Dr Mark Haskins
J J Sylvester Assistant Professor

E-mail address: mhaskin@math.jhu.edu
Telephone: 410-516-4047
Department: Mathematics
Office: Krieger 312


From September 2003 I will be a Hodge fellow at IHES outside Paris.

Research Interests

Singularities of calibrated varieties

Calibrated geometries are a special class of minimal submanifolds in Riemannian manifolds introduced in the early 1980s by Harvey and Lawson. A minimal submanifold, belying its name is merely a critical point of the volume - that it is not necessarily a minima of volume. Calibrated submanifolds have stronger minimzing properties - they minimize volume in their homology class. This property makes calibrated submanifolds more "rigid" than arbitrary minimal submanifolds.

Classical examples of calibrated geometries are the complex submanifolds of a Kaehler manifold, whose minimizing properties are well known. Another example is "special Lagrangian geometry" introduced by Harvey and Lawson. The flat version of the geometry exists on complex n-space and gives a special class of minimal n-dimensional submanifolds of complex n-space.

On a wider class of complex manifolds - the so-called Calabi-Yau manifolds - there is also a natural notion of special Lagrangian geometry. Since the late 1980s these Calabi-Yau manifolds have played a prominent role in developments in High Energy Physics and String Theory. In the late 1990s it was realized that calibrated geometries play a fundamental role in the physical theory, and calibrated geometries have become synonymous with "Branes" and "Supersymmetry".

Special Lagrangian geometry in particular was seen to be related to another String Theory inspired phemonenon, "Mirror Symmetry". Strominger, Yau and Zaslow conjectured that mirror symmetry could be explained by studying moduli spaces arising from special Lagrangian geometry.

This conjecture stimulated much work by mathematicians, but a lot still remains to be done. A central problem is to understand what kinds of singularities can form in families of smooth special Lagrangian submanifolds. A starting point for this is to study the simplest models for singular special Lagrangian varieties, namely cones with an isolated singularity. My research in this area ([2], [4], [6]) has focused on understanding such cones especially in dimension three, which also corresponds to the most physically relevant case.

Adiabatic limits in nonlinear geometric wave equations

Description of research to be posted here. See publication [5] below.

Discrete solitons

There are many systems of partial differential equations of interest in theoretical physics which possess soliton solutions (stable, smooth, localized lumps of energy). In applications in condensed matter and biophysics, solitons usually propagate through discrete spaces (crystal lattices for example), and it has long been recognized that spatial discreteness introduces crucial and highly complex phenomena into the soliton dynamics. In place of partial differential equations one gets infinite coupled systems of nonlinear differential equations. My research in this area has focused on the study of breathers (time periodic oscillatory solitons) in discrete systems [1] and [3].

Publications

[1] Breathers in the weakly coupled topological discrete sine-Gordon system (with Martin Speight), Nonlinearity 11 (1998) 1651.

[2] Constructing Special Lagrangian cones, PhD thesis, University of Texas, Austin, 2000, Advisor: K. Uhlenbeck

[3] Breather initial profiles in chains of weakly coupled anharmonic oscillators (with Martin Speight) Physics Letters A, 299 (2002) 549.

[4] Special Lagrangian Cones arXiv:math.DG/0005164, to appear American Journal of Mathematics (accepted Nov 2002).

[5] The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps (with M. Speight), arXiv:hep-th/ 0301148, Journal of Mathematical Physics. 44(2003) 3470-3495.

[6] The geometric complexity of special Lagrangian T^2 cones arXiv:math.DG/0307129. To appear Inventiones Mathematicae (accepted Oct 2003).

In Preparation

[7] Special Lagrangian cones with higher genus links. (with N.Kapouleas).


Collaborators

[1] Nikos Kapouleas Brown University.
[2] Ian McIntosh,University of York, England
[3] Martin Speight University of Leeds, England

Past Courses

Spring 2003:
Math 106: Calculus I for the Biological and Social Sciences
Math 112: Honors One Variable Calculus (2nd semester)

Fall 2002:
Math 111: Honors One Variable Calculus (1st semester)

Spring 2002:
Math 408: Geometry and Relativity
Math 421: Dynamical Systems and Chaos

Fall 2001:
Math 503: Undergraduate Research in Mathematics

Spring 2001:
Math 439: Introduction to Differential Geometry
Math 645: Riemannian Geometry

Fall 2000:
Math 421: Dynamical Systems and Chaos



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Mark Haskins