his month, in-cites talks with Dr. Dietmar Bisch, the Chair of the Vanderbilt University Department of Mathematics, about their highly cited work in this field. According to a recent analysis of the Essential Science Indicators database, Vanderbilt University now ranks in the top 1% in terms of total citations in the field of Mathematics. Their current record in this field includes 368 highly cited papers cited a total of 1,334 times to date. In addition to being the Department Chair, Dr. Bisch is also the Director of the Noncommutative Geometry and Operator Algebras (NCGOA) research group at Vanderbilt.

The Department of Mathematics at Vanderbilt has grown significantly in number and quality over the past 15 years. During the 1991-92 academic year, the Department of Mathematics had about 30 full-time faculty (tenured, tenure-track, and non tenure-track); for the 2006-7 academic year, we have about 50 full-time faculty (tenured, tenure-track, non tenure-track).

Several of the recent appointments were at the senior level. We now have two Distinguished Professors (Connes, McKenzie) and four Named Chairs (DiBenedetto, Olshanskii, Sapir, Schumaker). Also, three new research groups in noncommutative geometry and operator algebras, constructive approximation theory, and biomathematics came into existence during this time. High-profile mathematicians were hired in these areas of research. (See related links below^1,2,3 for more information.)

The department has a very active visitor and post-doctoral program. During the academic year 2006-07 we have 14 post-doctoral faculty and many (short-term and long-term) visiting professors in residence. The number of visitors to the department has increased dramatically over the last 10 years, mostly due to the fact that several top mathematicians were hired in recent years. The department now has seven invited speakers who spoke on their work at past International Congresses of Mathematicians among its faculty (Bisch, Connes, Kasparov, Olshanskii, Sapir, Yu, Jonnson [Professor Emeritus]).

Dollar amounts generated by external research grants in the Department of Mathematics at Vanderbilt University have grown by a factor of 5 over the last 10 years. Roughly 75% of the current faculty receives research support from the NSF, NSA, or NIH.

Mathematics is playing an increasingly important role in other areas of science, including biology, physics, economics, finance, and engineering. Vanderbilt has researchers in all these areas which have contributed to Vanderbilt’s visibility in terms of mathematics publications.

There are six research groups within the Department:

• Biomathematics
• Constructive Approximation, Computational Analysis (including spline theory, wavelets and signal processing, etc.)
• Noncommutative Geometry & Operator Algebras
• Algebra & Logic
• Topology & Group Theory
• Graph Theory & Combinatorics

Research vignettes for some of the faculty are presented below.

Dietmar Bisch’s current research is in operator algebras, more precisely the theory of subfactors. A subfactor can be viewed as a mathematical object encoding symmetry of a mathematical or physical problem, much like a group does. However, a subfactor is an infinite dimensional, highly noncommutative object, and the symmetry it represents is more general than group symmetry. Operator algebra methods can be used to decode this symmetry and one obtains finite dimensional data in this process, which can be described combinatorially and computed numerically. There are numerous fruitful connections of the theory of subfactors to statistical mechanics, algebraic quantum field theory, low-dimensional topology, representation theory, and other areas of mathematics and physics.

Philip Crooke’s primary area of research is biomathematics. Most of his collaborators come from Vanderbilt’s School of Medicine, although he does have collaborators at the University of Minnesota and the University of Pittsburgh. Presently, he has research projects that are addressing important questions in medicine. For example, one group is developing a mathematical algorithm that can predict different autoimmune diseases using gene expression data. Another group has created a mathematical model to predict the risk of breast cancer in women using easily obtained genetic data that is based on the metabolism pathway of estrogen. As a member of the Vanderbilt Integrative Cancer Biology Center, his team is perfecting mathematical models of solid tumor invasion.

Glenn Webb’s current research projects involve: (1) Mathematical models of tumor invasion—this research concerns the dynamic behavior of proliferating and quiescent tumor cell populations in spatial environments; (2) Mathematical models of bacterial evolution—this research concerns the role of contingency genes in the evolution of strep throat bacteria subject to mutation and selection in an infected host; (3) Mathematical models of antibiotic resistance—this research concerns the development and control of antibiotic resistant bacterial strains in hospital patient populations; (4) Mathematical models of prion replication—this research concerns the evaluation of hypothetical mechanisms involved in polymerization processes of prion population growth and the pathogenesis of transmissible bovine spongiform encephalopathies; (5) Mathematical models of the pandemic influenza—this research concerns the role of quarantine measures in the transmission of SARS and other viral epidemics; and (6) Mathematical models of age, size, and spatially structured populations—this research concerns the development of the general theory of the partial differential equations of these models.

Ralph McKenzie's work has been focused since 1980 on the classification of all finite algebraic systems, and on finding new ways to recognize and measure structure in finite algebraic systems. He believes that research into the interplay between, on the one hand, the existence or nonexistence of algorithms to recognize fundamental properties of finite algebras, and on the other hand, the levels of structural complexity manifested in the algebras, has the potential not only to expand our understanding of the structural possibilities in finite algebraic systems (which it has already done), but to produce fundamental breakthroughs in theoretical computer science.

The research team of Doug Hardin and Ed Saff, along with several graduate students, is pursuing research, supported in part by the NSF, in computational analysis, dealing with new and improved ways to distribute points uniformly on various types of surfaces. Their research has a significant number of applications. Among other things, their work is useful when trying to digitize curved surfaces for computer graphics and animations with greater efficiency, placing the elements of a sonar net on the ocean bottom in the best locations to detect the presence of submarines, and testing radar systems in aircraft to ensure uniform coverage.

The Vanderbilt University Department of Mathematics has a strong group in wavelet theory, frame theory, time-frequency analysis, sampling theory, compressed sensing and their applications to signal processing, image processing, and biomedical data analysis. Specific areas of application include learning theory, bioinformatics, and sigma delta quantization. The faculty researchers in this area are Professors Aldroubi, Hardin, and Powell.

The research group in Noncommutative Geometry and Operator Algebras (NCGOA) at Vanderbilt University consists of several regular faculty members (Bisch, Hughes, Kasparov, Yu, Zheng, Xia), post-doctoral fellows, visitors, and graduate students with scientific interests in noncommutative geometry, von Neumann algebras, the theory of subfactors, K-theory of operator algebras, operator theory, coarse geometry, index theory, analysis on manifolds, controlled topology, stratified spaces, harmonic analysis, quantum computing, and quantum information theory. The group is highly visible, has strong NSF support, and organizes a yearly spring school and international research conference in their research area. The school/conference is directed by Alain Connes.

The research team in group theory consists of several regular faculty members (Mihalik, Olshanskii, Sapir, Ratcliffe, Tschantz). The team is well funded, highly visible, and works on various important questions in algebra and geometric group theory. Research in group theory attracts many visitors, students, and post-doctoral researchers to Vanderbilt.

For more information on our departmental research projects, please visit the departmental Web pages^4,5.

Besides individual projects, research dissemination is performed through other activities of the department. For the past 20 years, the department hosted the Annual Shanks Lecture and accompanying Shanks conference⁶. In addition, several Shank workshops are held at the Department of Mathematics every year focusing on recent research developments. Furthermore, the journal Constructive Approximation (Editor-in-chief Ed Saff) is based at the Vanderbilt Department of Mathematics. It is ranked at #23 in Journal Citation Reports^®, with an impact factor of 0.909.

  What is your prediction for the state of our knowledge about this particular field 10 years from now?

Dietmar Bisch, Chair
Department of Mathematics
Vanderbilt University
Nashville, TN, USA

Related Links: [ Return ]
1	http://www.math.vanderbilt.edu/~ncgoa
2	http://www.math.vanderbilt.edu/~cca
3	http://www.math.vanderbilt.edu/~daphne/biomath/index.html
4	http://www.math.vanderbilt.edu
5	http://www.math.vanderbilt.edu/research.html
6	http://www.math.vanderbilt.edu/shanksconf.html
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