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Description: This course is the Topology
part of the qualifying sequence of Math 868-869: Geometry/Topology.
The topics to be covered will be selected from: Covering spaces,
fundamental group, van Kampen's theorem, homology and cohomology
theory, homotopy theory. The primary textbook is "Algebraic
Topology" by A. Hatcher.
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.
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Description: This is a course on complex
manifolds from a geometric analysis perspective. It will cover
complex tangent spaces and differential forms, ∂-bar
operators, Kahler structures, the Hodge theorem, curvature
and Chern forms. It will go on to cover Lefschetz theorems,
Serre Duality, the Hirzebrech-Riemann-Roch Theorem, and the
Kodaira Vanishing and Kodaira Embedding Theorems.
While this course is officially a sequel to Math 935, that
is not a prerequisite. Students should be familiar with manifolds,
differential forms, homology and cohomology. Beyond that,
the assumed background knowledge will be adjusted to the level
appropriate for the students in the class.
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 topics course in geometric
analysis will be run in a seminar style. We will cover topics
in variational theory and geometric flows.
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: This semsester we will
focus on the theory of algebraic surfaces. The topics to be
discussed in this part are: Intersection theory, adjunction
formula, Riemann-Roch theorem, Hodge Index theorem, Nakai-Moishezon
criterion, rational surfaces, Castelnuovo-Enriques criterion
for rationality, ruled surfaces, cubic surfaces, introduction
to other classes of surfaces (abelian, K3, Enriques and elliptic)
and (time permitting) Enriques/Kodiara classification of surfaces
or rational singularities and Artin's criterion for contractibility
or surfaces in char p. Reference books: Griffiths and Harris,
Beuaville, Barth-Hulek-Peters-Van de Ven, Hartshorne, Badescu.
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