CONTACT AND SYMPLECTIC GEOMETRY OF THREE AND FOUR MANIFOLDS

Instructor:

Course Description:

Symplectic geometry is the study of differentiable manifolds equipped with a closed, nondegenerate differential 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics and is a source of many rich and interesting geometric problems. Contact geometry is in many ways the odd-dimensional counterpart of symplectic geometry.

The short course for this project will focus on the contact geometry of three manifolds and the symplectic geometry of four manifolds. The first week of the course will be geared towards bringing students up to speed with the needed material to understand these subjects --- students will learn about smooth manifolds, the tangent bundle, vector fields, distributions, differential forms, and some basic theory of three and four dimensional manifolds. In the next two weeks, we'll cover the basics of contact and symplectic geometry. We'll start with the motivations from classical mechanics, and then cover some of the fundamental results of these subjects. Proofs will be given where appropriate, but students will also be exposed to some of the major results from this field --- for example, Gromov non-squeezing and the Weinstein conjecture in dimension three. The fourth week will be devoted to embedded contact homology and its applications. Part of this week will include a very basic introduction to the theory of pseudoholomorphic curves.

There are many open problems related to embedded contact homology that participants could become involved in during the research period. There will also be more basic research-type projects involving fundamental notions from contact and symplectic geometry (e.g. Legendrian knots).

Prerequisites: Any introductory course in geometry or topology. No experience with algebraic topology is necessary, but students should be comfortable with the most basic concepts in point-set topology (open, closed sets, continuous functions, etc.)

References:

Terry Tao, Differential Forms and Integration

Ana Canas da Silva, Symplectic Geometry

(

John Etnyre, Introductory Lectures on Contact Geometry (

Michael Hutchings, Taubes' Proof of the Weinstein Conjecture in Dimenson 3

(

John M. Lee, Introduction to Smooth Manifolds

HOMOLOGICAL ALGEBRA*

Instructor:

This course will define the basic notions of homological algebra. We will try to keep things less abstract and emphasize the computational aspect of the subject as well as various applications such as group cohomology and Hochschild cohomology.

For the research portion, we could explore a few different directions:

Hochschild cohomology: The subject is nice because there are always computations that students can explore with a computer:

a) An approachable problem might be to determine the Lie algebra and ring structure on Hochschild cohomology of algebras of certain classes.

More ambitious problems (we would begin by studying concrete examples) would be to look at:

b) Is the Hochschild cohomology of any artinian algebra finitely generated modulo a nilpotent ideal?

c) What does it mean for the Hochschild (co)homology to vanish in high degree for non-commutative algebras? For example, can we find a Frobenius algebra which has Hochschild cohomology which vanishes in high degree? See the work by Buchweitz et al. mentioned below to get a sense for the spirit of the question.

Matrix factorizations: This is a topic that is fashionable due to the role it plays in Mirror Symmetry. In our research, we will be focused on applications to commutative algebra. More precisely, we could look at:

a) Generalizations of matrix factorizations to the case of multiple functions, with possible application to the homological algebra of complete intersections.

Prerequisites: A course in abstract algebra. Students need not know any homological algebra, but should be comfortable with the basics of groups and commutative rings.

References:

Charles Weibel, an Introduction to Homological Algebra

Buchweitz, Green, Madsen, and Solberg. Finite Hochschild cohomology without finite global dimension.

http://arxiv.org/pdf/math/0407108v3.pdf

*This course will only run if we have at least 8 students who have decided to participate in the program. We will make every effort to make this happen.

Instructor:

__Daniel Cristofaro-Gardiner__Course Description:

Symplectic geometry is the study of differentiable manifolds equipped with a closed, nondegenerate differential 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics and is a source of many rich and interesting geometric problems. Contact geometry is in many ways the odd-dimensional counterpart of symplectic geometry.

The short course for this project will focus on the contact geometry of three manifolds and the symplectic geometry of four manifolds. The first week of the course will be geared towards bringing students up to speed with the needed material to understand these subjects --- students will learn about smooth manifolds, the tangent bundle, vector fields, distributions, differential forms, and some basic theory of three and four dimensional manifolds. In the next two weeks, we'll cover the basics of contact and symplectic geometry. We'll start with the motivations from classical mechanics, and then cover some of the fundamental results of these subjects. Proofs will be given where appropriate, but students will also be exposed to some of the major results from this field --- for example, Gromov non-squeezing and the Weinstein conjecture in dimension three. The fourth week will be devoted to embedded contact homology and its applications. Part of this week will include a very basic introduction to the theory of pseudoholomorphic curves.

There are many open problems related to embedded contact homology that participants could become involved in during the research period. There will also be more basic research-type projects involving fundamental notions from contact and symplectic geometry (e.g. Legendrian knots).

Prerequisites: Any introductory course in geometry or topology. No experience with algebraic topology is necessary, but students should be comfortable with the most basic concepts in point-set topology (open, closed sets, continuous functions, etc.)

References:

Terry Tao, Differential Forms and Integration

__http://www.math.ucla.edu/~tao/preprints/forms.pdf__Ana Canas da Silva, Symplectic Geometry

(

__http://www.math.princeton.edu/~acannas/Papers/symplectic.pdf__)John Etnyre, Introductory Lectures on Contact Geometry (

__http://people.math.gatech.edu/~etnyre/preprints/papers/contlect.pdf__)Michael Hutchings, Taubes' Proof of the Weinstein Conjecture in Dimenson 3

(

__http://arxiv.org/pdf/0906.2444v2.pdf__)John M. Lee, Introduction to Smooth Manifolds

HOMOLOGICAL ALGEBRA*

Instructor:

__Daniel Pomerleano__This course will define the basic notions of homological algebra. We will try to keep things less abstract and emphasize the computational aspect of the subject as well as various applications such as group cohomology and Hochschild cohomology.

For the research portion, we could explore a few different directions:

Hochschild cohomology: The subject is nice because there are always computations that students can explore with a computer:

a) An approachable problem might be to determine the Lie algebra and ring structure on Hochschild cohomology of algebras of certain classes.

More ambitious problems (we would begin by studying concrete examples) would be to look at:

b) Is the Hochschild cohomology of any artinian algebra finitely generated modulo a nilpotent ideal?

c) What does it mean for the Hochschild (co)homology to vanish in high degree for non-commutative algebras? For example, can we find a Frobenius algebra which has Hochschild cohomology which vanishes in high degree? See the work by Buchweitz et al. mentioned below to get a sense for the spirit of the question.

Matrix factorizations: This is a topic that is fashionable due to the role it plays in Mirror Symmetry. In our research, we will be focused on applications to commutative algebra. More precisely, we could look at:

a) Generalizations of matrix factorizations to the case of multiple functions, with possible application to the homological algebra of complete intersections.

Prerequisites: A course in abstract algebra. Students need not know any homological algebra, but should be comfortable with the basics of groups and commutative rings.

References:

Charles Weibel, an Introduction to Homological Algebra

Buchweitz, Green, Madsen, and Solberg. Finite Hochschild cohomology without finite global dimension.

http://arxiv.org/pdf/math/0407108v3.pdf

*This course will only run if we have at least 8 students who have decided to participate in the program. We will make every effort to make this happen.