Paul Bankston's Research Interests
- Model-Theoretic Topology: This is the study of the
ways in which Model Theory (in the sense of Mathematical
Logic) can affect General Topology. Model-theoretic notions have
long permeated Algebra, thanks to the pioneering efforts of Mal'tsev,
(Abraham) Robinson, and many others. But it is the prospect that
Model Theory can lend new perspectives to topological disciplines
that is of particular fascination to me.
One of the things I'm studying is model-theoretic categoricity in the
topological context;
trying to sort out a theory of what it means for a
topological space to
"really stand out in a crowd," to be identifiable within a
given class of spaces, using only first-order information about
the lattice of closed subsets of the space. What fascinates me
in particular is the combination of the descriptive qualities of
mathematical logic with the visual/geometric qualities of
topology.
For more specific information on this subject, either click on
some of the links in
my publications list, have a
look at some of my conference talk
abstracts
(courtesy of Topology Atlas), or peruse some
related work.
- Formal Aspects of Adjectives in Natural Language: This is a topic
that has long been of interest to me, involving the interface
between Natural Language and formal
approximations to it. First-order predicate language/logic
does a good job approximating many parts of (English) speech, i.e.,
nouns, verbs and prepositions, but fails dramatically when it comes
to, say, adjectives. In particular, it cannot account
for the fundamentally different ways in which the adjectives
"green" and "big" modify a noun such as "chair."
In the former case, we can easily view a world in
which the class of green chairs is the intersection of the class of
green things with the class of chair-things. By contrast, the way
"big" modifies a noun depends on the noun itself: a big chair is
microscopic when compared to the smallest of galaxies.
We investigate logical
languages inspired by this phenomenon; particularly those with
variables ranging over individuals and with variable-binding
operators akin to generalized quantifiers.
- For some new variations on an earlier theme of mine, click
here.