Abstracts:
"Zero to Infinity: Great Moments in the Evolution of Numbers"
Edward B. Burger
Williams College
Did ancient cavepeople have insights into infinity? How did prehistoric
people balance their checkbooks before we had the notion of "number"?
Was the Pythagorean Brotherhood really a "brotherhood"? Have you ever
prayed to "10"? Does the squareroot of 2 really exist? Can you count to
5? Will a randomly selected real number ever equal 1/3? Was Cantor as
kooky as his colleagues conjectured? If you've answered "yes" or "no" to
any of these questions, then this breathless journey through the history
of numbers is for you!

"How Always to Win at Limbo"
Edward B. Burger
Williams College.
Have you ever gone out with someone for a while and asked yourself: "How
close are we?" This presentation will address that question by answering:
What does it mean for two things to be close to one another? We'll take a
strange look infinite series, dare to mention a calculus student's
fantasy, and momentarily consider transcendental meditation. In fact,
we'll even attempt to build some very exotic series that can be used if
you ever have to flee the country in a hurry: we'll either succeed or
fail... you'll have to come to the lecture to find out. Will you be at
the edge of your seats? Perhaps; but if not, then you'll probably fall
asleep and either way, after the talk, you'll feel refreshed. No matter
what, you'll learn a sneaky way to always win at Limbo.

"Uncertain Determinism"
Eric Kostelich
Arizona State University.
Simple physical systems that obey Newton's laws can be
written as systems of differential equations whose solution, for a
given initial condition, is unique. In other words, if one can
specify the initial position and velocity of every component, then
the state of the system can be predicted, in principle, at every
instant in the future. Let us call such a system deterministic.
Suppose we want to predict the future state of a deterministic
system where the initial state must be measured somehow. No
measurement is perfect, but suppose that, given a positive integer
$d$ and enough time and money, it is possible to build an
apparatus to measure position and velocity with $d$ decimal digits
of accuracy. Furthermore, assume that the ultimate future state
of the deterministic system is one of two simple periodic motions.
Would it be possible, given these circumstances and unlimited
resources, to correctly predictsay 90 percent of the timewhich
of the two eventual behaviors one will observe given a suitable
measurement of the initial condition? I will describe a deterministic
system for which the answer is no.

"Mathematical Reasoning and Proving: Insights from the theory of Conceptual
Blending"
Michelle Zandieh
Arizona State University.
The purpose of this report is to describe how several seemingly different
phenomenon in mathematical reasoning can be seen through one unifying lens
with the use of Fauconnier and Turner's theory of conceptual blending. This
theory describes how humans reason and learn by combining familiar mental
spaces or frames into a new blended space or an integrated network of
blended spaces. I use examples of theoretical frameworks from the
mathematics education research literature, including data analyzed within
these frameworks, and show how they may be reframed using conceptual
blending. For this presentation these examples will draw mainly from
undergraduate students working to construct a proof that two parallel
postulate of Euclidean geometry are equivalent. We will see blending
occurring within verbal, symbolic and pictoral reasoning. By explicating
these examples I hope to illustrate how conceptual blending may play a role
in our understanding of how our students learn and how we may use this
knowledge to aid in curriculum design.

Download the full list of abstracts here.

