time and place |

warm up question for the next lecture |

Although the course is a continuation of MAT621-spr09, it can be
taken independently. It consists of lots of separate stories, so the
title "Topics in Topology" is appropriate. The first of the topics
is orientations. The notion of * orientation* appears in Mathematics
here and
there, but many issues related to it are not covered in textbooks properly.
Even the very definition of orientation is formulated incorrectly in most of
standard textbooks.

**Goals.**
The goal of the course is to provide a broad
introduction to the **Low Dimensional Topology**.
This subject developed tremendously during the last 40 years.
New techniques both emerged inside of it
and came from other fields such as Quantum Physics, Hyperbolic
Geometry and Algebra.

**Objects.** Low Dimensional Topology studies a
great variety of objects:
classical knots and links, immersions of curves to surfaces,
manifolds of dimensions up to 4, surfaces in 3- and 4-manifolds, etc.
Each of the objects can be presented in many ways. For example,
a classical link can be presented by a planar diagram, as a
closed braid, by an algebraic equation, as a Legendrian lifting
of an arc immersed generically in a disk, etc.
In the course a systematic study of the objects and their geometric,
combinatorial and algebraic descriptions will be undertaken. We
will pay attention also to the objects which appear in the context of
Singularity Theory and Algebraic Geometry (both complex and real).

**Invariants.** The objects will be studied together
with their topological invariants.
The invariants will be developed and introduced gradually,
as the geometric problems require.
Nowadays there is a great variety of invariants.
Some of them comes from the classical Algebraic Topology, but
the nature of many others has not yet found a satisfactory
interpretation in a traditional topology.

**Prerequisites.** Most of the objects and their
invariants can be introduced in
quite a naive way. The corresponding fragments of the course
will be very elementary and independent from each other.
However, for a complete understanding of the material a knowledge
of elementary topology is required and some familiarity with
homology would be very desirable. On the other hand, I promise
to provide at a first request a short outline of the necessary theory.

**Style.** The material will be presented in small fragments.
Each of them will be made as much independent from the others as possible.
Special efforts will be made to connect the fragments to each other and to
other pieces of mathematics.