Project VIGRE for Advanced Math Education

The National Science Foundation has awarded the University of Arizona in Tucson a $3.5 million grant over the next five years to improve advanced mathematics education under an ambitious project called VIGRE. The grant will be shared by the Department of Mathematics and the Program in Applied Mathematics. The acronym VIGRE stands for 'Vertical InteGration of Research and Education' and in this context, vertical means across academic ranks—faculty, graduate students, and undergraduates.

Each year, the project will support 12 graduate students, two postdoctoral fellows, and 20 undergraduates. Undergraduates will have opportunities to pursue research on topics like cryptography, mathematical modeling, the mathematics of fluids, and number theory. For graduate students, the grant provides funding so they can focus more on doing high-level research in current mathematical fields and improve their professional and communication skills.

VIGRE will also aim for “horizontal integration” with other areas of research and other departments on campus and also going beyond geographical and institutional boundaries to provide support to other organizations involved in mathematics education. The program includes research training at the highest level, a spectrum of teaching and outreach activities, and a wealth of professional development opportunities, all combined in a carefully designed and collegial framework.

SToMP

“Receiving the information we need is not the problem­s--sorting it and deciding what is useful without being totally overwhelmed is the primary challenge,” says Robert Ghrist, an associate professor of mathematics and a researcher at the Coordinated Science Laboratory at Illinois. He continues, “Imagine, for example, that you have thousands upon thousands of mobile video cameras and one of them catches something important. What do you do now? And, to make it interesting, let's assume that you don't have GPS, range finders, orientation sensors, or a compass. What now?”

The answer to this “What now?” question is the goal of a new research project called SToMP (short for “Sensor Topology & Minimal Planning”) funded by the Defense Advanced Research Projects Agency (DARPA). This $7.98 million project will cover three phases of development over four years. As a multi-year, interdisciplinary project, SToMP will implement global topological tools to dramatically reduce the amount of sensing complexity needed to solve problems across a variety of Department of Defense applications involving sensor networks, autonomous systems, and configurable sensor platforms.

Public video cameras, RFID-enabled warehouses, and mixture monitors in a chemical plant are all examples of extensive sensor networks that may one day feed enormous amounts of information into self-monitoring systems that will pluck useful kernels from the chaff. In short, SToMP is all about integrating local readings from many sensors into a global picture and bring out important information out of that mass of seemingly random data. One potential application could be to detect holes in the coverage of a cell phone network. Topology mathematics can map the twists and curves of the holes. Once topology has captured sensor information showing where holes exist, it can map them out in a way that provides the guidance needed to fix them. Another such example could be the proper use of a grid of, say, 1,600 motion sensors, each occasionally offering irrelevant feedback such as leaves dropping from trees. In a display of such continuous feedback, a human eye may miss intruders proceeding through a monitored area. But topology algorithms resulting from the research will spot them,

Another feature of the project is its emphasis on minimality. The "minimal planning" part of the project reflects its goal of building the smallest sensor network necessary to get a job done, as opposed to overinvesting in sensor placement.

Institutes involved in project SToMP are University of Illinois, Bell Labs/Lucent, Arizona State University, Rochester University, Carnegie-Mellon University, Melbourne University, the University of Pennsylvania, and the University of Chicago.

History: Mathematics of Mapping

"The act of mapping is as profound as the invention of a number system ... The combination of the reduction of reality and the construction of an analogical space is an attainment in abstract thinking of a very high order indeed ..."

- Arthur Robinson, "Early Thematic Mapping in the History of Cartography"

In 1569, the Flemish Cartographer Gerardus Mercator first tried to create a map of the world on a flat surface, as opposed to a globe and the mathematical challenge he faced seemed to be unsurmountable: How can a curved surface be accurately represented on a flat surface? In the end he came up with equations of projection which enabled cartographers to create charts which sailors could easily navigate. With that huge benefit of those years came the very uncomfortable feeling of visual distortion. Mercator's projection method showed, for example, Greenland to be larger than South America, when, in fact, it is approximately of the size of Mexico.

After about 400 years a renowned cartographer named Arthur Robinson first took up that challenge of bringing some sense of truth in that totally mathematical representation of Mercator. In his procedure of projection he followed a reverse direction - He started with a sketch of a map that, in both shape and size, appeared to be more accurately representing the world than that coming out of Mercator's method. Thereafter he calculated the mathematical representation of this map. The poles appeared to be much less distorted in Robinson's map.

His projection method is used by maps from the company Rand McNally who commissioned Robinson to work on this problem in 1963.

Robinson died on October 10th, 2004 at the age of 89.

Numerical Mathematics Consortium Updates

The Numerical Mathematics Consortium today announced the latest revision to a technical specification introduced earlier this year that defines an open mathematics semantics standard for numerical algorithm development. This update includes newly ratified functions from classes that include polynomials and vector analysis. In addition to the new function definitions, the consortium resolved significant technical issues that simplify ratification of new functions.

The industry for mathematical software has been lacking a unified and standardised policy for a long time. So, people involved in making and using these softwares are coming together to formulate a consistent policy for numerical programming. The founding companies of the Numerical Mathematics Consortium -- which include INRIA, Maplesoft, Mathsoft recently acquired by Parametric Technology Corporation, and National Instruments -- established the organization in 2005 to create a specification that facilitates reuse and portability of numeric algorithms. To reach this goal, the organization is initially focusing on standardizing a core set of mathematical functions that can be used in a wide variety of application areas such as industrial control, embedded design and scientific research, as well as be easily reused by researchers and developers in industry and academia.

The newly resolved technical issues address practical topics related to algorithm design and compliance with the standard. The resolved issues cover topics such as when to specify vector orientation, how to support vectorization, what it means to be compliant and how to choose a semantic representation. Settling these technical issues provides guidelines that improve the rate of progress for new function adoption.

The latest revision of the standard, which documents the newly resolved technical issues and ratified functions, is available for download from the Numerical Mathematics Consortium Web site, http://www.nmconsortium.org/. This site provides additional information about the standard and offers information about how organizations and individuals can get involved.