For more information about this meeting, contact Stephen Simpson.
|Title:||Calibrating the complexity of mathematical proofs and constructions|
|Seminar:||Department of Mathematics Colloquium|
|Speaker:||Richard Shore, Cornell University|
|We will discuss two related measures of complexity for mathematical theorems and constructions. One asks what proof techniques (or formally axioms) are needed to prove speci
c theorems. The other asks (for existence
proofs) how complicated (in the sense of computability) are the objects that
are asserted to exist.
For this talk we will consider some illustrative examples from Combinatorics. In particular, we will consider several theorems of matching theory
such as those of Frobenius, (M. and P.) Hall and König. While in the
finite case these theorems seem both different and yet somehow the same, an
analysis of the countable case in terms of computability or provability clearly
distinguishes among them and assigns precise levels to their complexity.
At the most complicated level we will consider lies the König Duality
Theorem: Every bipartite graph has a matching such that one can choose
a vertex from each edge of the matching so as to produce a cover, i.e. a
set with an element from every edge. This theorem cannot be proven using
algorithmic methods even when combined with compactness (König's lemma
for binary trees) or full König's lemma. We will show that it requires highly
nonelementary methods as typified by constructions by transfinite recursion,
choice principles and, for some versions, even more.
If time permits, we may also mention the calibration of some results of
Ramsey theory that lie at the other (low) end of our classification scheme:
Ramsey's theorem for n-tuples for different n and some consequences such as
the theorems of Dilworth and Erdos-Szekeres. (Every in
finite partial order
has an in
finite chain or antichain and every infinite linear order has an in
ascending or descending sequence.)
We will not use, or even consider, any formal systems and no knowledge
of logic is presupposed. We will work instead with an intuitive notion of what
it means for a function to be computable, i.e. there is a computer program
that calculates it given time and space enough and no mechanical failures.
We will also explain the relevant combinatorial notions.|
Room Reservation Information
|Date:||01 / 21 / 2010|
|Time:||04:00pm - 05:00pm|