Problems in Recursive Analysis and Geometry
			   Anil Nerode
		         445 Malott Hall
		        Cornell university
		         Ithaca NY 14851
		    email anil@math.cornell.edu or
		      anil@hybrid.cornell.edu

Abstract:
At present, recursive analysis, topology, and geometry are much simpler
subjects technically than recursive algebra.  It is notorious that
recursive analysis thus far gets its main counterexamples from pure coding
arguments, coding r.e. and inseparable r.e. pairs, without use of the
priority method. 

Many of the results in recursive algebra using the priority method stem
from determining the effective content of structural theorems in algebra.
For a simple instance, any two countably infinite dimensional
algebraically closed fields of the same characteristic are isomorphic, but
George Metakides and I exhibited in 1977 recursively presented ones with
very different recursive properties: the standard one with a recursive
transcendence base, another where the only r.e.  algebraically closed
subfields are finite dimensional. The standard one was constructed by
classical algebra in Froelich- Shepherdson in 1954, the second required
substantial constructive polynomial ideal theory and the priority method. 

We think that the full range of prioric methods will be required to
analyze the corresponding type of theorem in functional analysis,
topology, and geometry. Theorems that characterize separable topological
structures up to homeomorphism, or separable Banach spaces up to isometry,
or differentiable structures up to diffeomorphism, are all fair game, along
with all universality theorems. 

We wish to point out a common misunderstanding of Brouwer's thesis that
all constructive functions on the interval are continuous. It has often
been misinterpreted to mean that, for instance, there is no sense in
trying to define recursive functions on the interval which are not
continuous. But recursiveness of a function is defined to mean it is an
effective limit with effective rate of convergence of a recursive sequence
from a separable base of given functions.  The notion of limit depends on
which function space topology is intended, be it sup, L1, L2, or something
else.  In fact many of the counterexamples in recursive analysis depend on
a construction being recursively continuous in one topology and not so in
another. See Pour-El Richard's book. Thus should not hesitate to develop
the effective content of those branches of modern analysis, such as
stochastic processes and stochastic differential equations, which deal
with measurable functions. 

We believe the reason these areas have not been developed is that
logicians are usually better trained in algebra than the other subjects.