Models of Spherical and Hyperbolic Geometry
by Richard Goldthwait, Youngstown State University
The geometry of Euclid provides the foundation for trigonometry and supports a variety of topics in algebra and the calculus
(e.g. slope of a line, length of arc, etc.)
Euclidean geometry is also an indispensable tool in engineering and the sciences.
But two other geometries - spherical geometry and hyperbolic geometry - also play significant roles in mathematics and science.
In this workshop we will describe these two geometries. We will also draw some interesting comparisons between them and compare each to classical Euclidean geometry.
Finally, we will discuss some scientific and technical applications of both spherical and hyperbolic geometry.
Mathematical Modeling of the Brain
by Jozsi Jalics, Youngstown State University
About a trillion neurons in the human brain interact in complicated ways
to perform innumerable complex functions each moment. Neuronal disorders
such as schizophrenia, Parkinson's disease, and epilepsy are caused by
the abnormal firing activity of certain neurons. We will discuss the
vital role that mathematical modeling plays in understanding the complex
patterns of activity present in the brain and explore a neuronal model
that employs the unit circle.
Coordinates On A Spacetime Light Cone
by Steve Kent, Youngstown State University
Light rays, or more generally electromagnetic radiation, form light cones in
4-dimensional spacetime and are important in studying areas of mathematical
physics such as Yang-Mills theory (a mathematical generalization of electricity
and magnetism) and Einstein's relativity. The sphere is often used to mathematically
model directions of this radiation emanating from a point, and form cross-sections of these light cones.
Simpler coordinates allow for simpler study of these objects. We will investigate how to
put complex number coordinates on a sphere using "stereographic projection" onto the complex plane.
This workshop will be accessible to students familiar with geometry and trigonometry.
Are You In The Zone?
by Jay Kerns, Youngstown State University
Any American sports fan is likely to have heard a basketball player referred to as a "streak shooter"
or a baseball player referred to as a "streak hitter". Such people are sometimes said to have the
"hot hand" or be "in the zone". Is this phenomenon real? Or could these long streaky sequences be
happening by chance alone? In this workshop, we will investigate some of the models and methods
statisticians use to distinguish
between streaky and random sequences, and we will finally be able to decide: are you in the zone??
Taking the Mystery out of i
by David Pollack, Youngstown State University
At one point in history,
when people thought of numbers as magnitudes, the equation
x+1=0
must have seemed impossible to solve.
Someone may have declared that its solution is -1,
but most people would have wondered
what that meant.
How could you add 1 to an amount and end up with nothing?
The number line enhanced
our ability to reason by
providing a visual model for numbers and their
operations that made sense for
negatives as well as for positives.
At a later point in history, the equation
x^2+1=0
was considered, and it too seemed impossible to solve.
Your algebra book declared that
the solution is i, but you must have wondered what that means.
How can you multiply a number by itself
and end up with -1? Again the difficulty is resolved
with an extended visual model; the (complex) number plane takes
the “imaginary” out of imaginary numbers and their operations.
The purpose of this session is to discuss a
visual interpretation of numbers and their
operations in which the equation
i^2=-1
makes just as much sense as the equation -1+1=0.
How America Elects Its President: Mathematics Tells Us that the
Process May Not Be Satisfying
by Nathan P. Ritchey, Youngstown State University
Did you know that American citizens do not have the Constitutional
Right to vote for the U.S. President? In fact, the Electoral College
elects the U.S. President and the mathematics lurking behind the scenes
suggests that an election calamity could occur. In this workshop we
will explore some mathematical problems that result from the election
process. We will also explore some ways to "fix" the process.
The Mystifying Number Predictor
by Thomas Smotzer, Youngstown State University
This talk will explain the mathematics behind a few known number
guessing magic tricks.
Fun With Probability
by Gary Stanek, Youngstown State University
Activities will focus on several interesting probability questions,
including the famous "Birthday Problem." (You will be able to amaze your friends!) No
prior knowledge of probability is needed.
Keeping Secrets
by Tom Wakefield, Youngstown State University
How do internet sites keep credit card data safe? How do spies communicate
messages without being caught? Cryptography is the study of hiding
information. In this workshop, we will explore various techniques used to
keep messages secret.
What do the Swine Flu and a Murder Mystery have in
Common? (They use the same Math!)
by George Yates, Youngstown State University
Mathematics has played an important role in advancing epidemiology,
genetics, ecology, physiology and other areas of biology. Workshop
participants will explore the dynamics of population growth and the spread
of deadly diseases such as the H1N1 flu. Several activities will be
performed to measure the growth of a population and the spread of a disease.
The participants will also develop models for these populations and compare
their models to data observed during the workshop. Finally, the
mathematical models discussed will be used to determine if a murder
suspect's alibi assures that he could not have committed the crime.
A graphing calculator is not required but may be useful for this workshop.
