Shabnam Beheshti
(1) B.Sc. Mathematics, Physics. McGill University, Montreal QC (2000)
(2) M.Sc. Pure Mathematics. Texas Tech University, Lubbock, TX (2002)
Title of M.Sc. Thesis: Weierstrass Data and Symmetries of Minimal Surfaces;
discussion of the classification of Minimal Surfaces using the classical
methods of Weierstrass, with new treatment of surface deformations.
(3) Ph.D. Candidate, Pure Mathematics. University of
Massachusetts, Amherst, MA (2007)
Thesis Topic: Solitons, Sigma-Models and Two-Dimensional Gravity; relating
dilatons from generalised black hole metrics to soliton metrics arising from
the sine-Gordon equation; construction of harmonic maps.
Research Interests:
I am interested in studying the roles of
analysis and geometry in physics. Recent work with my advisor,
Floyd Williams, has involved investigating the interconnections
between harmonic maps (sigma models), soliton metrics, String Theory
and the BTZ black hole.
A soliton metric is a metric of the form ds^2=cos^2(u/2)dx^2 +/-
sin^2(u/2)dt^2, where
u(x,t) is a solid wave solution of a
sine-Gordon equation:
u_xx +/- u_tt = +/- A sin(u).
I have been exploring the relationship between soliton metrics and black
holes in various models
for classical gravity in two spacetime dimensions.
One natural question is whether it is possible to find a general method for
constructing maps between soliton metrics and black hole metrics;
often, this involves solving a complicated system of highly nonlinear
differential equations. As a simplification, I use the two-dimensional
Jackiw-Teitelboim model for gravity to explore some concrete examples for
which such maps can be explicitly found.
In addition, I have been examining a generalisation of the J-T action
integral to extend some known results in the J-T case to the
String-Inspired Theory (SIG) and the Spherically Symmetric Gravity (SSG) model.
I also study the construction of harmonic maps from soltion metrics to the
2-Sphere, leading us in turn to sigma models from black hole metrics to S^2.
This leads us to a concrete connection between sigma-models, solitons and
2-dimensional gravity.
In two dimensions, the fact that solutions to the Einstein are conformally
flat translates in string theory to guaranteeing that the classical dynamics
of the string is independent of the world-sheet geometry of the string. In
the future, perhaps it will be possible to investigate the extent to which
this fails at the quantum level by comparing some of the basic maps and
constructions of above.
Talks, Preprints, and Publications
Explicit soliton-black hole correspondence for static
configurations
Authors: Shabnam Beheshti and Floyd
Williams
Reference: Accepted for publication in Journal of Physics
A:Mathematical and Theoretical. Preprint available on the
arXiv.
From Solitons to Dilatons: Relating Sine-Gordon Equations to Two-Dimensional Gravity.
(11 Apr 2006) Geometry Seminar, University of Connecticut. Preprint
available.
Double Bubbles in the Three-Torus
Authors: Miguel Carrion Alvarez, Joseph Corneli, Genevieve Walsh, and
Shabnam Beheshti
Reference: Published in
Journal of Experimental Mathematics.
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