Stony Brook undergraduate mathematics majors have the opportunity to
participate in the National Science Foundation (NSF) supported Grant for
Vertical Integration of Research and Education in the Mathematical Sciences
(VIGRE) by joining a research group
or by establishing an approved program of study, scholarship and research
with a faculty member or a VIGRE postdoctoral fellow. For the academic year
2001/2002 two research groups
(Riemann surfaces and Kleinian groups, Complex dynamics)
and the three VIGRE post doctoral fellows (Suzanne Hruska, Matthew Kudzin
and John Terilla) will accept students. Other faculty members may also provide
research opportunities for undergraduates. Although most projects described
here require that students have completed at least two years of college level
mathematics, many faculty members have problems that are accessible to
curious and hard working students without this background.
The SYMPLECTIC GEOMETRY GROUP consists of Dusa McDuff, Ely Kerman and several graduate students. There is a weekly seminar on Mondays, that is mostly a reading seminar but has occasional visitors. There is also a Symplectic Topology and Dynamics seminar that is run jointly with the Courant Institute at New York University (NYU) and meets every three weeks or so either at Stony Brook or at NYU.
Symplectic geometry grew out of the study of energy preserving
systems in classical physics - such as the motion of the planets around the
sun - and it still has great relevance to modern day physics. It is an even
dimensional geometry; in 2-dimensions one studies motions that preserve
area, but in dimensions 4 and above the geometric significance of a
symplectic structure is less transparent. There are a variety of
open problems in 2 and 4 dimensions that are relatively easy to
understand, and where it would be interesting to work out examples.
Prospective students should have taken either a
topology class (such as 362 or 364) or some analysis in many variables
(such as 322). If you are interested in this area, come to talk to Dusa
McDuff.
RIEMANN SURFACES AND KLEINIAN GROUPS. The group consists of
Chris Bishop, Irwin Kra, Bernard Maskit and Yair
Minsky. Its work concerns problems in complex analysis, particularly:
Function theory on compact Riemann surfaces.
Kleinian groups.
Two and three dimensional topology and geometry.
Other members of the
department have expertise in related areas. Among these are Mark
de Cataldo and Sorin Popescu
whose main areas of research are Algebraic Geometry.
Bishop works on a variety of problems involving the
geometry of conformal and quasiconformal mappings. Among the
topics involved are: harmonic measure, Brownian motion,
random walks, Hausdorff and other dimensions, groups
of Möbius transformations, the Riemann mapping theorem,
quasiconformal and quasisymmetric mappings, hyperbolic geometry.
Undergraduate projects could include expository papers on
related topics, learning about existing software to
compute Riemann mappings, circle packings, or limit sets
or writing new code. Students would have to know the contents of the
basic undergraduate courses on real and complex analysis. Knowledge of some measure theory would be useful.
Kra's past work was in the area of moduli of Riemann surfaces
and Kleinian groups. He has studied Poincaré series and Eichler
cohomology. He is currently interested in using theta functions and theta
constants to obtain applications of complex analysis to number theory and
combinatorics. The work involves both scholarly investigations and computer
experimentations that are particularly appropriate for projects for
undergraduates. Kra is also interested in mathematics curriculum reform at
the college and pre-college level. To work with Kra, students
should have completed the equivalent of MAT 310, 320 and 342. Students who
have not completed these courses and are willing to learn new mathematics
and experiment with computations are encouraged to contact Kra.
Maskit's work is in the area of uniformizations and moduli of
Riemann surfaces, hyperbolic geometry, and low dimensional topology and
geometry. The basic objects of study are groups of two-by-two matrices with
real or complex entries. To work with Maskit, students should have
completed the equivalent of MAT 310, and either MAT 342 or MAT 364. In addition
to his work described above, Maskit is also interested in
mathematical pedagogy, primarily concerning grades 7-12, and college
level. His interest is in research that is experimental and/or statistical,
rather than philosophical.
Minsky has been working in the area of 2 and 3 dimensional
topology and geometry. He is interested in hyperbolic structures on
3-manifolds, deformation spaces of Riemann surfaces, transformation groups of
surfaces, geometric methods for studying groups, and the interface between
hyperbolic geometry and complex dynamics. To work with Minsky,
students should have completed the equivalent at least one of MAT 342, 362
or 364.
de Cataldo works on the topology and geometry of algebraic
varieties. Students who wish to work with de Cataldo
should first complete the equivalent of MAT 313 and
either MAT 342 or MAT 364.
Popescu works on problems in areas of complex algebraic
geometry, commutative algebra, combinatorics and their computational
methods. Among research topics are modular curves, moduli of
abelian varieties, syzygies and their relevance in geometry and
combinatorics, toric varieties, hyperplane arrangements. The computational
aspects involve the design of new algebraic algorithms and their
methods. Among research topics are modular curves, moduli of
abelian varieties, syzygies and their relevance in geometry and
combinatorics, toric varieties, hyperplane arrangements. The computational
aspects involve the design of new algebraic algorithms and their
implementation in current systems as Macaulay2, Macaulay, Maple,
Mathematica, etc. Undergraduate mini-projects could include
experimenting with concrete open problems related to one of the above
topics and/or writing new code in the currently used computer algebra
systems (Macaulay2, Maple, Mathematica) or in C/C++, Java.
Students who wish to work with Popescu should first
complete the equivalent of MAT 313.
COMPLEX DYNAMICS The group consists of Suzanne Hruska,
Suzanne Hruska, Mikhail Lyubich (on leave dring the 2002/03
academic year), John Milnor and Scott Sutherland.
A description of the opportunities for undergraduates to work
with members of this group is in preparation. Hruska's interests are
described below.
VIGRE POST DOCTORAL FELLOWS These fellows provide opportunities
for research in their respective area of expertise and their general
involvement in the VIGRE grant may also result in projects appropriate for
undergraduates.
Hruska works in complex dynamical systems, with an emphasis
on using a computer to assist with rigorous proofs and for inspiration and
exploration of unknown phenomena. Students interested in working on
computational dynamical systems projects with Hruska should have completed the
second-year courses (a calculus sequence that includes linear algebra). MAT
331 or some other experience with programming would be helpful.
Kudzin's current research is in Riemannian geometry. He is
working on group actions on manifolds and spaces of non-negative curvature.
He is looking for students with an interest in either geometry or topology.
Some exposure to the subject (MAT 362 or 364) would be helpful, but is not
necessary.
Terilla's research areas include deformation theory, mirror
symmetry and quantum computing. Students who have had some abstract algebra
and who are interested in mathematical physics, topology, and combinatorics
may wish to work with Terilla.
MATHEMATICS FACULTY Many members of the department welcome
undergraduates to participate in their research. We are working on a fuller
description of the various opportunities. At the moment only David Ebin's
and Santiago Simanca's areas of activity are described.
Ebin works on partial differential equations, particularly
those which come from fluid dynamics. He has worked on the initial value
problems for the Euler and Navier-Stokes equations, on the relations
between incompressible and incompressible flows, on fluids with free
surfaces, etc. Recently he has been involved in a laboratory project to
measure the sound created by a moving fluid.
Simanca's past work has been in the areas of geometric
analysis and partial differential equations. He has been involved with both
theoretical and applied mathematics projects, including some that required
heavy use of computers for numerical calculations. He is currently
interested in non-linear equations that arise in Riemannian geometry, and
mathematics education. To work with Simanca, students should
have completed the equivalent of MAT 310, 320 and 342.
Application procedure Mathematics majors interested in the
program should contact:
Irwin Kra
Math 4-111
2-8273
irwin@math.sunysb.edu
or
Dusa McDuff
Math 3-111
2-8288
dusa@math.sunysb.edu
Applied Mathematics and Statistics majors interested in the
program should contact:
Alan C. Tucker
Math P-138
2-8365
atucker@ams.sunysb.edu
Normally, qualified students will be referred to an appropriate faculty
mentor. The mentor and student will prepare a plan for the student's
participation in the VIGRE program. The plan may involve a semester or more
of course work to bring the student to a level where he or she can
undertake a scholarship and research assignment. The final product of an
undergraduate's
participation in the Research-like Experiences under the VIGRE grant is a
written report approved by the mentor. The stipend for undergraduate
students accepted into the program is $2,000 per summer and $1,000 per
semester during the academic year. The stipend is funded by the NSF grant;
recipients must be US citizens or permanent residents.
10/14/02 version.