Stability and its Ramifications. M.S. Narasimhan. Interaction between Algebraic Geometry and Other Major Fields of Mathematics & Physics. Main theme :
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Stability and its Ramifications
notion of “stability”,which arose in moduli problems in algebraic geometry (classification of geometric objects), and its relationship with topics in partial differential equations, differential geometry, number theory and physics…
withf (x,y) = 0,
where fis a polynomial in two variables, and R a rational function of x and y.
purely algebraic treatment of the work of Riemann
The Rosetta Stone
Picard Poincare Hodge Kodaira Spencer Atiyah Singer
DIFFERENTIAL ANALYSIS ON MANIFOLDS
DEEP RESULTS IN
Now restrict to:
Particular area of
Suppose the (full or special) linear group) G acts linearly on a vector space V and S(V) the algebra of polynomial functions on V .
HILBERT: The ring of G-invariants in S(V) is finitely generated.
“It is not mathematics; it is theology.”
P(ss) the set of semi-stable points in P(V).
Then a "good " quotient of P(ss) by the group exists (and is a projective variety, compact, in particular)
-- classification problem in algebraic geometry .
In order to get moduli spaces one has to restrict to the class of good objects
SeshadriHitchin Kobayashi Donaldson