COURSE DESCRIPTION FOR ALGEBRA II (V3A2) WINTER

COURSE DESCRIPTION FOR ALGEBRA II (V3A2) WINTER

COURSE DESCRIPTION FOR
ALGEBRA II (V3A2)
WINTER SEMESTER 2016/17
PROF. DR. MICHAEL RAPOPORT
The subject matter of the course is Algebraic Number Theory. This means that we will study
the arithmetic of the rings of integers in finite extensions of Q or Fp [X], and their local analogues
Qp or Fp [[X]]. Highlights will be: the prime ideal decomposition in Dedekind rings, the finiteness
of class groups, the Dirichlet unit theorem, Hilbert ramification theory, density of ideals in an
ideal class, the Dedekind zeta function and the distribution of prime ideals, the class number
formula.
The prerequisites are:
• Basic concepts of Commutative Algebra, in particular localization, completion, going up/down
(the book ”Commutative Algebra” by Atiyah/Macdonald contains more than necessary).
• Later some knowledge of Complex Function Theory.
References
[1]
[2]
[3]
[4]
J. Neukirch: Algebraic number theory, Springer
E. Hecke: Lectures on the theory of algebraic numbers, Springer
G. Janusz: Algebraic number fields, Am. Math. Soc.
P. Samuel: Algebraic theory of numbers, Houghton Muffin Co.
1
Set Theory / Mengenlehre (V3A4/F4A1-4)
Winter Semester 2016/17
Prof. Dr. Peter Koepke
Contents
Sets are ubiquitous in present-day mathematics. Basic structures are
introduced as sets of objects with certain properties. Fundamental notions like
numbers, relations, functions and sequences can be defined from sets. Set
theory, together with formal logic, is thus able to provide a universally
accepted foundation for mathematics.
Set theory also comprizes a theory of the (mathematical) infinite through the
study of infinite sets and their combinatorics. Generalizing the finitary
arithmetical operations leads to an infinitary arithmetic of cardinal numbers
which has surprising properties. For the smallest infinite cardinal ℵ0 which is
the cardinality of the set of natural numbers we have: ℵ0+ℵ0 = ℵ0, ℵ0xℵ0 =
ℵ0, whereas the value of 2ℵ0 is (provably!) undetermined by the common
principles of set theory.
The lecture course Set Theory will cover the following basic material:
The Zermelo-Fraenkel axioms of set theory; relations, functions, structures;
ordinal numbers, induction, recursion, ordinal arithmetic; number systems:
natural, integer, rational, real numbers; the axiom of choice and equivalent
principles; cardinal numbers and cardinal arithmetic; sets of real numbers,
Borel sets, projective sets, regularity properties.
Possible further topics:
The Hausdorff paradox which gives an impressive example of a set which is
not Lebesgue-measurable; infinitary combinatorics.
The initial development of Zermelo-Fraenkel set theory is rather canonical
and is portrayed in similar ways in many books on set theory; references will
be given. Lecture notes will be made available.
V3B1 PDG UND FUNKTIONANALANALYIS, PDE AND
FUNCTIONAL ANALYSIS
HERBERT KOCH
Contents
Distributions and Fourier transform, examples of Banach spaces: Lebesgue spaces
and their duals, spaces of continuous functions and measures, Sobolev spaces. The
structure theorems of functional analysis and applications: The theorem of HahnBanach, Banach-Steinhaus and the uniform boundedness principle, closed graph
and open mapping theorem. Spectral theory of compact operators. The Lemma of
Lax-Milgram and elliptic PDEs. Examples of bounded operators and applications
to PDE.
Literature
(1) Alt: Linear functional analysis / Lineare Funktionalanalysis. Springer.
(2) Lieb and Loss: Analysis. AMS
(3) Werner: Funktionalanalysis. Springer
Required and recommended background
• Required: Good knowledge of Lebesgueintegral.
• Recommended: Basic knowledge of PDEs (harmonic functions, heat equation, Poisson problem)
Time and place
Wednesday 10 (c.t.) - 12, Friday 8 (c.t.) - 10, (Kleiner Hörsaal, Wegelerstr. 10)
Contact Herbert Koch
Mathematisches Institut,
Endenicher Allee 60,
53115 Bonn,
Room 2.011,
Tel.: -3787,
[email protected]
1
V3B3 – Global Analysis I
Prof. Dr. Matthias Lesch
Winter term 2016/17
Global Analysis I is the first course in a series of courses on the analysis of
manifolds.
Global analysis uses from geometry, topology and partial differential
equations to study global behavior of (partial) differential operators.
In this course we will cover:
- Differential forms and Stokes' Theorem
- vector fields and dynamical systems
- vector bundles and differential operators
- de Rham Cohomologie and Hodge theory
Prerequisites: Real Analysis and Linear Algebra, basic knowledge of
differential topology and functional analysis is helpful but not mandatory.
The course comes with a recitation class and weekly exercises.
V3C1 – Linear and Integer Optimization
Dr. Ulrich Brenner
Winter term 2016/17
Goals:
Understanding the basic connections between the theory of polyhedra and
the theory of linear and integer programming.
Knowledge of the most important algorithms, ability to model and to solve
practical problems as mathematical optimization problems.
Implementation of algorithms.
Topics:
Modelling optimization problems as (integral) linear programs, polyhedra,
Fourier-Motzkin Elimination, Farkas' Lemma, duality theorems, simplex
algorithm, network simplex, ellipsoid method, conditions for integrality of
polyhedra, TDI-systems, total unimodularity, cutting plane methods.
Requirements:
The modules “Lineare Algebra” and “Algorithmische Mathematik”
Recommendable books:
- A. Schrijver, Theory of Linear and Integer Programming,
Wiley, New York, 1986.
- V. Chvátal, Linear Programming, Freeman, New York, 1983.
- B. Gärtner, J. Matousek, Understanding and Using Linear Programming,
Springer, Berlin, 2006.
- B. Korte, J. Vygen : Combinatorial Optimization: Theory and Algorithms.
Springer, 2012.
- R. Ahuja, T. Magnanti, J. Orlin : Network Flows. Prentice-Hall 1993
Topology I — Course Description
Prof. Dr. Carl-Friedrich Bödigheimer
winter term 2016/17
(1) Content
The course is an introduction into singular homology theory. So it is already
algebraic topology, although this word occurs as a course name only later in the
master program; it is part of a longer series, namely (0) Introduction into Geometry
and Topology (summer term 2016), (1) Topology I (winter term 2016/17),
(2) Topology II (summer term 2017), (3) Algebraic Topology I (winter term 2017/18),
(4) Algebraic Topology II (summer term 2018).
We will define the singular homology groups H∗(X) of a space X and prove that they
constitute a homology theory and satisfy the axioms of Eilenberg and Steenrod.
Our next goal is to compute the homology groups of some important spaces like
spheres, surfaces and projective spaces or some 3-manifolds. To achieve this we
will need to define simplicial homology groups and therefore simplicial complexes.
As an application we define degrees of maps between spheres, indices of vector
fields, winding numbers of curves on surfaces, linking numbers of curves in space
etc.. We will use it for applications like the fundamental theorem of algebra, or
Brouwer’s fixed point theorem, as well the invariance of domain, of dimension and of
boundary of manifolds.
The course Topology II will introduce singular cohomolgy and study the interrelation
between homology and cohomology and the multiplicative structures in cohomology.
Our goal will then be the Poincare duality for manifolds.
(2) Prerequisites
The course assumes a good understanding of set theoretic topology (as much as is
covered in the book by Bredon listed below), including a basic knowledge of
manifolds, and also a good knowledge of the fundamental group and covering
spaces (see again the book by Bredon and he book by Hatcher). We need the
beginnings of group theory and modules over commutative rings (mainly principal
ideal domains) will be used; for both topics see chapters I - III in the book by Lang.
(3) Recommended Literature
There are many very good textbooks covering the topic. The books below cover
singular homology and cohomology, thus the content of the course Topology I
(winter term 2016/17) as well as the next course Topology II (summer term 2017).
• G. E. Bredon: Topology and Geometry. Springer Verlag (1993).
• A. Dold: Lectures on Algebraic Topology. Springer Verlag (1973).
• A. Hatcher: Algebraic Topology. Cambridge University Press (2002).
• S. Lang: Algebra. Addison-Wesley (1993 ).
Bachelorstudiengang Mathematik
Rheinische Friedrich­Wilhelms­Universität Bonn
Code:
V3F2
Title:
Introduction to Stochastic Analysis
Lecturer:
Dr. Matthias Erbar
The lecture will focus mainly on the following topics:
•
Brownian motion
•
martingales in discrete and continuous time
•
continuous semimartingales
•
stochastic integration and Ito's formula
•
stochastic differential equations
Possible references are:
•
•
•
D. Williams : Probability with martingales, Cambridge UP 1991
M. Steele : Stochastic calculus and nancial applications, Springer 2001
I. Karatzas, S. Shreve : Brownian motion and stochastic calculus, Springer 1991
Prerequisites:
being familiar with concepts from measure theoretic probability theory such as
•
Probability spaces, expectation and variance, notions of convergence of random variables
•
laws of large numbers, characteristic functions, central limit theorem
•
conditional expectations
desirable is familiarity with concepts from stochastic processes such as
•
conditional expectation
•
Markov chains, ergodicity
•
Brownian motion
helpful might be also knowlede in linear functional analysis and partial differential equations
possible reference for this are:
•
•
•
•
J. R. Norris : Markov chains. Cambridge UP 1997
R. Durrett : Probability: Theory and examples. Duxbury Press 1995
L. Breiman : Probability. Addison­Wesley 1968
M. Reed und B. Simon: Methods of modern mathematical physics, Volume 1: Functional Analysis,
Academic Press 1981