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Description: This course will be an
elementary introduction to the mathematical theory of knots
with emphasis in some developments that occurred in the last
twenty years. It is designed primarily for undergraduate students
that have completed the basic sequence of calculus courses
and have had some background in writing proofs. Students must
have completed MTH 309 or MTH 314, MTH 310, and MTH 320 (or
the honors equivalent).
We will describe the different classifications of knots, their
properties, various ways for measuring their complexity and
some applications of knot theory to biology and physics. Since
some open problems in this area can be explained at an elementary
level, students will also have the opportunity to get a taste
of what it is like to do research in mathematics.
Description. This course is an introduction
to manifolds; their topology and geometry. We begin with the
definition, properties and examples of a manifold, its tangent
and cotangent bundles and tensor bundles. We will then study
vector fields, differential forms, integration on manifolds,
Riemannian metrics and general vector bundles. Finally we
will study Morse theory as time allows. Texts are M. Spivak,
'A comprehensive intro. to differential geometry, Vol. 1'
and J. Milnor, 'Morse Theory'. |
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Description: The aim is to give an introduction
into theory of Lie groups and Lie algebras. The theory of
Lie groups and Lie algebras and their representations plays
a central role in modern mathematics and theoretical physics.
It is not an exaggeration to say that knowledge of Lie algebras
and Lie groups should be a basic requirement for any professional
mathematician.
Following different sources (Curtis
"Matrix groups", B.Hall "Lie Groups, Lie Algebras,
and Representations. An Elementary Introduction" , Onishchik,
Vinberg "Lie Groups and Algebraic Groups", and others)
we will start with a simple matrix group (such as the group
of rotations in 3D space). Using similar examples we consider
basic notions and theorems of the subject. Then we continue
with representations of Lie algebras, Universal enveloping
algebras, classification theorems, connection between Lie
groups and Lie algebras, infinite dimensional representations
etc.
This course is intended
in the first place for graduate students. Everybody who is
interested in the subject is WELCOME to join.
Description: This course will cover
basic results on Riemann surfaces (1-dimensional complex manifolds),
calculus on Riemann surfaces, harmonic theory, the Riemann-Roch
theorem, the Abel and Jacobi theorems, the uniformization
theorem, etc. By and large I will take the analytic approach
to the subject, but I will also try to make connections as
much as possible with other areas such as algebraic and differential
geometry.
Description: Symplectic Field Theory
(SFT) is a general theory of algebraic invariants for symplectic
and contact manifolds. It generalizes Floer-theory, Gromov-Witten
theory, and contact homology and is constructed by measuring
moduli spaces of pseudoholomorphic curves. In order to describe
the structure of these moduli spaces a novel nonlinear Fredholm
theory has to be developed. We will discuss the compactness
results in SFT which generalize Gromov's compactness theorem.
After that we know what the underlying sets of the SFT moduli
spaces and their ambient spaces look like. These ambient spaces
do not carry any smooth structure in the classical sense,
for example they are not homeomorphic to open sets in Banach
spaces. We will introduce the concept of a polyfold and of
Fredholm theory on polyfolds. We will show how the spaces
occuring in SFT fit into this framework.
Description: This is an elementary course
about the geometry and topology of manifolds of dimension
< 9. The grading will be based on student presentations
of the assigned topics, and doing the occasional homework
assignments.
Description: The first part of this
course will introduce the theory of algebraic curves.The topics
which will be discussed are: definitions of curves, morphisms
of curves, various invariants attached to curves, Hurwitz
theorem, line bundles on curves (or invertible sheaves), Riemann-Roch
theorem (for line bundles), Picard groups, Jacobians, elliptic
curves, and (time permitting) the Torelli theorem, and an
introduction to moduli spaces of curves. Reference books:
books by Miranda, Fulton, Griffiths and Harris and an article
by James Milne on Jacobian varieties.
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