# 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

Kisin.

*For More Information:* Paul Norbury tel. 8344 5534 http://www.ms.unimelb.edu.au/~pnorbury/colloquium/colloquium-2005.html