Update (August 2015)
My research interest has
shifted to Higher Algebraic K-theory and related topics
(namely, Grothendieck Witt theory and Witt theory).
Other
than that, instead of working on affine schemes (commutative
rings), I consider wider variety of algebraic objects (schemes).
Some of the information below would be out dated.
However, I would be building on my expertize in the following.
What would it take to get a PhD with Satya?
Following a short list of papers and books that will take you to the forefront of research:
- Mandal, Satya, Projective modules and complete intersections,
Lecture Notes in Mathematics, 1672.
Springer-Verlag, Berlin, 1997. viii+114 pp. ISBN: 3-540-63564-5
- Bhatwadekar, S.M., Sridharan, R.:
The Euler class group of a Noetherian ring.
Compos. Math. 122, 183-222 (2000)
-
Mandal, Satya,
Homotopy of sections of projective modules,
Algebraic Geom. 1 (1992), no. 4, 639--646.
What kind of Research I do?
We will use the notation A for a commutative noetherian ring.
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At a basic level, I work on projective modules and minimal number of generators of ideals (and modules). Given an ideal I in A and an integer r ≥ 1, we ask whether there is a projective module P of rank r that maps onto I?
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At a slightly more sophisticated level, we work on a so called "Obstruction Theory".
I will try to explain it, with some over simplification (i.e. hiding some of the details). We define obstructions groups E(A). Given a projective A-module P, we define an obstruction class e(P ) in E(A).
For an ideal I, we also define a class [I] in E(A).
We ask, if e(P )= [I], whether P maps on to I and conversely?
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There is a parallel theory of obstructions that is much more abstract and sophisticated.
Among my projects is to try to reconcile these two theories.
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Most of what I do would be obvious when the ring is local. So, our rings are non-local.
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