Thursday, July 29, 1-2 pm.

Eric Brown, Princeton University.
From Spikes to Speed - Accuracy Via the Brainster

Recent brain recordings suggest a link between different firing patterns
in the brainstem nucleus locus coeruleus (LC) and different levels of
performance in simple cognitive tasks.  I will describe mathematical
models for these firing patterns, based on phase reduction and a
probability density formulation.  Then, in an extension of previous work,
I will discuss a possible role for the LC in optimizing speed and accuracy
in decision tasks, via release of neuromodulators which dynamically adjust
the sensitivity (i.e. gain) of neural populations.  This joint work with
Jeff Moehlis, Phil Holmes, Mark Gilzenrat, and Jonathan Cohen.
Tuesday, July 20, 1-2 pm.

Paul Ellis, Mathematics Graduate Student, Rutgers University.
Knot Theory

So the idea in Knot theory is to find invariants.  This lets
us prove that 2 distinct knot are, in fact, distinct.  The hard part
is to prove that an object is a knot invariant.  I will discuss 1
Combinatorial invariant and 2 Algebraic invariants of knots.
Tuesday, July 13, 1-2 pm.

Amy Cohen, Professor of Mathematics, Rutgers University.
Korteweg-deVries, Solitons, and all that

This talk will be an introduction and survey of the mysteries of the
Korteweg-deVries equations (KdV).  KdV was originnaly formulated in 1895
to model the long-time behavior of certain water waves on canals.  It
succeeded in "predicting" the "great waves of translation" first described
by John Scott Russell in 1844.  In the 1960's KdV reappeared in the
attempt to understand numerical simulations of certain phenomena in atomic
physics, simulations that unexpextedly did not show the expected
"equipartition of energy".  Matematical investigation of those simulations
spurred many preogressively more "pure" mathematical investigations of
"completely intergrable systems" as well as a theory of "non-linear
Fourier transfors" which turned out to be useful for problems in fiber
optics and in ocean waves.
Tuesday, July 6, 1 - 2 pm.

Andrew Sills, Mathematics Department, Rutgers University.
An Introduction to WZ Theory

A geometric series is a series a(0) + a(1) + a(2) + ... in which the
ratio a(k+1)/a(k) of consective terms is constant for all k=0,1,2,3,...
In contrast, a hypergeometric series is a series in which the ratio of
consective terms is not constant, but rather a rational function.

Hypergeometric series were first studied by Gauss, and over the centuries
many identities involving hypergeomtric series were discovered.
In the course of studying such series, it became standard to look for,
by ad hoc methods, recurrence relations satisfied by the series.

A major advance was made in the 1940's by Sister Mary Celine Fasenmyer, who
discovered an ALGORITHM for finding a recurrence relation satisfied by
a given hypergeometric term.  In the early 1990's Herbert Wilf (U. Penn)
and Doron Zeilberger (now at Rutgers) discovered a faster algorithm
which accomplishes the goal of Sister Celine's algorithm, and further
allows a computer to rigorously prove (not just test a large number of
cases, but actually PROVE) hypergeometric identities completely

Tuesday, June 30, 11:30am -12:30pm.

Alexander Soifer, DIMACS, Princeton University, University of Colorado.
Good Ole Euclidean Plane, Don't We Know All About It?

Define a Unit Distance Plane as a graph U2 on the set of all points of
the plane R2 as its vertex set, with two points adjacent iff they are
distance 1 apart. The chromatic number of U2 is called the chromatic
number of the plane. It makes sense to talk about a distance graph when
its set of vertices belongs to a metric space, and two points are
adjacent iff the distance between them belongs to a given set of

We will discuss the problem of finding the chromatic number of the plane,
including the recent Conditional Chromatic Number Theorem,  obtained by
Saharon Shelah and the presenter, which described a setting in which the
chromatic number of the plane takes on two different values depending
upon the axioms for set theory. We will look at examples of distance
graphs on the real line R, the plane R2, and the  n-dimensional Euclidean
space Rn, whose chromatic number depends upon the system of axioms we
choose for set theory.
Tuesday, June 30, 1 - 2 pm.

Michael Saks, Mathematics Department, Rutgers University.
How fast can NP-Complete problems be solved in worst case?

It is widely believed that any algorithm that solves an NP-Complete
problem must have running time that, in worst case, is an exponential 
function of the size of the input.  Researchers have developed ad hoc
algorithms for specific problems that work surprisingly well in practice.
These algorithms tend to be rather complicated and it is difficult to
prove anything about their worst case behavior.
In this talk, I'll discuss the problem of developing algorithms
for NP-complete problems whose PROVABLE worst case running time is as
small as possible.  I'll concentrate on  three computational problems:
(1) hamiltonian cycle, (2) chromatic number,  and (3) Boolean CNF

Tuesday, June 22, 12:30 - 2 pm.

Lara Pudwell, Mathematics Department, Rutgers University.
Pebbling and Cover Pebbling

Pebbling is a game played on the vertices of a graph with "pebbles".
Cover pebbling is an extension of this idea. I will explain how to
determine both the pebbling number and the cover pebbling number of
a graph and will discuss the relationship between these two values.

Tuesday, June 15, 12:30 - 2 pm.

Dinesh K. Pai, Computer Science Department, Rutgers University.
The HAVEN (Haptic, Auditory, and Visual Environment)

Humans experience the world with all their senses, including vision,
touch, and hearing.  Therefore, human interfaces should provide
correlated multisensory information that is both realistic and
responsive to interaction. I will describe how we can construct such
environments with integrated graphics, haptics, and sounds.  I will
show how we can construct physically based models suitable for
multisensory interactive simulation, and how these models could be
acquired from the real world.  Finally, I will describe the HAVEN, our
new Haptic, Auditory, and Visual Environment.