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Description: This course will be an
introduction to the subject of metric spaces and point-set
topology. We will explore the notions of what it means for
points in an arbitrary space to be "close" to each
other, what it means for a function from one space to another
to be continuous, and how exactly a coffee cup is the same
as a donut. Our textbook will be Michael Gemignani's "Elementary
Topology."
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 310 or MTH 320). One of the principal
goals of the course is to improve proof-writing skills.
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. |
Courses
Fall 2005
Courses
Spring2006
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Description: This course is an introduction
to Riemannian geometry. We begin with the notion of a Riemannian
metric on a smooth manifold, its associated connection, the
Levi-Civita connection, and the curvature of the connection.
The remainder of this course and Math 931 are explorations
of the curvature and its relation to the topology and geometry
of the manifold and analysis on the manifold. In Math 930
we exploit the geometry of geodesics to explore some of these
relationships. We hope to conclude with the classical and
fundamental Toponogov comparison theorem.
Description: This is a year long course
in Algebraic Topology: Topics will be selected from: Cohomology
theory: (Dualities, Local coefficients, Kunneth Formula, universal
coefficient Theorem. Examples and applications), homotopy
theory, cellular and simplicial approximations, the Whitehead-Hurwitz
theorem, fiber bundles, homology and homotopy exact sequences,
relation of homotopy and cohomology for CW-complexes, Potsnikov
Towers, obstruction theory, characteristic classes and, time
permiting, a few elements from spectral sequences.
The material selected is presented with an eye towards knot
and 3-manifold theory and complemented with a selection of
topics from 3-dimensional topology aiming to indicate how
some of the theories above reflect and are used in that dimension.
Description: This course will be navigated
through several active topics of low dimensional topology
and geometry, taking account of students' interest as we go
along. We will start with topology of calibrated submanifolds
inside the manifolds with G_2 and Spin(7) holonomy (these
require some Lie group theory). These are special kind of
3 and 4-manifolds. We then go on to study symplectic and contact
structures and various Floer homology theories (such as Heegard-Floer
homology) on 3 and 4-manifolds. We will apply these structures
to hands-on constructions on 3,4-manifolds (using framed links).
A lot of these topics will be covered with the audience participation,
students will be assigned many topics to present in class,
so come prepared to do a lot of work.
Description: This course is a two semester
introduction to surface mapping class groups. The topics to
be discussed in this course include: the Thurston theory of
surface diffeomorphisms, subgroups of surface mapping class
groups, automorphisms of surface mapping class groups, the
complex of curves, automorphisms of the complex of curves,
and related topics, as time allows.
Reference books:
J. S. Birman, Braids, links and mapping class groups, Ann.
Math. Studies 82, Princeton University Press, 1974.
N. V. Ivanov, Subgroups of Teichmuller modular groups, Translations
of Mathematical Monographs, A.M.S., Providence, RI, 1992.
N. V. Ivanov, Mapping class groups, in Handbook of geometric
topology, North-Holland, Amsterdam 2002, pp. 523-633.
Description: Algebraic geometry is one
of the oldest and deepest areas of mathematics. It began as
the study of the zero sets of polynomials of two or more variables;
that study leads to many interesting relations between algebra
and geometry. This course starts with introduction to classical
algebraic geometry: We begin by introducing affine and projective
varieties, study their basic properties and eventually concentrate
on the study of algebraic curves. We then discuss methods
of modern algebraic geometry which are also important tools
for studying number theory: schemes, sheaves and sheaf cohomology.
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