Description: 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,
Postnikov 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 investigate
some of the basic properties of these two positive curvature
conditions. We will, in particular, study the role of minimal
hypersurfaces and the Bochner formula in the study of positive
scalar curvature and the role of minimal surfaces in the study
of positive isotropic curvature.
Description: This is a course in General
Relativity from a geometric perspective. The goal is to understand
gravitational radiation in terms of geometric analysis (in
particular, to understand the Peeling Theorem without recourse
to conformal compactifications, and to understand the relation
between ADM and Bondi mass).
Students should be familiar with the objects of Riemannian
geometry: manifolds, vector and tensor fields, differential
forms, metrics and curvature, etc. and have a solid knowledge
of linear algebra.
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 will be an
introduction to the theory of smooth 4-manifolds. We will
study the classical theory of 4-manifolds including Wall's
theorems and Rohlin's theorem along with an introduction to
Kirby calculus techniques. Then we will study Seiberg-Witten
invariants and applications to the construction and classification
of 4-manifolds.
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.
Description: This course is an introduction
to moduli spaces in algebraic geometry. We will discuss moduli
problems and various solutions such as coarse and fine moduli
spaces, and briefly algebraic spaces and stacks. The emphasis
will be on examples such as moduli of curves, and vector bundles
on curves. Also Hilbert schemes will be constructed using
geometric invariant theory.
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