Mathematics Symposium: "Mathematics Awareness Month at UNO"
Mark J. Ablowitz,
"Waves, Waves, Waves, ... Waves Everywhere!"
Waves are common. Water waves, ocean waves, light waves … so typical but yet so complex. Waves and their dynamics
are widely studied by scientists. In fact, mathematicians are in an advantageous position. Even though the underlying physical
descriptions may be different, in many cases the dynamics satisfy the same or very similar equations! In this lecture we will take
a tour of some wave phenomena. This will include: water waves in channels, tsunamis, great undersea waves, waves in fiber
optics, and waves constructed with only “0’sand 1’s” (the “game of life”). Although these systems may seem very
different, we will expose their important common elements. The lecture will be general in nature, without any advanced
mathematics required. At times some equations will be presented as a vehicle to explain and illustrate the fundamental
similarities between the systems.
"Geographic Information Science for Studying Oceans and Complex System Dynamics
Our scientific understanding of oceanic, climate, and lithospheric processes
is depended upon mathematics, spatial modeling, and enabling technologies of remote
sensing, geographic information system technology, and scientific visualization.
Geographic information science represents the integration of developments in these and
other disciplines, such that scientists are able to study the spatio-temporal scale dependencies
of processes and complex feedback mechanisms. This talk will introduce important
concepts of geographic information science and provide example of how mathematics
and technology is used to collect data , extract environmental information, and model
complex dynamic systems. Some examples include assessment of oceans and climate,
thematic mapping, and topographic evolution.
"Modeling Fossil Plant Form and Function"
Biologists and mathematicians have been attracted to the geometric features and repeating patterns of plants and algae for centuries. These features are linked to the developmental processes of plants, which are controlled by genetics and the surrounding physical world. Mathematical principles can be used to model and quantify plant form, which can be especially useful when studying ancient plants with no modern representatives. This presentation will explore three examples from the field of paleobotany; studying the biomechanics of fossil leaves using finite element analysis, using elliptic Fourier analysis to quantify leaf shape for identification, and the use of fractals in characterizing plant architecture.
"The Logarithmic Spiral and the Golden Mean"
The oceanic chambered snail Nautilus has the shape of a logarithmic spiral with golden mean proportions. First, the meaning of these terms is explained. Then the mathematics of this relationship are discussed in a simple geometric manner.
"Solitary Waves in Shallow Water - A Nonlinear Phenomenon"
When John Russell observed a mass of water rolling along in the Union Canal near Edinburgh, Scotland in 1838, the nonlinear science of solitons was born. I will talk about the Korteweg-de Vries equation which encodes such a wave and the unusual behavior of the solitary wave solutions it possesses. The mathematics will be mostly informal with lots of pictures.
"Creatures of the Ocean: Will They Become Extinct?"
In this talk we will show how a special type of stochastic processes, called Branching Processes, can be used to model the perpetuation of species. Extinction probabilities will be introduced and a formula for their computation will be determined. We will illustrate by an example how this formula can be used to model the chances of extinction/survival.