Courses WS 18/19

Advanced topics in analysis V5B7
Topic: Introduction to dynamical systems and ergodic theory
Prerequisits: Solid knowledge in first year calculus and linear algebra
and same basic knowledge in measure theory, functional analysis
and point set topology
The class is accessible for students in the bachelor program
Literature:
K. Oliveira, M. Viana, Foundations of ergodic theory,
Cambridge University Press 2016.
Begin: Mo, October 8, 2018
Week 1: A short review of sigmaalgebras, measure spaces, measurable
and measure preserving maps
Example: Circle rotations
Poincare recurrence theorem and its improvement by Kac
Chapter 1.1, 1.2, 1.3.3 in the book by Viana and Oliveira
Week 2: Borel regular measures; recurrent points in the topological
sense;
minimal invariant sets, examples
Return maps are measure preserving, inducing
Examples: Interval exchange transformations
Chapter 1.2 and part of Appendix A.3, Sections 1.4.11.4.3 of the book
Week 3: The space P(X) of probability measures on a compact
metric space X;
the action of a continuous map on P(X);
the weak* topology on P(X)
Chapter 2.1 and 2.2 of the book
Week 4: The space P(X) of probablity measures
on a compact metric space $X$ is metrizable and compact
ergodic invariant measures
torus rotations with rotation vector of entries independent
over Q are ergodic
every orbit of such a rotation is dense
Chapter 2.2, 2.3 and 4.1 of the book
Week 5: The Kingman subadditive ergodic theorem
torus rotations
torus rotations with rotation vector of entries independent
over Q are uniquely ergodic
full Bernoulli shift
Chapter 4.34.5 of the book
Week 6: The Kingman subadditive ergodic theorem continued
The Neumann ergodic theorem
Example: Expanding maps of the circle and their symbolic
coding
Chapter 4.2 of the book
Week 7: Examples of ergodic measures:
Bernoulli shift
Torus automorphisms
Chapter 4.2.3, 4.2.5, 4.2.6 of the book
Week 8: Ergodic decomposition of invariant measures
Disintegration of measures with respect to a partition
The theorme of Rokhlin
Chapter 5.1.1 5.1.4 of the book
There is no class on Monday, Dec. 3, and on Wednesday, Dec. 5.
Courses SS 18

Advanced Geometry II V4D4
Mo 10.15h12h, SR 0.008
We 10.15h12h SR 0.006
Tutorial: Tu 12.15h14h, SR 1.007, Tutor: Andreas Pieper
Begin: April 16, 2018
Topic: Kähler geometry
Prerequisits:
Most of the material of the class on Advanced Geometry I
as summarized below
Exercise sheet 1
This sheet will be discussed on April 24 and will not be collected.
Week 1 4: Sheaves
Literature. The book by Griffith and Harris (p.3449, 139142)
The book by Huybrechts (Appendix B)
Chapter 4 of the book by Huybrechts
Exercise sheet 2
Due on April 25
Exercise sheet 3
Due on May 2
Exercise sheet 4
Due on May 9
Week 5. Chern classes of complex line bundles: Definition,
realization as de Rham cohomology classes of the curvature of
a connection.
The Ricci form as the curvature form of the canonical
bundle of a Kähler manifold
Literature: Section 4.4 of Huybrecht's book
Section 4.4 of Ballmann's book
Exercise sheet 5
Due on May 30
Exercise sheet 6
Due on June 6
Exercise sheet 7
Due on June 13
Exercise sheet 8
Due on June 20
Exercise sheet 9
Due on June 27. Exercise 4 has to be postponed to the following week, sorry!
Exercise sheet 10
Due on July 4.
Literature:
P. Griffith, J. Harris, Principles of algebraic geometry,
Wiley 1978.
C. Voisin, Hodge Theory and Complex Algebraic Geometry I,
Cambridge Univ. Press 2002.
R.O. Wells, Differential Analysis on complex manifolds,
Springer Grad. Texts in Math. 65, 1973.
S, Kobayashi, K. Nomizu, Foundations of differential geometry II,
Interscience, Wiley 1969.
Wiley 1978.
W. Ballmann, Lectures on Kähler manifolds, Eur. Math. Soc. 2006
A. Besse, Einstein manifolds, Springer Ergebnisse
D. Huybrechts, Complex geometry, Springer 2005
W. Barth. C. Peters, A. Van de Ven,
Compact complex surfaces, Springer 1984
F. Zheng,
Complex differential geometry, AMS Studies in advance math.,
AMS 2000

Graduate seminar on differential geometry S4D2
with Werner Ballmann and Anna Siffert
More information
Organizational meeting: April 9, 2018
Courses WS 17/18

Advanced Geometry I V4D2
Second final: Tuesday, March 6,
oral examination
Topic: Introduction to Kähler geometry
Summary:
Week 1: Basic properties of holomorphic maps; complex
manifolds.
Week 2: Holomorphic vector bundles,
the holomorphic tangent bundle,
types of differential forms, Chern connection
Sections 2.2 and 3.2 in Ballmann's book
Week 3: Chern connection and its curvature, examples,
line bundles
p.3132, p.1415, Section 3.4 in Ballmann's book
Week 4: Divisors and line bundles, the Picard group, curvature
Section 3.4 in Ballmann`s book, Section 5.6 in Jost's book
Section 2.2 in Huybrechts's book contains a purely sheaftheoretic
account which I did not pursue
Week 5: Degree of divisors and line bundles on Riemann surfaces
Relation to curvature
Kähler manifolds: Characterization by LeviCivita and Chern connection
Section 5.6 in Jost's book
Proposition 2.18 (p.20) in Ballmann's book (the signs in the proof are wrong)
p.4647 in Ballmann's book (the formula on p.47 is wrong)
Chapter 3 in Voisin's book is recommended for reading
Week 6: Various characterization of Kähler manifolds, Examples
Chapter 3 in Voisin's book
Part of Chapter 4 in Ballmann's book (Hopf manifold)
Week 7: Positivity of line bundles revisited.
Kähler metrics on blowups of points and on projective
bundles
Chapter 3 in Voisin's book
Week 8: Hodge star operator and beyond
Week 9:
The Hodge star operator, Dolbeault cohomology, the
Laplacian, harmonic (p,q), forms, formulation of the
main theorem of Hodge theory
Everything can be found in Griffith Harris's book (which is full of
small mistakes)
Week 10: Applications of the Hodge theorem to compact Riemann surfaces:
Computing the dimension of the vector space of
holomorpic oneforms, construction of meromorphic functions.
A good source is Chapter 7 of Donaldson's book.
Week 11: The Riemann Roch theorem
The Kodaira
embedding theorem for surfaces.
Chapter 7 of Donaldson's book, chapter 9 of Jost's book
Week 12: Cohomology of Kähler manifold
Prop. 3.14, Section 6.1.1, 6.1.2 and part of 6.1.3 of Voisin's book
Week 13: Hodge theory for differential form with values
in a holomorphic line bundle
The Kodaira vanishing theorem
Section 3 and 5 of Ballmann's book
Week 14: The Kodaira embedding theorem
Found in the book by Griffith and Harris
Literature:
W. Ballmann, Lectures on Kähler manifolds, Eur. Math. Soc. 2006
D. Huybrechts, Complex geometry, Springer
R. Narasimhan, Several complex variables, Chicago Lect. in Math. 1971.
R.O.~Wells, Differential Analysis on Complex Manifolds,
Springer Graduate Texts in Math. 65 (1980)
Chapter IX of S. Kobayashi, K. Nomizu,
Foundations of Differential Geometry II, 1969.
J. Jost, Compact Riemann Surfaces, Universitext, Springer
P. Griffith, J. Harris, Principles of Algebraic Geometry,
Wiley 1978.
C. Voisin, Hodge Theory and Complex Algebraic Geometry I,
Cambridge Univ. Press 2002.
R. Bott, L. Tu, Differential forms in algebraic topology,
Springer Grad. Text, 1982.
S. Donaldson, Riemann surfaces, Oxford Grad. Text in Math. 22, 2011.
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