REU 2019

Overview of the project:

Geometry is the study of structure of space. Symplectic geometry is the geometry of even dimensional smooth manifolds equipped with symplectic structures. A symplectic structure is a closed, nondegenerate 2-form, w. A 2-form is closed means that the symplectic area of a surface with boundary does not change as the surface moves, provided that the boundary is fixed. The nondegeneracy condition means that for any direction v at least one of the family of 2-planes spanned by v and a varying other direction w has nonzero area.

Symplectic geometry first surfaced in the study of classical mechanical systems such as the planetary systems. In 19^th century, Hamiltonian discovered deep mathematical symmetries between an object’s position and momentum by reformulating Newton’s laws of motion. It began to take its modern shape with Arnold, Gromov and others in 60’s and 70’s. They asked : What are the geometric differences between the transformations that preserve volume and those preserve the symplectic form? More recently, in 1985, Gromov developed a whole new set of tools, called pseudo-holomorphic curves, for studying this question and proved the absolutely foundational result of symplectic geometry known as the non-squeezing theorem. Since symplectic geometry is similar to complex geometry, Gromov used that to prove his celebrated theorem. This was the first theorem which distinguished volume preserving maps from symplectic maps.

What will the research involve?

In this project our goal is mainly to learn about this new geometry and its development: First, we will get familiar with all the necessary background in mathematics and physics. Second, we will explore specific symplectic manifolds and work out many of the details on familiar spaces such as even dimensional Euclidean spaces. Then after analyzing some well-known theorems (such as Darboux theorem, nonsqueezing theorem and embedding theorems) we will work to get an answer to a specific question such as “When does one region embed symplectically into another?”. There are some partial answers for this question but in general the answers unknown.

Prerequisites for applicants:

This project involves wide variety of topics in mathematics and physics. Students must be very comfortable with Calculus, Linear Algebra and some aspects of Euclidean Geometry such as working with coordinate systems and higher dimensional Euclidean spaces. To have some knowledge in basic Differential Geometry and Physics would be helpful but is not required.

How to apply:

For application information and instructions, please visit the GVSU Summer Mathematics REU home page .