The mathematical research in this project involved the topology of spaces related to String Theory in physics, i.e., theories that describe all particle (including gravity) interactions in the universe.   The spaces studied include smooth and symplectic 4-manifolds, 6-dimensional Calabi-Yau manifolds, and embedded knots and links in 3-manifolds. 

A 3-manifold is a space that locally looks like the familiar space in which we live. If one imagines replacing every point in the 3-manifold with a circle or line in a nice way, one gets an example of a 4-manifold.  Four manifolds are particularly useful for modeling physical phenomena if they also have a symplectic structure. The PI investigated the shapes of 4-manifolds by constructing new examples and distinguishing them from each other by using invariants of 4-manifolds derived from the count of solutions of certain systems of nonlinear partial differential equations on the manifold.  As part of this grant, the PI  

• Produced a series of research papers on small symplectic 4-manifolds related to the “symplectic Poincaré conjecture,” including the construction of a symplectic 4-manifold with special properties that is homeomorphic but not diffeomorphic to .

• Created a new type of construction called a telescoping triple that was used to fill out much of the remaining geography of simply connected, minimal, closed symplectic 4-manifolds with odd intersection forms.

• Wrote a series of papers deriving upper bounds on the minimum Euler characteristic of all symplectic 4-manifolds with a prescribed fundamental group.

Symplectic Calabi-Yau manifolds are symplectic manifolds with vanishing first Chern class.  They were first introduced to study mirror symmetry in physics.  As part of this project, the PI

• Introduced a new surgery operation on symplectic manifolds called coisotropic Luttinger surgery that generalizes Luttinger surgery.

• Constructed infinitely many distinct symplectic Calabi-Yau 6-manifolds that are not of the form  for  a closed symplectic 4-manifold and  a closed surface.

Mathematicians have long been interested in creating knot invariants.  In some models of the universe, particles are thought of as knotted strings in the visible 3-dimensional space, with invariants of knots giving information about the particles.  One important way to check for an invariant is to show that it does not change under three simple Reidemeister moves of 2-dimensional pictures of knots.  Since its discovery in the 1920’s, Reidemeister’s theorem has helped mathematicians and physicists find and prove many invariants of knots.  The problem with Reidemeister moves, however, is that they are inherently 2-dimensional: it is far more natural to work with knots in 3-dimensional space than work with 2d pictures of them.

The PI proved the first 3-dimensional version of Reidemeister’s theorem using a new representation of knots that the PI created called a cube diagram.  The 3-dimensional moves are called cube moves and, importantly, there are only two of them.  In a series of papers, the PI 

• Described a Heegaard-Floer homology theory for cube diagrams and proved that it was an invariant of knots using cube moves.
  
  
• Generalized cube diagrams to a representation of knotted tori in  called hypercube diagrams.
  
  
• Created a new construction of Legendrian tori in  using Lagrangian hypercube diagrams.

The first broader impacts project was the completion of a highly rated textbook series for prospective teachers.  The next project raised the mathematical achievement o...

Please report errors in award information by writing to: awardsearch@nsf.gov.