Shabnam Beheshti
Mailing Address:
Department of Mathematics -- Hill Centre
110 Frelinghuysen Road
Rutgers University
Piscataway, NJ 08854-8019
Office: #214, Hill Center, Busch Campus
Telephone: 001 848 445 7261
e-mail: beheshti [at] math [dot] rutgers [dot] edu
Research Experiences for Undergraduates (REU)
The Mathematics Department, in joint with the Center for Discrete Matheamtics (DIMACS) as well as several
other groups hosts a summer research program for undergraduates. Full details can be found on both the
DIMACS REU homepage and the Math REU homepage.
Summer 2012
One aspect of my research during summer 2012 has been to better understand the connections between
symplectic geometry, knot theory and analytical dynamics; study of this topic was largely motivated from
attending the IAS Summer Program on 21st Century Geometry.
For the DIMACS/Math REU program, I gave a series
of six
lectures on
these topics: three talks were on Legendrian Knots and Contact Structures, and three more were on Hamiltonian Dynaics and Symplectic Structures. My two
Mathematics REU students, have explored different aspects of these themes and their projects are briefly
described below.
Knot Theory and DNA Topology: Michael Boemo
This project involves understanding knots in 3-space as a tool for modeling DNA coiling. In particular,
we are investigating
- Effects of mutation on DNA curvature: we estimate, numerically and analytically, a possible lower bound
for change in bond angle when swapping nucleotides. It would be useful to also determine whether the
estimate is robust under long-term changes.
- Legendrian Knots and their use in DNA modelling: front projections of Legendrian knots encode all the
necessary information required to reconstruct the knot in 3 dimensions. We are exploring torus knots as
Legendrian knots in order to connect with the current DNA Topology literature.
- Knot pinching: it is common for DNA to recombine in such a way that two points on a coil bond, forming
a "pinched" knot. A more difficult question to consider is whether there is a well-defined notion of a
(non-injective) knot invariant.
Further details of the project can be found on Mike
Boemo's REU
page.
Analytical Dynamics and Materials Science: Michael Geis
This project involves understanding the role of symplectic structures arising in analytical dynamics and
nonlinear partial differential equtions (PDEs), in general. Part of our task is to study Lagrangian and
Legendrian subspaces of a symplectic vector space, and as such, we are also studying the differential
geometry of surfaces (exterior products, tangent spaces, extrinsic/intrinsic curvature).
Long-term goals of the project include
- connecting the geometric ideas learned with Noether's Theorem on conserved currents, as
well as understanding the Euler-Lagrange equations geometrically
- studying the properties of minimal curves and surfaces: a minimal surface in 3-space is a surface
having zero mean curvature. It turns out that this special class of surfaces has a rich structure, which
can be studying from a variety of perspectives. We would like to understand how the least area-least
energy principles connect with problems in dynamics (and potentially materials science).
- determining the presence of symplectic structures in
materials science models;
it would be particularly interesting to formulate carefully the PDEs involved in a particular interface
problem (could involve bio-heat transfer, fluid mechanics, etc.)
Further details of the project can be found on Mike Geis'
REU
page