Speaker: John Goodrick,
http://www.math.umd.edu/~goodrick/
Place: Monroe Hall (
Title: The Schröder-Bernstein property for ordered structures and
generalizations, part I
Abstract: We say that a theory has the Schröder-Bernstein property if any two of its models which are
elementarily bi-embeddable are necessarily isomorphic. Part of my thesis
showed that the theory of any infinite linearly ordered structure does not
have the Schröder-Bernstein property by
a modification one of Shelah’s “many-model arguments”— essentially one can build Skolem
hulls over a collection of sufficiently different-looking orderings and prove
that some of the resulting models are bi-embeddable but nonisomorphic.
The same arguments just as easily show that a theory does not have the Schröder-Bernstein property if there is an infinite subset of
any model which is orderable by a single formula (i.e., the theory is unstable)
or even by a formula in L_{infty, omega} (e.g.,
if the theory has OTOP). If there is time, I will also discuss how a Dushnik-Miller type argument gives an alternate
construction of counterexamples to the Schöder-Bernstein property.
Speaker: Jennifer Chubb, GWU
http://home.gwu.edu/%7Ejchubb/
Place: Monroe Hall (
Title: Computability, topology, and ordered groups, part III
Abstract: In this talk, we will see the details of a proof
that is a good example of a standard type of computability theoretic
construction. We will, following
Speaker: Jennifer Chubb, GWU
http://home.gwu.edu/%7Ejchubb/
Place: Monroe Hall (
Title: Computability, topology, and ordered groups, part II
Abstract: A countable group is computable if its
universe is computable and there is an algorithm for computing “products” in
the group. This second talk will include a basic introduction to computability
theory and will focus on algorithmic properties of orderings on computable
groups. We will see some examples of standard computability theoretic
constructions, including the construction of an orderable computable group
admitting no computable ordering.
Speaker: Jennifer Chubb, GWU
http://home.gwu.edu/%7Ejchubb/
Place: Monroe Hall (
Title: Computability, topology, and ordered groups, part I
Abstract: A
group is left-orderable if there is an ordering of its elements that is invariant
under multiplication from the left, and bi-orderable if there is an ordering
that is simultaneously left- and right-invariant.
There are a number of interesting questions surrounding these objects
concerning the cardinality of the collection of orderings for a given group, as
well as the topological structure of this collection endowed with a very
natural topology.
This is the first in a series of three talks surveying
the theory of ordered groups, in large part from a computability theoretic
point of view, and their applications in topology.
Speaker: John Goodrick,
http://www.math.umd.edu/~goodrick/
Place: Monroe Hall (
Title: The Schröder-Bernstein property, part II
OTHER LOGIC TALKS
Graduate
Student Seminar
Friday, December 5, 2008
Monroe
Hall (
Speaker: Valentina Harizanov, GWU
Title: Turing
degrees of complexity
Abstract: In 1936 Alan Turing introduced the notion of
an ideal computer and gave a negative answer to Hilbert's decision problem. Two
years later in his dissertation Turing defined the notion of a Turing machine
augmented with the so called oracle which provides external information during
the computation. This led Emil Post to develop in 1944 a powerful notion of
Turing degree as a measure of relative algorithmic complexity of sets of
natural numbers and problems they encode. There are uncountably
many Turing degrees, they are partially ordered and form an upper semilattice. We will show how some familiar mathematical
objects can have certain Turing degrees by encoding sets of natural numbers
into them. Although the talk will be based on topics of current research
interest, it will not require prior knowledge of computability theory.
Monroe
Hall (
Speaker:
Jennifer Chubb, GWU
Title: Model
theory and computability
Abstract: The notion of computability has its origins in the work of
Alan Turing, who in 1936 made precise the notion of machine computation. In
computable model theory we are interested in determining the properties of
mathematical structures and their theories that are accessible to us in an
effective way. A mathematical structure is computable if its domain, relations,
and functions can all be described by algorithms. The computability of a
structure by no means implies that everything we might want to know about it is
algorithmically accessible. I will provide a brief introduction to both model
theory and computability and describe some examples. Then, I will describe some
of the types of questions of interest in this area, and present some results
from recent research in computable model theory. This talk will be accessible
to undergraduates.
Special Topology/Logic Lecture
Speaker: Ivan Dynnikov,
http://higeom.math.msu.su/people/dynnikov/
Place: Bell Hall (
Title: A geometric approach to braid conjugacy
Abstract: I
will speak about an algorithm that is conjectured to solve the conjugator search problem for braids in polynomial time. It
is based on geometric presentation of braids as homeomorphisms of a punctured
disk rather than algebraic one.