Professional mathematics since 1984
I am on family leave for the Spring 2017 semester, taking care of my baby daughter. Please contact me by email.
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Direct involvement in the education of mathematics majors is extremely important to me. I have frequently taught such courses as Differential Equations, Linear Algebra, and Advanced Calculus. During 1999 and 2000, I taught the Foundations in Mathematics course (Math 2600) twice. During the Fall 1995 term, I taught the senior seminar, Math 5900, with the topic Introduction to Hilbert Spaces. I taught the seminar again in 2001 and in 2004 with Mathematical Cartography as the theme. More recently, Al Marrero and I have co-taught the seminar under the title Research Experience, where our goal has been to introduce every math major to the world of mathematical research through reading, studying, and explaining articles in real mathematics journals and then formulating and working to solve original problems inspired by the reading. More recently, I developed my interest in the mathematics of medical imaging processes into both a course and a text (The Mathematics of Medical Imaging: A Beginner's Guide, 2nd edition, Springer, 2015. Among the graduate courses I have taught are Linear Algebra, Advanced Calculus, and Real Analysis. I also recently designed and ran a course in Applied Linear Algebra, in which we read journal articles concerning contemporary applications of linear algebra to problems in data compression, reduced-rank approximations to high-dimensional problems, ratings and rankings of sports teams and web pages, information retrieval, and asset-pricing.
In an attempt to make mathematics more interesting and current to liberal arts students, I have created my own version of Math 1210 (Mathematical Concepts) in which we discuss, read about, and write about the mathematics of symmetry, modular arithmetic and the Chinese Remainder Theorem, the wonderful world of prime numbers, the powerful cryptosystems that are used to protect private information on the Internet, and the twists and turns of geometry involved in making flat maps of our round world.
Finally, both as a change of pace and to broaden my own horizons as a professor, I have taught the Core Humanities Seminar in Modern Thought (now known as the Augustine in Culture Seminar - ACS) through the Honors Program, focusing on the influence of scientific thinkers such as Descartes, Pascal, and Einstein on our modern view of the world.
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My original area of research is the theory of operators on Hilbert spaces which was once described by Louis de Branges (in the October 1994 Bulletin of the American Math. Society) as "the field of mathematics that has the strongest interaction with the scientific and technological developments which are characteristic of the twentieth century." Beginning with my Ph.D. thesis on problems of best approximation from compact perturbations of nest algebras, my research focused mainly on algebras of operators and specifically on nest algebras. In the mid-1990s, I wrote two papers which together provide complete characterizations in algebraic (as opposed to topological) terms of the Bourgain algebra and higher order Bourgain algebras associated to a nest algebra.
Since the mid- to late-1990s, my scholarly efforts have become more diversified. First, inspired by the Cartographiometry course I developed with Dr. Elaine Bosowski, I have been led to explore the connections between mathematics and cartography beyond the basic level we had introduced in our course. First, Elaine and I wrote a paper (appearing in the journal Cartographica) on the application of mathematics to certain elements of map analysis that we felt would be accessible to most professional cartographers. (The date of the issue is misleading. Though labeled as "Winter 1997", it actually appeared in late 1998, after Dr. Bosowski's untimely death.) Next, I contributed an article to the SIAM Review based on material from the Cartographiometry course. Taking a somewhat different tack, I then explored how a cartographic point of view can provide insight into mathematical problems. For instance, my Polya Award-winning paper Conformality, the exponential function, and world map projections (published in The College Mathematics Journal, volume 32, November 2001) takes a cartographic approach to the problem of constructing conformal mappings of surfaces of revolution into the plane and demonstrates without the use of any complex analysis the important mathematical fact that the complex exponential function is conformal. I believe this point of view makes the subject of conformal mapping more intuitive and more accessible to college mathematics teachers and their students. This work culminated with the publication, in 2002, of my book, Portraits of the Earth: A Mathematician Looks at Maps, by the American Mathematical Society.
Another main direction of my research has been a collaboration with my colleague Al Marrero that began in the autumn of 1996, when we read and mulled over several papers of mutual interest. Since then, we have published eight papers together.
Also, as described above, I developed an interest in the mathematics of medical imaging processes that led me to write the book The Mathematics of Medical Imaging: A Beginner's Guide, published by Springer (2nd edition, 2015 now available).
Most recently, I have been collaborating with Jesse Frey, a brilliant young statistician in my department. We have worked together on several problems related to the use of judgment ranking in the estimation of various population parameters. We have published four papers together.
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I was honored to receive the 2002 George Polya Award for Expository Writing, awarded by The Mathematical Association of America. The award was made on August 2, 2002, at MathFest 2002 in Burlington, Vermont.
Click here for a photo of me accepting the award. (Thanks to Dennis DeTurck for taking the photo.)
**2002 MAA Polya Prize for expository writing**
Conformality, the exponential function, and world map projections, College Math Journal, Volume 32, Number 5, November 2001, pp. 334--342. pdf file
The William Lowell Putnam Mathematical Competition, run by the Mathematical Association of America, is a prestigious competition for undergraduate students of mathematics. Begun in 1938, the competition is designed to stimulate a healthy rivalry amongst the colleges and universities of the United States and Canada.
The competition is held each year on the first Saturday in December and is open to regularly enrolled undergraduates in the United States and Canada who have not yet received their degrees. The five highest ranking individuals are designated as Putnam Fellows and their departments receive cash awards. Each year, one Putnam Fellow receives a scholarship to attend graduate studies at Harvard or Radcliffe. The Elizabeth Lowell Putnam Prize is awarded for "particularly meritorious" performance by a woman contestant. Locally, the Department of Mathematics & Statistics at Villanova offers a prize of a one-year student membership in the Mathematical Association of America to the highest ranking Villanovan contestant.
Participating in the Putnam Competition is an illuminating experience -- please join us this year!
Villanova Putnam Hall of Fame 1988-present
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Pi Mu Epsilon is a national honorary mathematics society. Each year, the best junior and senior (and occasional brilliant sophomore) mathematics majors are inducted into Villanova's Chapter. For more information, see me or the Department Chair. Visit the National Office of Pi Mu Epsilon's official website.
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|Exploration of the geometry involved in making maps of the world led to my collaboration with Dr. Elaine F. Bosowski, late of Villanova's Geography Department. Together we developed an introductory level interdisciplinary Math/Geography course which we dubbed "Cartographiometry". In the course, which was offered in 1996 and 1997, we examined the historical relationship between mathematics and geography, especially cartography, from Eratosthenes to the present. We discussed the general mathematical problems posed by making flat maps of the sphere and introduced in this context such basic concepts of non-Euclidean geometry as curvature and geodesics. From a cartographic perspective, we investigated the uses of maps and the evolution of map-making as well as the various elements of Cartographic Communications Systems. The students also learned the mathematical and cartographic tools needed for the construction of maps having specific desired properties, such as preserving areas or shapes of land masses, and then actually constructed these maps from scratch in a laboratory setting using the computer algebra system Maple as an aid.||
In memoriam Elaine Bosowski (1952-1998), stricken by cancer in the prime of her life. The joy, inspiration, and love she brought to everyone she knew and everything she did remain vibrant and alive.
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Outside of mathematics, I have a variety of interests that keep me going. Foremost among these are my sons, Max and Simon. They are amazing people and a ton of fun to hang out with. Max was in two bands while a student at Alfred University: including Steamwhistle Cartle, whose first gig was to open for Everclear at Alfred University on November 20, 2009. His latest group is Tektonic. Simon recently graduated (class of 2014) from Dickinson College, where he was a mathematics major (!) and co-captain of Dickinson's Ultimate team, the Jive Turkeys.
I spent over a quarter of a century playing Ultimate (frisbee), which I first learned as an undergraduate at Brandeis University. I co-captained the championship team in the first annual Philadelphia Ultimate Summer League back in 1985 and was thrilled to be on the championship team once again in 1997. In 1990, in Oslo, Norway, I was on the U.S. National team that won the World Championships in the Masters Division (30 years old and up). I retired from club play back in 1993, but still keep a disc in my office, just in case the opportunity arises! For local Ultimate info, contact the Philadelphia Area Disc Alliance (PADA); For the national and international scene, check out the USA Ultimate web site.
I got talked into running the Philadelphia Marathon in November of 1996, even though I hadn't really trained properly for such an event. I finished completely exhausted but with a respectable time of 3:20:51. Mostly I became inspired to take up running as a sport, rather than as a means of keeping fit, for the first time in my life. Some of my personal bests are: 1:04:27 for 10 miles (Broad Street Run, Philadelphia, PA, 1997); 30:45 for 5 miles (Red Rose Run, Lancaster, PA, 1997); 1:21:59 for the half-marathon (Philadelphia Distance Run, 1997); 18:12 in the 5K (Bryn Mawr Shut Up and Run, 1997); 38:55 in the 10K (Philadelphia Zoo Run, 1999); 3:12:21 in the marathon (Philadelphia Marathon, 1997). I also ran the 102nd (1998) and 103rd (1999) Boston Marathons. I remain a member of the Bryn Mawr Running Club.
Alas, I can no longer run, a long-term consequence of having blown out my ACL back in 1986, playing Ultimate. (Insufficient cartilage remains in my right medial meniscus to support the constant impact of running.) So I have taken up road cycling with enthusiasm, logging at least 5000 miles every year since 2008 and reaching 6000 miles or more four times. 2015 was my record year, with 7,059 miles of road riding logged. (That's 11,358 kilometers -- well more than the distance from the equator to the pole along a meridian! It's also very nearly 38 light-milliseconds.) Click here to see a photo of my ride.
My commitment to working for peace, justice, and equality began in the late 1960s and early '70s when I accompanied my mother to anti-Vietnam War protests and my family supported the United Farm Workers' grape boycott. Subsequent involvement in the anti-apartheid, anti-nuclear weapons, and Central American solidarity movements, among other things, has strengthened my convictions and my confidence in the power of mass action as a force for democracy and progressive change.
In June, 1996, I participated in the first (and only, as it turned out) Philadelphia-to-D.C. AIDS Ride. Along with 1900 other everyday people, I bicycled 235 miles from Philadelphia to Washington, D.C., over seemingly endless hills, across rivers, in and out of valleys, through the beautiful countryside of Pennsylvania and Maryland, to heighten awareness of the urgency of the AIDS epidemic and to raise money for local service organizations providing vitally needed assistance and information to people with AIDS, their friends, lovers, and families. It was an exhilarating experience filled with the spirit of dedication, courage, commitment, and love that each and every rider and support crew member brought to it.
I also enjoy hiking, cross-country skiing, and backpacking. During a sabbatical stay in England in 1995, I developed a fondness for the British countryside (not to mention afternoon tea and orange marmalade), and especially the English Lake District.
Speaking of England, check out the cryptic crossword puzzles in The Times (London) -- as addictive as they are inscrutable!
Apart from being a rank amateur on several musical instruments (several brass instruments, penny whistle, guitar, and electric bass), I have a decent collection of record albums (and a turntable to play them on!), CDs, and tapes, mainly jazz, folk, and "alternative" rock.
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