Math  Matters  Published by the Department of Mathematics, University at Buffalo

Inside this issue

1. Message from the chair
2. International Activities
3. Myhill Lecture Series
4. Faculty News
5. Student News
6. AMS Meeting
7. "Spotlight On" Feature
8. Freshmen Calculus On-line
9. Alumni Notes- Many Thanks

Message from the Chair...

We had a number of activities that went on in the Spring term in the Department that I wanted to bring up in this space. We successfully hosted a regional conference of the American Mathematical Society this April, and you can read more about it. Holding such an event is work for many people, but it also increases the visibility of the Department within the mathematics community. We also hosted a major set of lectures in our annual John Myhill Memorial Lecture Series (see page 3 for details). A disappointment we faced this spring concerned budget challenges at the university level which did not allow us to do any faculty hiring this year, despite the increasing demands on our program. However, we were able to hire a couple of instructors for the spring term that not only were involved in our instructional commitments, but participated in our research activities. We also got the go-ahead to transfer a senior mathematical statistician over from the medical school so that the Department could start teaching some calculus-based statistics courses now that a department of statistics no longer exists at UB. This coming fall, students entering UB have access to their own computing resources. Although the Department is not directly participating in organized instructional efforts coordinated though the Office of Vice Provost for Instructional Technology, we are running some pilot activities. Among them are making available online resources for calculus, and making streaming video excerpts of a calculus class available to student groups as an extra information resource. We also will have a faculty member, Dr. Bruce Pitman, involved in the educational initiative for the new Center for Computational Research. This center puts UB into the top 10 institutions in the country in terms of computing resources. We have also moved forward with proposals for new interdisciplinary programs with the Computer Science, Economics, and Finance departments. We will give more details as these program proposals get approved. Our Course Preview and other information activities continue to be successful, and one of our students, Ms. Cynthia Rudin, will be honored as Outstanding Sciences Graduate for 1999. As a follow-up to a previous news item, the foundation and framing of the new north campus mathematics building has been completed. A target date for completion is early April, 2000. Mathematics still represents a terrific major, with opportunities to go into many areas after graduation. In a Wall Street Journal publication called Jobs Rated Almanac, this year 250 jobs were rated on six criteria, and mathematician was rated fifth from the top, with 6 of the top 10 occupations rated requiring some mathematics background.

         - Jon Bell, Chair
Previous Message from the Chair
TOC

International Prominence for

UB Mathematicians

Members of the faculty of the Department of Mathematics frequently receive invitations to travel abroad and lecture to audiences at foreign institutions. Most recently, the following faculty have returned from such speaking engagements:

Dr. Lewis A. Coburn visited China in April where he presented two talks: "The Berezin-Toeplitz Quantization", at Peking University in Beijing, and "The Segal-Bargmann Space" at Beijing Polytechnic University. He also spoke at York University in Toronto, Canada on "The Measure Algebra of the Heisenberg Group".

Dr. Thomas Cusick presented a seminar on "A new lattice-based public key cryptosystem" at the University of Waterloo in Ontario, Canada.

Dr. Brian Hassard, who was on a sabbatical in Europe during part of the 1998-99 academic year, presented the following lectures: "Precise Solution of Partial Differential Equations", University of Ulm and "Existence, Uniqueness and Analyticity of the family of solutions of the Planar Benard Problem", Technical University of Munich, both in Germany. He also gave a lecture on "Precise Solution of Partial Differential Equations" in the Theoretical and Applied Mathematics Department, Technical University of Vienna in Austria.

Dr. John Ringland, who was also on a sabbatical, participated in the Special Year on Dynamic Systems at Tsing Hua University in Taiwan. He gave weekly seminars on "Computation in Dynamical Systems".

Dr. Xingru Zhang participated in a Topology Seminar at the University of British Columbia, Canada by presenting a talk entitled "The A-polynomials of knots and Dehn surgery".

M.I.T. Mathematician Speaks at

12^th Annual Myhill Lecture Series

"Quantum Cohomology and Enumerative Geometry" was the featured topic of the 1999 Myhill Lecture Series which took place March 24-26, 1999. The distinguished speaker, Professor Gang Tian from the Massachusetts Institute of Technology, presented three talks over the three day event. Professor Tian, a Simons Professor of Mathematics at M.I.T., holds the Oswald Veblen Prize in Geometry (1996) and is the 1994 recipient of the Alan Waterman Award, the highest honor awarded by the National Science Foundation.

The Myhill Lecture Series was created by the Department of Mathematics in 1987 to honor the distinguished career of Professor John R. Myhill who served on the faculty as full professor, 1966-1987. Topics presented in the lecture series are unrestricted, reflecting Dr. Myhill's mathematical interests; and the event is highly anticipated and well attended by the mathematical community. Professor Mohan Ramachandran served as the organizer for this 12^th successful event.

Invited Lectures Span the U.S.

Mathematicians from the UB Math Department continue to be active speakers, presenting invited lectures on a variety of mathematical topics at universities across the nation.

• James J. Faran, V gave an invited seminar at Rutgers University entitled, "A Synthetic Frobenius Theorem".
• William Lawvere, an invited speaker at the University at Chicago, presented "This Thing is Bigger Than Both of Us" at the MacLane Seminar. He also spoke at the Mathematics Department Colloquium on "Calderon's Complaint, Grothendieck's Gift, and Pedagogical Progress".
• Bruce Pitman presented a number of invited talks:

"Oscillations in a model of tubuloglomerular feedback" Applied Math Seminars at the California Institute of Technology.

• Brian Spencer spoke at the Fall Meeting of the Materials Research Society held in Boston. The title of the talk was "Morphological Instability in Coherently Strained Alloy Films".

Ph.D.'s, M.A.'s Awarded

Two Ph.D's and five Master's Degrees were conferred in the current academic year.

Jun Ling, Ph.D., "A Bound for the First Fundamental Gap". Major Professor: Dr. Yieh-Hei Wan.

Roberto Raimondo, Ph.D., "Compact Toeplitz-type Operators and the Berezin Transform on the Bergman Spaces of the Complex Balls and Multiply-connected Domains". Major Professor: Dr. Lewis A. Coburn.

Rebecca Metcalf, M.A., Project: "Web-based Support for Introductory Calculus: A Critical Assessment and Implementation". Major Professor: Dr. Richard E. Vesley.

Linda Roycroft, M.A., Project: "Numerical Simulation of Strained Film Growth". Major Professor: Dr. Brian Spencer.

Master's Degrees with Comprehensive Examination awarded to Xueting Liu, Gabriela Stanica and Yulai Xie.

UG Math Office Names

1999 Outstanding Graduates

Undergraduate mathematics majors have distinguished themselves both within the department and the university with outstanding academic achievements. Dr. Richard Vesley, Director of Undergraduate Studies, relates that the quality of the 1999 graduating class was of such superior caliber that the decision to name the annual outstanding senior in mathematics was an especially difficult one. The Undergraduate Studies Committee, after much deliberation, announced Ms. Cynthia Rudin as the 1999 Outstanding Senior in Mathematics.

Ms. Rudin, who also earned a B.A. degree in Music and a minor in Computer Science, was chosen by Dean Kerry S. Grant to receive the College of Arts & Sciences Outstanding Senior Award in Science and Mathematics. She also received the 1999 State University of New York Chancellor's Award for Student Excellence which was presented to her by President William Greiner at the annual Commencement on Sunday, May 16. Ms. Rudin was also named as the Outstanding Senior in the Music Department.

Cynthia has an impressive accumulation of academic achievements at UB which include a number of research papers, and summer internships at Haystack Observatory at M.I.T. and the Supercomputing Program for Undergraduate Research (SPUR) at Cornell University. She has to her credit a prestigious list of scholastic awards which include a Goldwater Scholarship, Phi Beta Kappa, Golden Key Award, Grace Capen Award, Phi Eta Sigma, and numerous others. Ms. Rudin will attend Princeton University for graduate studies in Applied and Computational Mathematics.

Mr. Joshua Berne, a double major with Computer Science, was one of the three finalists for the department's outstanding senior award. He was awarded Outstanding Senior in Computer Science. Josh will be attending the University of Chicago for graduate study in mathematics.

Mr. Michael Buice, a double major with Physics, was also a math department finalist. He received the Outstanding Senior Award in the Physics Department. Michael has received a National Science Foundation Graduate Fellowship and will be attending the University of Chicago for graduate study in Physics.

Montague Award Recipient Named

by UG Studies Committee

Mr. Ian Blumenfeld, a junior in the preparation for graduate study program in mathematics has been named as the 1999 Harriet F. Montague Award winner. Ian carries an overall GPA of 4.0 and has already completed advanced coursework at the graduate level in mathematics. Mr. Blumenfeld best exemplified the intent of the award, which is "to be given to a mathematics student at the junior level who shows intellectual and creative promise in mathematics."

Department Announces Recipients of

Gehman Memorial Scholarship

Mr. Zachary Mekker, undergraduate major, and Mr. Anurag Agarwal, Ph.D. student, are the 1999 recipients of the Harry Merrill Gehman Memorial Scholarship. This annual award is given to students at the undergraduate and graduate level, who exhibit academic achievement and whose intent is to pursue a career in teaching mathematics. Mr. Mekker '99, has completed his major in the mathematics concentration for teacher certification and the Teacher Education Program: BRIET which will qualify him for provisional NYS teacher certification. Mr. Agarwal is completing a Ph.D. dissertation under the supervision of Professor Thomas Cusick.

Two Math Majors Rank in Top 500

in Math Competition

Joseph N. Lillo '00 and Joshua M. Berne '99, achieved ranking in the top 500 in the 59^th William Lowell Putnam Mathematical Competition. In the December 1998 competition, 2581 students from 419 colleges and universities in both Canada and the U.S. participated. The Putnam competition is considered to be the most rigorous test of mathematical reasoning ability for college students. Solving the problems requires incredible ingenuity and insight on the part of the contestant.

The problems and solutions for the Putnam exam are regularly published in the American Mathematical Monthly along with lists of the top ten teams and complete details of the competition.

Math Graduate Student Honored

by UB Graduate School

Mr. Anurag Agarwal, a Ph.D. student completing his dissertation under Dr. Thomas Cusick, was honored for exceptional competence in teaching at a ceremony hosted by the Graduate School. Dr. Kerry S. Grant, Dean, College of Arts and Sciences presented the Graduate Student Excellence in Teaching Award, a check and a certificate, to Mr. Agarwal. The ceremony, held at the Center for the Arts on March 26, honored 15 award winners and 5 students who achieved honorable mention.

The Graduate Student Excellence in Teaching Award was established jointly by the Graduate School and the Graduate Student Association. Criteria for selection of winners include teaching competence, mentorship, academic standards and requirements, and professional growth.

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AMS @ UB The 943rd Meeting of the American Mathematical Society

Michèle Audin from L'Université Louis Pasteur

Alexander A. Voronov from Michigan State University

Russel E. Caflish from UCLA and E. Bruce Pitman, SUNY at Buffalo

Gregg J. Zuckerman from Yale University

Mathematicians attending the AMS meeting in Diefendorf Hall, pause for a group picture during a refreshment break between sessions
Dr. David J. Triggle SUNY Distinguished Professor, Provost and Dean of the Graduate School SUNY at Buffalo

150 Talks Concentrate an Abundance of Mathematics at UB Math Department

The recent AMS Meeting hosted by UB's Department of Mathematics provided attendees with a wide and abundant choice of mathematical topics for discussion. UB's Provost, Dr. David Triggle welcomed the participants prior to the first of four invited addresses. The AMS Invited Lecturers were Michèle Audin from l'Université Louis Pasteur, who spoke on Integrable systems and spaces of curves; Russel E. Caflisch from UCLA, who spoke on Atomistic, continuum and bulk models for epitaxial growth; Alexander A. Voronov from Michigan State University, who spoke on Operad Theory and Some Applications; and Gregg J. Zuckerman from Yale University, who spoke on Harmonic Algebra.

One of the few disappointments of the meeting was the unexpected, unfortunate and yet unavoidable absence of Jeffrey H. Smith from Purdue University, who had also been invited to give a plenary address.

Nine special sessions were held over the two day period: Knots and 3-Manifolds; Representations of Lie Algebras; Complex Geometry; Integrable Systems; Smooth Categories in Geometry and Mechanics; Thin Films: Solid and Liquid; Combinatorics and Graph Theory; Mathematical Physics; Operads, Algebras and Their Applications.

UB Department of Mathematics Well Represented at AMS Meeting

UB faculty and a few graduate students were among the presenters in the approximately 150 scientific talks at the 943^rd AMS Meeting. Department Members presenting papers were:

Faculty:

Dr. James J. Faran, V, A Synthetic Approach to Characteristic Cohomology

Dr. F. William Lawvere, Smoothness of Cubical Splines

Dr. Adam S. Sikora (Visiting Instructor), Kauffman Bracket Skein Module at 4^th Root of Unity

Dr. Xingru Zhang, On Simple Points of Character Varieties of 3-manifolds

Graduate Students:

Cezar Joita, On the Projection of Runge Domains

Florin F. Nichita, On some Duality Theorems, (a paper co-authored with Sorin Dascalescu, University of Bucharest and Dr. Samuel D. Schack, SUNY at Buffalo)

Department members involved as Special Sessions Organizers were:

Dr. Mohan Ramachandran, Dr. Thang T.Q. Le, Dr. William W. Menasco, Dr. Jonathan Dimock, Dr. F. William Lawvere, Dr. E. Bruce Pitman and Dr. Brian Spencer

Department Earns Praise from AMS

AMS Associate Secretary, Lesley Sibner said in a message to Dr. Jon Bell, Department Chair, and Dr. James J. Faran, V, Associate Chair and Meeting Organizer for the Department of Mathematics, "... The local organization and participation was truly extraordinary ... Everyone I spoke with had a wonderful time and found it extremely valuable professionally. ... We will be delighted to look forward to other meetings at Buffalo in the future!"

Spotlight On.. by Dr. William Menasco

William Menasco is a full Professor in the Department of Mathematics. He came to UB in 1984 after a postdoctoral position at Rutgers University. He received his Ph.D. from the University of California at Berkeley in 1981. Professor Menasco's research interests are in knot theory, 3- & 4-manifolds, and geometric topology. His most celebrated work is a collaboration with Morwen Thisthlewaite (University of Tennessee) proving the Tait Flyping Conjecture for alternating knots and links.

A Circular History of Knot Theory

In the nineteenth century physicists were speculating about the underlying principles of atoms. In 1867, Lord Kelvin put forward a comprehensive theory of atoms which, through heuristic reasoning, seemed to explain several of the essential qualities of the chemical elements. Kelvin's theory conjectured that atoms were knotted tubes of ether. (To a topologist a knot in 3-space is any closed loop having no self-intersections and a link is any collection of non-intersecting closed loops.) The topological stability and the variety of knots were thought to mirror the stability of matter and the variety of chemical elements.

What the physicists abandoned, intrigued mathematicians, then and now, and the basic question is still the same: How do we tell when two knots are isotopically the same? (Research tip: Sometimes the most interesting problems can be found in someone else's trash.) This failed atomic theory also left in its wake the riches of Tait's tabulation -- 163 knot projections -- and a rudimentary understanding of isotopic sameness in terms of how one projection could be continuously manipulated to look like another. This understanding of projection manipulation was summarized in a set of conjectures for knot projections, the famous Tait Conjectures.

To attack the Tait Conjectures and the basic question of sameness of knots, topologists developed knot invariants. An early example of a successful knot invariant is the Alexander polynomial, discovered by J. W. Alexander in 1927. The Alexander polynomial for the knot labeled 3₁ (the trefoil) is -t²+t-1 and the polynomial for 4₁ (the figure-eight) is -t²+3t -1. Since these two polynomials are different we know their associated knots are different. The Alexander polynomial was remarkable for how successful it was in distinguishing the knots in Tait's orginal table and it gave witness to how thorough a researcher Tait was. (Historical note: The last of the few duplications in the Tait/Little table was found in 1974 by Kenneth Perko, a New York lawyer and part-time topologist, while he was manipulating loops of rope on his living room floor. If a lawyer can do research in knot theory, it can't be that hard.) Unforunately, there are many knots with equivalent Alexander polynomial that can be shown to be isotopically different through the uses of other invariants.

So the search was on for more sensitive knot invariants that would detect when two knots were different. This led to alternate understandings of the notion of sameness. In particular, to a topologist there is no difference between the loops representing 4₁ and 5₁. What is different is the space away from these loops, that is the complement of the knot. Two topological spaces are homeomorphic if there is a bijective invertible continuous function that maps one space to the other. Thus, we have an alternate notion of sameness: If two knots/links have homeomorphic knot/link complements then they are homeomorphic knots/links. Now, it would seem that homeomorphic sameness would be weaker than isotopic sameness. And in fact, for link complements it is -- there exist examples of links that are not isotopic, but have homeomorphic complements. But for knots a seminal result of Cameron Gordon and John Luecke showed that two knot are homeomorphic if and only if they are isotopic. In the vernacular of the knot theorist, a knot determines its complement.

Understanding that the principle object of study is the knot complement places knot theory inside the larger study of 3-manifolds. A 3-manifold is a space which locally (assume you are near sighted) looks like standard xyz-space and knot complements are readily seen as examples of 3-manifolds. It was through the study of 3-manifolds that in the 1970's knot theory began returning to its ancestoral roots in physics. To understand this we have to flashback to the 1860's work of Bernhard Riemann. Riemann was interested in relating geometric structures to the forces in physics. Building on Gauss' work, Riemann investigated three different geometric structures for 3-dimensional spaces -- elliptic, euclidean, and hyperbolic. (Einstein's Theory of Relativity was built on Riemannian geometry.) Each of these distinct structures can be characterized by the behavior of triangles in planes. In elliptic 3-space, the interior angles of a triangle in a plane have a sum greater than 180 degrees. In Euclidean 3-space, the sum is 180 degrees and in hyperbolic 3-space the sum is less than 180 degrees. In 1978, William Thurston established sufficient conditions for when a 3-manifold possesses a hyperbolic structure. Surprisingly, except for a well understood subclass of knots, all knot complements possess a complete hyperbolic structure. (The beauty of Thurston's work is captured in the video Not Knot that is distributed by the American Mathematical Society and has been frequently viewed at Grateful Dead concerts.)

Thurston's work on hyperbolic structures firmly re-established knot theory's connections with physics. In the 1980's, through some totally unexpected routes, knot theory made further connections with its ancestral roots. In 1987 Vaughan Jones discovered a totally different polynomial invariant from that of Alexander using the theory of operator algebras. Within a short period of time, more than five new polynomial invariants generalizing the Jones polynomial were discovered. (One of these polynomials was simultaneously discovered by five different mathematicians and its name is an acronym of their last names -- HOMFLY.) Moreover, Jones' polynomial quickly led to proofs that established all of Tait's original conjectures on knot projections.

With this proliferation of new polynomials it was natural to ask whether any of these invariants had a natural extensions to all 3-manifolds. Two facts worked in favor of having such extensions: 1) all 3-manifolds can be describe in terms of knots and links via an operation called Dehn surgery; 2) there exists a set of moves, the Kirby calculus, that allows one to move between differing Dehn surgery descriptions of the same homeomorphic 3-manifold. Using the Kirby calculus as a means to generalizing the polynomial invariants, Edward Witten, a theoretical physicist, proposed new invariants for 3-manifolds. His invariants came out of the theoretical area of physics know as quantum field theory. These new invariants can be realized as certain averages of link polynomials obtained from a given Dehn surgery representation of the manifold.

Starting with the flawed theory of Kelvin's knotted vortex to the work of Thurston, Jones and Witten, knot theory has circled back to its ancestral origins of theoretical physics.

Math Department On-Line for

Implementation of Access 99

The Department of Mathematics is taking advantage of the University's Access 99 program to enhance significantly its core calculus course, MTH 141. Under the Access 99 program, each student is to have access to a computer with certain basic capabilities, including use of the internet. Several special features of MTH 141 will be introduced next fall involving the internet.

First, a new text will be used, which has an associated internet home page maintained by the publisher, Prentice-Hall. Students will find animated and interactive examples of text material, supplemental true-false tests, and other material on this web site.

Second, an electronic help facility is being set up, under which a graduate student will be available during certain evening hours to help students by answering e-mailed questions about specific problems.

Third, an edited and compressed version of highlights from lectures in the course will be put in video on the net, particularly in aid of students who are having difficulty with the course. The instructor featured in these lectures will be Professor Ann Piech.

The second and third features are being supported by the University's technology initiative; the second through a grant by the Vice-Provost for Technology, and the third through aid from the Computing Node in Science and Engineering.

Many thanks to the following UB alumni and friends who were recent donors to the University at Buffalo Foundation Mathematics Resources Fund (UBF-Math Fund).

Your generous donations in this general fund help finance student awards and help support departmental teaching and research missions. If you prefer, your donation can be marked for special purposes, such as the Harry Merrill Gehman Endowment Scholarship and the Harriet F. Montague Award.

Ms. Judith F. Alter ('73)

/Matching gift by TRW Foundation

Mr. Robert L. Cooper ('76)

Mr. Kenneth S. Murchison, Jr. ('91)

Dr. Clifford H. Spiegelman ('70)

Mr. Frank H. Sterzinar ('85)

Mr. Lawrence B. Stone ('76)

Dr. Marlene G. Zimmerman ('78)

/Matching gift by Lockheed Martin Corp.

Gehman Memorial Endowment Donation

Dr. Joseph E. Kist ('52)