| Description: In this course we
will explore ways to define a "space" so that we
recover the notions of limits and continuity. This leads to
the subject of metric spaces and point-set topology. We will
then examine some of the many consequences of the continuity
of functions defined on these spaces. These consequences rest
on properties of a space more basic than its geometry, and
it is these properties we aim to study.
Math 461 is designed primarily for undergraduates intending
to go on to graduate school in mathematics, statistics, or
engineering. Some beginning graduate students in these areas
may also find it interesting. Students will need some background
in writing proofs (e.g. MTH 320).
Description: Real algebraic sets
are the sets that are solutions of polynomial equations F(x,y,z,...)=0
in Rn. For example, x2+y2-1=0
gives a circle in R2. The question of "which sets
are algebraic sets" is one of the main themes of algebraic
geometry. For example, three rays meeting at the origin in
R2 can never be an algebraic set. On the positive
side, by some elementary modifications, we can change the
shapes of algebraic sets. For example, in the equation of
the circle, by replacing y by y/x and clearing denominators,
we can turn the circle into the shape of the figure eight,
with equation x4+y2=x2. This
is a simple version of the process known as 'blowing up' in
algebraic geometry. In this class we will study such things.
|
Courses
Fall 2005
Courses
Spring2006
Courses
Fall 2006
Courses
Spring 2007
|
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. The text is
John Lee's 'Introduction to Smooth Manifolds', Springer.
Description: This is an introduction to complex geometry.
We will cover the following topics in varying details:
holomorphic functions of several variables, complex manifolds,
holomorphic vector bundles and Chern classes,
Kahler metrics, sheafs and cohomology, Hodge theory, Riemann-Roch
theorem and proof for curves.
Description: This is a year long course
in Algebraic Topology. Topics to be covered are: Cohomology,
universal coefficient theorem, Kunneth formula, Poincare duality.
deRham theory, homotopy theory, cellular and simplicial approximations,
Whitehead and Hurewicz theorems, fiber bundles, Postnikov
towers, obstruction theory, characteristic classes and spectral
sequences.
Description: Basic lectures on geometry
and topology (enough to keep audience interest at elevated
level), followed by some lectures on varying topics from the
topology of $4$-manifolds to topological aspects of mirror
symmetry and manifolds with special holonomy. Students will
be required to give talks on selected topics and be encouraged
to present them in written form.
Description: Thurston's theory of (self-)diffeomorphisms
of surfaces provides a classification of surface diffeomorphisms
roughly similar to the Jordan normal form of matrices. This
theory turns out to be a key tool in low-dimensional topology
and also has deep connections with analysis (with the theory
of the quasiconformal maps) and the theory of dynamical systems
(pseudo-Anosov homeomorphisms, discovered by Thurston, turned
out to be an important example in this theory). The
goal of the course is to provide an elementary introduction
to this theory starting with an introduction to plane hyperbolic
geometry. The course assumes familiarity with a basic course
in topology (differential manifolds and fundamental groups),
but not much more. It may be of interest not only to topologists.
|