Vortex Equations over Riemannian Surfaces

Khoa Lu Nguyen

Mentor: Professor Christopher Woodward

 

 

Hello! My name is Khoa Lu Nguyen. I am an incoming senior at Massachusetts Institute of Technology. Reading the title, you may have already guessed that I am a pure math major. My field of interests include differential geometry, complex geometry, algebraic topology and geometric analysis.

 

This summer 2009, I participate in the REU at the Mathematics Department of Rutgers University. Together with Amanda Hood and Joseph Shao, I work on vortex equations, a generalization of Yang Mills equations. The Yang Mills equation originated from physics, specifically particle physics and field theories. The well-known Maxwell equation in electromagnetism is a trival case of the Yang Mills equation on trivial S1-principle bundle over R4.

 

The problem I am working on concerns the regularity of the solutions. The regularity of a solution gives you detail about what function spaces the solution is in. The solutions of vortex equations are critical points of an energy functional. From the Dirichlet variational principle, we have the solutions are in some certain Sobolev spaces. Using tools from PDE and estimates from the equations, we improve the regularity of the solutions.

 

I hope to show that under some gauge transformation, the solutions can be smooth. This generalizes the regularity results by Cieliebak, Gaio, Riera and Salamon.

 

Here is my log of progress:

 

Week 1 (June 1-June 7)

-          Had a meeting with Professor Woodward, discussing about what to read

-          Read chapter 9 of Introduction to Mechanics and Symmetry by Marsden and Ratiu about basic Lie group.

-          Read parts I, II, III, V of Lectures on Symplectic Geometry by Silva about basic  symplectic theory, Darboux’s theorem, Moser trick, almost complex structure.

 

Week 2 (June 8- June 14)

-          Had a meeting with Professor Woodward and discussed about associated vector bundles, connections, curvature.

-          Read chapters I, II of Foundation of Differential Geometry by Kobayashi and Nomizu about principle G-bundle, connection, curvature, structural equations.

-          Read parts V, VI of Lectures on Symplectic Geometry by Silva about almost complex structure, Dolbeault operator, variational methods.

-          Had another meeting with Professor Woodward about the presentation on Friday.

 

Week 3 (June 15- June 21)

-          Read chapters II, III of Foundation of Differential Geometry by Kobayashi and Nomizu about holonomy theory, curvature, linear connection.

-          Read parts  VIII, X of Lectures on Symplectic Geometry by Silva about moment maps, Atiyah-Bott symplecitc structure on the space of connections, …

-          Had a meeting with Professor Woodward and discussed about classical Yang Mills equations, critical points of Yang Mill energy functional, …

 

Week 4 (June 22-June 28)

-          Reviewed almost complex structure, Dolbeault operators, …

-          Started to read simple sections, mainly tried to understand results and formulations in the papers by Oscar Prada.

-          Had a meeting with Professor Woodward and discussed about complex manifold, vector bundle, the correspondence between Dolbeault operators on vector bundle and covariant derivative, complexified Lie groups, induced complex structure on complexified principal bundle, …

-          Understood the 1-1 correspondence between covariant derivatives on vector bundle and connections on frame bundles.

 

Week 5 (June 29- July 5)

-          Read Bradlow’s lecture notes (1-40) on his Gauge Theory class

-          Had a meeting with Professor Woodward and discussed about complex vector bundles, relations between Hermitian metrics, connection on unitary frame bundles, holomorphic structures on complex vector bundles.

-          Studied Chern classes and started to read the computations in Bradlow’s paper.

-          Read the foundation section in Ph.D thesis The Geometry of Vortex Equations by Oscar Prada.

-          Had another meeting with Professor Woodward and discussed about how complexfied gauge transformation can make curvature flat.

-          Had another meeting with Professor Woodward and discussed about stable vector bundle and Morse applications on Yang Mills solutions.

 

Week 6 (July 6-July 12)

-          Professor Woodward traveled to France for a conference

-          Reviewed PDE materials from Partial Differential Equation by Jost about Sobolev space, Dirichlet principle, elliptic bootstrap, …

-          Reviewed some Yang Mill computations. Did similar computations for vortex equations.

-          Read part VI of Lecture notes on Symplectic Geometry by Silva and some parts of  chapter IX of Foundations of Differential Geometry vol.2 by Kobayashi and Nomizu about complex manifolds, integrability, Kahler manifold, basic Hodge theory results, …

 

Week 7 (July 13-July 19)

-          Mainly worked on the presentations.

-          Reviewed learned materials: connections, curvature, moment maps, Hamiltonian actions, ….

-          Did some small regularity computations for a trivial case.

 

Week 8 (July 20-July 24)

-          Read the regularity section in the paper by Cieliebak, Gaio, Riera, Salamon about local slice theorem, regularity results.

-          Tried to use these ideas and results to prove the case for vortex equations.

 

Here are our presentation slides

 

References

·  Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture notes in mathematics 1764, Springer-Verlag, 2001.

·  Kobayashi and Nomizu Foundations of Differential Geometry Volume I. Wiley Classics Library, 1996.

·  Kobayashi and Nomizu Foundations of Differential Geometry Volume II. Wiley Classics Library, 1996.

·  Marsden J. and Ratiu. S.T Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, 17, Springer-Verlag, 1994.

·  Jost J. Partial Differential Equations. Graduate Texts in Mathematics, 214, Springer Verlag, 2001.

·  Cieliebak K., Gaio A.R., Riera I.M., Salamon D. The symplectic vortex equations and invariants of Hamiltonian group actions.” J. Symplectic Geom. Volume 1, Number 3 (2002), 543-646.

·  Garcia-Prada, Oscar. "A Direct Existence Proof for the Vortex Equations Over a Compact Riemann Surface." Bull. London Math. Soc. 26 (1994): 88-96.

·  Bradlow S. “Vortices in Holomorphic Line Bundles over Closed Kahler Manifolds.” Communications in Mathematical Physics 135 (1990), 1-17.