Schedule:
2:00-3:00 Peter Burton (Caltech)
3:15-4:15
Garrett Ervin (UCI)
4:15-5:00 Coffee Break
5:00-6:00 Siddharth Bhaskar (UCLA)
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Abstracts:
Peter Burton
Title: Structure on the space of actions modulo weak
equivalence
Abstract: We discuss topological, convex and algebraic
structure on the
space of measure-preserving actions of a countable group modulo
weak
equivalence. We will define a natural Polish topology on this
space and
address the questions of how to represent its convex structure as
induced
from a Banach space and whether this structure forms a Poulsen
simplex. We
will also introduce a stronger (nonseparable) topology in which it
forms a
topological semigroup.
Garrett Ervin
Title: Linear orders that are isomorphic to one of their
finite powers
Abstract: In the early 1950s, Sierpinski asked whether
there exists a
linear order that is isomorphic to its lexicographic cube but not
isomorphic to its square. The analogous question has been answered
for
various other kinds of structures: it is known that there are
groups that
are isomorphic to their cube but not to their square (even
countable
ones), and similarly for modules, Boolean algebras, and Banach
spaces. In
this talk, I will show that if such a linear order exists, it is
necessarily uncountable. I will also give a general
characterization of
structures X satisfying equations of the form A x X = X, and show
how this
can be used to construct structures that are isomorphic to their
n-th
power, for any n.
Siddharth Bhaskar
Title: Recursion versus tail recursion over abstract
structures
Abstract: There are several ways to understand
computability over
first-order structures. We may allow functions given by arbitrary
recursive definitions, or we may restrict ourselves to “iterative”
functions computable by nothing more complicated than while loops.
In the classical case of recursion over the natural numbers, these
two
notions of computability coincide. However, this is not true in
general. We ask whether there is a model-theoretic classification
of
structures over which iteration is as powerful as recursion.
In this talk I will discuss some conditions which affect this
outcome
one way or the other. I will also give a few examples of
“intermediate” structures for which the question of recursion vs.
iteration reduces to hard open problems in computational
complexity.
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