School Seminars and Colloquia

Real algebraic geometry via points over finite fields.


by Professor Gus Lehrer

Institution: University of Sydney
Date: Wed 26th October 2005
Time: 11:00 AM
Location: Russell Love Theatre, Richard Berry Building, University of Melbourne

Abstract: Suppose {fi(x1, . . . , xn) | i = 1, . . . , r}
is a collection of polynomials with real coefficients. The question of
describing the set (variety) X(R) of points (a1, . . . , an) in Rn which
are common zeros of these polynomials goes back to antiquity, and even the
case n = 1, r = 1 is non-trivial. It is well known that X(R) has finitely
many connected components, but even counting them can be interesting, as
the case when X is a hyperplane complement (the "cheese-cutting problem"
in 3 dimensions) shows. There is no complete result giving either the
number or topological nature of the connected components of X(R). In this
talk I shall show how some light can be thrown on these questions when the
coefficients of the fi are algebraic numbers, by comparing the properties
of the sets X(k) for different rings k, particularly when k = Fq, a finite
field. The basic ideas needed for these comparisons come from various
filtrations (particularly the Hodge and weight filtrations) of the
cohomology of X, regarded as a scheme. Examples will be given from among
easy low dimensional varieties, hyperplane complements, toric varieties,
the moduli space of curves, the variety of regular ("general position")
elements in a Lie group or Lie algebra, and the fibres of the
Grothendieck-Springer resolution. Some of this work is joint with Mark

For More Information: Paul Norbury tel. 8344 5534

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