(senior
thesis defense)

5:15–6:15p.m.

Speaker: James Clark, GWU

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Complexity of Orders on Computable Groups*

Abstract: A left order on a group is a linear ordering of its
elements, which is left-invariant under the group operation. If a left order is
also right-invariant, it is called a bi-order. Orders on groups are often
identified with their positive cones. Certain algebraic conditions determine
such cones. For an orderable group (even magma), there is a natural
topology on the set of its orders. For computable groups, we also investigate
computability theoretic complexity of their orders.

(senior
thesis defense)

5:15–6:15p.m.

Speaker: Milica Taskovic, GWU

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Axiom of Choice across Mathematical Disciplines*

Abstract: Ever since its establishment, the axiom of choice
(AC) has been one of the most controversial axioms in set theory. The main
criticism of AC is that it claims the existence of an object without explicitly
defining the object in the language of set theory. Even so, AC has been used in
proofs of many known theorems across all mathematical disciplines. However, its
use is often disguised since it is equivalent to hundreds of other mathematical
statements. In this talk we will discuss the use of AC in undergraduate
mathematics courses, and some counterintuitive examples it creates.

6:15–7:15p.m.

Speaker: Alexandra Shlapentokh, East Carolina University

http://personal.ecu.edu/shlapentokha/

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Easy as Q: HilbertÕs Tenth Problem for subrings of Q and number fields*

Abstract: We show that there are subrings of *Q*, ÒcloseÓ to *Q*, where HilbertÕs Tenth Problem is Turing equivalent to
HilbertÕs Tenth Problem over the rational numbers. These results
complement results of Poonen and others showing that there are subrings ÒcloseÓ
to *Q*, where HilbertÕs Tenth Problem
is equivalent to the problem over *Z*.

5:15–6:15p.m.

Speaker: Wesley Calvert, Southern Illinois University

http://lagrange.math.siu.edu/Calvert/

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Optimal characterization of learnability*

5:15–6:15p.m.

Speaker: Rumen Dimitrov, Western Illinois University

http://www.wiu.edu/users/rdd104/home.htm

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Algorithms in computable vector spaces*

5:15–6:15p.m.

Speaker: Leah Marshall, GWU

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Computable categoricity of computable partial injection structures** *

Abstract: We will review and continue our discussion from the
December seminar. A computable structure *A* is computably categorical if for every isomorphic computable
structure *B*, there is a computable
isomorphism from *A* to *B*. A computable injection
structure is a structure consisting of a computable set and a computable
injective (1–1) function. We will present some recent results generalizing the
notion of computable injection structures to include partial computable
functions. We will also present recent results regarding computable
categoricity of computable partial injection structures.

5:15–6:15p.m.

Speaker: Valentina Harizanov,
GWU

http://home.gwu.edu/~harizanv/

Place:
Monroe Hall (2115 G Street), Room 267

Title:
*Orderable groups** *

Abstract: We investigate properties of orders
on groups that respect the algebraic
structure. There is a natural topology on the set of such orders, and this
space is compact even for magmas. For some groups, this space is homeomorphic
to the Cantor set. For familiar computable groups, we investigate the structure
and complexity of orders and their connection with computable trees. Many
questions remain open.

4:00–5:00p.m.

Speaker: Russell Miller, City University of New York

http://qcpages.qc.cuny.edu/~rmiller/

Place: Monroe Hall (2115 G Street), Room 267

Title: *The theory of fields is
complete for isomorphisms*

Abstract: We
give a highly effective coding of countable graphs into countable fields.
For each countable graph *G*, we build
a countable field *F*(*G*), uniformly effectively from an
arbitrary presentation of *G*.
There is a uniform effective method of recovering the graph *G* from the field *F*(*G*). Moreover,
each isomorphism *g* from *G* onto any *G*' may be turned into an isomorphism *F*(*g*) from *F*(*G*)
onto F(*G*'), again by a uniform
effective method so that *F*(*g*) is computable from *g*. Likewise, an isomorphism *f* from *F*(*G*) onto any *F*(*G'*)
may be turned back into an isomorphism *g*
with *F*(*g*)=*f*. Not every
field *F* isomorphic to *F*(*G*)
is actually of the form *F*(*G*'), but for every such *F*, there is a graph *G*' isomorphic to *G* and an isomorphism *f*
from *F* onto *F*(*G*'), both computable in
*F*.

It follows that many computable-model-theoretic properties which hold
of some graph *G* will carry over to
the field *F*(*G*), including spectra, categoricity spectra, automorphism spectra,
computable dimension, and spectra of relations on the graph. By previous
work of Hirschfeldt, Khoussainov, Shore, and Slinko, all of these properties
can be transferred from any other countable, automorphically nontrivial
structure to a graph (and then to various other standard classes of
structures), so our result may be viewed as saying that, like these other
classes, fields are complete for such properties.

This work is joint with Jennifer Park, Bjorn Poonen, Hans Schoutens,
and Alexandra Shlapentokh.

5:00–6:00p.m.

Speaker: Victoria Lebed, Advanced Mathematical
Institute, Osaka City University**, **Japan

http://www.math.jussieu.fr/~lebed/index_ENG.html

Place: Monroe Hall
(2115 G Street), Room 267

Title: *Towards topological
applications of Laver tables*

Abstract: Laver tables are certain finite shelves (i.e., sets endowed with a binary
operation which is distributive with respect to itself). They originate from
set theory and have a profound combinatorial structure. In this talk I will
discuss our dreams regarding potential braid and knot invariant constructions
using Laver tables, and also present some real results in this direction, such as
a detailed description of 2- and 3-cocycles for Laver tables. The rich structure
of the latter promises interesting topological applications.

(Joint
work with Patrick Dehornoy)

*Logic in Baltimore*

2014 Joint Math Meetings, Baltimore Convention Center: January
15–18, 2014

http://jointmathematicsmeetings.org/meetings/national/jmm2014/2160_intro

__AMS-ASL Special Session on Logic and
Probability__

__AMS Special Session on Computability in
Geometry and Topology__

Association for Symbolic Logic Winter Meeting: January 17–18, 2014

3:45–5:00p.m.

Speaker: Rumen Dimitrov, Western Illinois University

http://www.wiu.edu/users/rdd104/home.htm

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*The open problem regarding the automorphisms of L*(Q_inf)*

4:00–5:00p.m.

Speaker: Leah Marshall, GWU

Place: Monroe Hall
(2115 G Street), Room 267

Title:
*Computable categoricity*

Abstract: A computable structure *A* is computably categorical if for every isomorphic computable
structure *B* there is a computable
isomorphism from *A* to *B*. We will present some recent results
regarding computable isomorphism including an example of a computable structure
that is not computably categorical.

5:30–6:30 p.m.

Speaker: Valentina Harizanov, GWU

http://home.gwu.edu/~harizanv/

Place: Monroe Hall (2115 G Street), Room 267

Title: *C.e. and co-c.e. structures and their isomorphisms*

4:30–5:30 p.m.

Speaker: Mietek Dabkowski, University of Texas at Dallas

http://www.utdallas.edu/math/faculty/dabkowski.html

Place: Monroe Hall (2115 G Street), Room 267

Title: *Orderable groups and their spaces of orders*

Abstract: A left order on a group *G* is a linear order of the domain of *G*, which is left-invariant under the
group operation. Right orders and bi-orders are defined similarly. We
investigate computability theoretic and topological properties of spaces of
left orders on computable orderable groups. Topological properties of spaces of
orders on groups were first studied by A. Sikora who showed that for free
abelian groups of finite rank *n* >1
the space of orders is homeomorphic to the Cantor set. We study groups for
which the spaces of left orders contain the Cantor set and we establish that a
countable free group of rank *n* ³2 and
fundamental groups of oriented surfaces have a bi-order in every Turing degree.

5:30–6:30 p.m.

Speaker: Jennifer Chubb, University of San Francisco

Place: Monroe Hall (2115 G Street), Room 267

Title: *Orderings of algebraic structures*

Abstract: A partial left ordering or bi-ordering of an algebraic structure is a
partial ordering of the elements of the structure, which is invariant under the
structure acting on itself on the left or, respectively, both on the left and
on the right. I will discuss algorithmic properties of the orderings admitted
by a computable structure, and consider some general questions.

5:30–6:30p.m.

Speaker: Jozef Przytycki,
GWU

http://home.gwu.edu/~przytyck/

Place: Monroe Hall
(2115 G Street), Room 267

Title: *Quandles and codimension two
embeddings*

Abstract: Distributivity has been an
integral part of logic for a long time. An attempt to decouple them in linear
logic applied to quantum mechanics was not successful. Distributivity in
topology is a more recent development and can be dated to the PhD dissertation
of Joyce in 1979, in which quandles were applied to knot theory. The next push
came with construction of homology theory for quandles by Fenn, Rourke, and
Sanderson (between 1990 and 1995). In 1998, Carter, Kamada, and Saito

discovered how to use homology of quandles to study classical and higher-dimensional
knots. From that time we observe an exponential growth of the topic, and it is
my pleasure to report that it was achieved partly by work of our students. In
this I talk will offer a gentle introduction to this unusually successful use
of distributivity in

knot theory.

References

1. J.S. Carter, S. Kamada, and M. Saito, Surfaces in
4-space, Encyclopaedia Mathematical Sciences, Low-Dimensional Topology
III, Gamkrelidze,

and Vassiliev, editors, 2004, 213pp.

2. D. Joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory,
Ph.D. dissertation, University of Pennsylvania, 1979.

3. J.H. Przytycki, Distributivity versus associativity in the homology theory
of algebraic structures, Demonstratio Math., 44 (2011), pp. 823–869;

e-print: http://front.math.ucdavis.edu/1109.4850