomadze also proved the Reciprocity Law
for residues: if one fixes all elements of the flag, except
for X_k, where 0<k<n, and consider all possible choices of X_k, then
only finitely many of these choices give non-zero residues, and
the sum of these residues is zero.

Parshin's constructions are completely algebraic. In fact, they
work in very general settings, not only over complex numbers.
However, in the complex case one would expect a more geometric
variant of the theory.

In my thesis I study Parshin residues from the geometric point of
view. In particular, the residue is expressed in terms of the
integral over a smooth cycle.  Parshin-Lomadze Reciprocity Law for
residues in the complex case is proved via a homological relation on
these cycles.

The thesis consists of two parts. In the first part the theory of
Leray coboundary operators for stratified spaces is developed.
These operators are used to construct the cycle and prove the
homological relation. In the second part resolution of singularities
techniques are applied to study the local geometry near a
complete flag of subvarieties.

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