http://home.gwu.edu/~harizanv/Logic%20Seminar%20F09.html

4:30–5:30 p.m.

Speaker: Joe
Mourad,
Georgetown University

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

Title: *Applications
of Fixed Point Behavior in ÒEcologicalÓ Computability, *III*, the case N ³3*

4:30–5:30 p.m.

Speaker: Joe
Mourad,
Georgetown University

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

Title: *Applications
of Fixed Point Behavior in ÒEcologicalÓ Computability, *II*, the case N =2*

4:30–5:30 p.m.

Speaker: Joe
Mourad,
Georgetown University

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

Title: *Applications
of Fixed Point Behavior in ÒEcologicalÓ Computability, *I*, the case N* =1

Abstract: This will be the first in a
series of three talks, which will build on themes that we touched on in the
Fall 2011 series. There we used the fixed point
theorems to talk about self-identifying machines, which needed to locate
themselves given a changing (evolving) environment. We used a map of North
Carolina to identify the fixed point of that particular map (it went through
Rock Springs North Carolina and, if you recall, lies somewhere southeast of
Durham.) We will review these results and explore applications of such ideas to
philosophy of mind/cognitive science. In doing so we will explore metatheoretic issues dealing with the application of
rigorous formal methods to both mainstream experimental science and Òfolk
scienceÓ. We will make the transition to the case of *N* =2. Although familiarity with the fixed point
theorems in both logic and analysis is helpful, no prior knowledge will be
needed.

11:00 a.m.–12:00 noon

Speaker: Doug Cenzer, University of Florida

http://www.math.ufl.edu/~cenzer/

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

Title: *Effective Categoricity
of Injection Structures*

Abstract:**
**An injection structure *A* consists of a set *A* together with a one-to-one function *f* mapping *A* to *A*. These are among the most fundamental
structures in mathematics. The *orbit*
of an element *a* of *A* is the set of images *f *^{n}*(**a)* and also the preimages *f *^{-n}*(a)* of *a*.
Orbits are of three types. They may be finite, where *f *^{n}*(**a)* = *a* for
some *n*. They may have type w and consist of an
element *a* not in the range of *f*, together with its images *f *^{n}*(**a) *for every natural number *n*. Finally, they may have type *Z*
and consist of an element *a* together
with an infinite sequence of images *f *^{n}*(a)* and also an infinite
sequence of preimages *f *^{-n}*(a)*. By an effective structure
we will mean that *f* is computable and
that *A* is either computable, c.e. or co-c.e. set of natural numbers. A computable injection structure *A* is
said to be *computably categorical* if
for any computable structure *B* isomorphic to *A*, there is a computable isomorphism. We show that *A* is
computably categorical if and only if it has only finitely
many infinite orbits. Any c.e. injection
structure *A* is isomorphic to some computable structure *B* and, furthermore,
there is a computable isomorphism from *B* onto
*A*. We also consider
Delta-2 and Delta-3
isomorphisms.

(jointly with Knots in Washington)

Time: TBA

Speaker: Wesley Calvert, Western Illinois University

http://www.math.siu.edu/calvert/index2.html

Place: Rome
Hall (801 22^{nd} Street), Room 459

Title: *Computing Distances in Graphs*

Abstract:**
**How can we
compute the distance between vertices in a graph, given only data on adjacency?
If there are infinitely many vertices, this problem may be unsolvable, but it
can still be approximated in an interesting way. It turns out that distances in
graphs capture this sort of approximation exactly, in that any function that
can be approximated can be approximated by a graph. I'll give examples that
aren't obviously graph-like. This is joint work with Jennifer Chubb Reimann and Russell Miller.

4:30–5:30 p.m.

Speaker: Kai
Maeda, GWU

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

Abstract:
We
investigate Turing degrees of non-associative structures and their
isomorphism types. These structures include racks, quandles
and crossed sets. We will first review their definitions and establish basic
properties.

4:30–5:30 p.m.

Speaker: Kai
Maeda, GWU

Place: Monroe Hall (2115 G Street), Room 253 (Math Lab)

Abstract:
I will establish several results about enumeration reducibility in
computability theory. I will then show how they can be used in computable model
theory to produce a countable structure the isomorphism type of which does not
have a Turing degree.

4:30–5:30 p.m.

Speaker: Kai
Maeda, GWU

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

Abstract:
I will introduce one of RichterÕs theorems, which uses enumeration
reducibility to show that a structure does not have the degree of its isomorphism
class. I will then apply this result to structures not previously studied
in computability theory.

4:00–5:00 p.m.

Speaker: Kai
Maeda, GWU

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

4:00–5:00 p.m.

Speaker: Valentina Harizanov, GWU

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

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

4:00–5:00 p.m.

Speaker: Valentina Harizanov, GWU

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

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

4:00–5:00 p.m.

Speaker: Fredrick
Nelson, Howard University

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

Abstract:** **The Holm curve *H* is given by an equation of the form *ky**(y+r+s)(y-r+s) *= *lx(x+r+s)(x-r+s)*, where
*r* and *s* are relatively prime integers with *r* > |*s*|, and *k* and *l* are distinct square-free relatively prime positive integers. *H* is an elliptic curve passing through
each the nine points *(-r-s,-r-s),
(-r-s,0), (-r-s,r-s), (0,-r-s), (0,0), (0,r-s),
(r-s,-r-s), (r-s,0), *and* (r-s,r-s). *The other rational points on *H* are associated in a precise (but
possibly non-effective) manner with pairs *(M,N)* of *q*-congruent numbers
where cos(*q*) = *s/r* and *M/N* = *k/l*. Conjecturally,
the group of rational points on every Holm curve is torsion-free so
its rank is at least 1. It is natural to ask whether rank-1 Holm curves
exist. We answer this question in the
affirmative and discuss an effective characterization of the *p*-adic
filtration on the Weierstrass curve *E* corresponding
to *H*, which yields infinitely many
pairs of *q*-congruent
numbers with ratio *k/l*.

4:00–5:00 p.m.

Speaker: Joe
Mourad,
Georgetown University

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

Title: *Paradox, Fixed Points, and
Computability*

4:00–5:00 p.m.

Speaker: Joe
Mourad,
Georgetown University

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

Title: *ÒEcologicalÓ Computing, part
II*

4:00–5:00 p.m.

Speaker: Joe
Mourad, Georgetown University

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

Title: *ÒEcologicalÓ Computing, part
I: Fixed Points*

11:10a.m.–12:10p.m.

Speaker: Jozef Przytycki, GWU

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

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

Title: *Homology of Distributive Lattices*

Abstract: While homology theory of associative structures,
such as groups and rings, was extensively studied in the past, beginning with the
work of Hopf, Eilenberg,
and Hochschild, homology of non-associative
distributive structures, such as quandles, has been
neglected until recently. Distributive structures have been studied for a long
time. In 1880, C.S. ~Peirce emphasized the importance of (right-) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years
ago by Fenn, Rourke, and
Sanderson. We develop this theory in the historical context and propose
a general framework to study homology of distributive structures. We illustrate
the theory by computing some examples of 1-term and 2-term homology, and then
by discussing 4-term homology for Boolean algebras and distributive lattices.
We will start with a gentle introduction to distributive lattices and Boolean
algebras (and their generalizations) for topologists, and with homology theory
of distributive structures for logicians. We will end by outlining
potential relations to Khovanov homology, via the
Yang-Baxter operator.

11:00a.m.–12:00 noon

Speaker: Kai
Maeda, GWU

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

Abstract:** **RichterÕs degree of a countable
algebraic structure is a computability theoretic measure of complexity of its
isomorphism class. It has been shown that some classes, such as abelian groups or partially ordered sets, have arbitrary
degrees, while other structures, such as linear orderings or trees, can only
have degree **0**. In this presentation,
I will explain how we measure this degree of complexity of isomorphism types
and will survey some known results. Finally, I will further extend these
results to structures not previously studied in logic.