Lehrveranstaltungen

Preliminary Program

This master course is a first introduction to the topics of differential geometry. We will discuss differentiable manifolds as geometric objects in an intrinsic approach. A particular emphasis will be put on the global calculus on manifolds, showing how the coordinate-based calculations can be minimized as far as possible. After manifolds, we discuss vector bundles as the next important ingredient in differential geometry. Integration on manifolds will be presented in two ways, using an orientation and without orientation. We will see the most important cohomology theories attached to manifolds. If time permits, we will also give a short introduction to Lie groups

Literature

A detailed list of references will be available in the WueCampus course.

Prerequisites

A solid knowledge of the bachelor courses on analysis and linear algebra is required, knowledge in topology will be useful.

WueCampus

Preliminary Program and Prerequisites

This master course is a first introduction to the topics of Lie theory. We will discuss the theory of finite-dimensional Lie groups their Lie algebras and if time permits basic representation theory. The main focus will be on matrix Lie groups, i.e., closed subgroups of the general linear group. Apart from a certain amount of mathematical maturity, the prerequisites for this course are a familiarity with the rudiments of group theory, linear algebra 1+2 , the basic aspects of analysis in several real variables and elementary concepts of differential geometry. Some basic knowledge in the theory of ordinary differential equations and interest in theoretical physics might not harm.

We recommend to register to the parallel master course on differential geometry, as well.

Literature

In due course a detailed list of references will be available in the WueCampus course, see also the first lecture 16.10.24, 14:15, SE30.

WueCampus

Preliminary Program

This course is a first introduction to functional analysis. It can be used either as bachelor course or as master course (then with higher level in the tutorials and the exams). The topics we cover Banach spaces and continuous maps between them, Hilbert spaces with their geometry and operators on them. If time permits we will give an introduction to spectral calculus of self-adjoint operators.

Literature

A detailed list of references will be available in the WueCampus course.

Prerequisites

A solid knowledge of the bachelor courses on analysis and linear algebra is required, knowledge in point-set topology as well as in abstract measure theory will be useful.

WueCampus

Preliminary Program

This master course is a first introduction to geometric mechanics. We will discuss classical Newtonian mechanics using differential geometric methods within the context of symplectic geometry. The Hamiltonian approach to mechanics based on cotangent bundles as phase spaces will be central for this approach. The Lagrangean formulation of mechanics will be formulated on the tangent bundle of the configuration space. The comparison between the two will give, among many other aspects, a new look on Riemannian geometry through the symplectic looking glass. But geometric mechanics will not stop at these two (equivalent) versions but requires more general symplectic manifolds as phase spaces. In particular, the presence of symmetries and conserved quantities will lead to phase space reductions which yields quite generic symplectic manifolds as result. Finally, Poisson manifolds provide a yet more general framework of great importance when it comes to questions like quantization.

Literature

A detailed list of references will be available in the WueCampus course.

Prerequisites

The course can be see as a continuation of the lecture Differential Geometry. Some good knowledge in basic notions of differential geometry (manifolds, Cartan calculus) is required. If in doubt, please get in contact early, then we can arrange the details.

WueCampus

Program

In this RiG  we will investigate the basic notions from sheaf theory: the agricultural background will not be touched so much, instead we focus on the mathematical notions of germs, sheaves, and  stalks. The basic idea behind sheaves is to provide a quite general framework of how local data can be patched together to yield global data. Typically, the opposite direction is easy, one just restricts the global data to the local situation. The opposite direction requires gluing and consistency between local data on different neighbourhoods. The possible obstructions are then controlled by sheaf theory.

Sheaves come typically with additional structures, leading to sheaves of groups, of rings, of algebras and modules. Then algebraic operations can be carried over to sheaves of such algebraic  structures as well, leading to important constructions of new sheaves out of previously given ones.

The goal of this RiG will then be to establish the cohomology theory of sheaves and provide various applications. This way, obstructions for local-to-global constructions are best encoded. Cech-cohomological methods will provide efficient and essentially combinatorial methods to actually compute cohomologies.

Prerequisites

For this RiG some background knowledge in topology is certainly very useful and should be acquired quickly if not yet present. Most of the examples we will investigate come from (smooth)  differential geometry. To understand the resulting motivations it will be thus very convenient to have a basic knowledge of differentiable manifolds and vector bundles over them. Alternatively,  motivations and important examples come from complex analysis and in particular from holomorphic functions of several variables.

Literature

A detailed list of references will be available in the WueCampus course.

WueCampus

Preliminary Program

This master course is a first introduction to principal fiber bundles, connections and curvature.

Literature

A suitable list of references will be available in the WueCampus course.

Prerequisites

The course can be seen as a continuation of the master lecture Differential Geometry taught in WS 2023/24. However, many aspects of that course will be  reviewed and last but not least deepened and further extended. Certainly, good knowledge in calculus and linear algebra is indispensable.

More details will be discussed during the first lecture, Friday 19. April, 10:00, SE31.

Exercises start Thursday, 25. April, 12:00, SE40.

WueCampus

Aufbauend auf LA 1-2 werden weiterführende Themen der Linearen Algebra behandelt.

Die Teilnehmer halten einen ca. 60 min Vortrag incl. Diskussion, ev. Handout,

sowie geben vor dem Vortrag eine 4-seitige PDF-Zusammenfassung ab.

Es herrscht Anwesenheitspflicht.

Anmeldung:

per email (einzeln, via stud-mail-account!) bis 18.4.24, 18:00, zwingend erforderlich.

Vorbesprechung incl. Themenvergabe:

Donnerstag 25.4.24, 14:00, S0.106.

Notenvergabe am Semesterende.

WueCampus

Program

In this RiG  we will investigate more applied topics from differential geometry building on the course Differential Geometry, taught in WS 2023/24.

In particular, we concentrate on pseudo-Riemannian and semi-Riemannian differential geometry. 

Registration, by email (stud-email !), is indispensable. Deadline is Thursday, 18. April, 2024, 18:00.

Details about location, format, contents, etc.. will be give on Friday 19. April on the WueCampus course page.

WueCampus

Fortsetzung der Analysis 1 Grundvorlesung aus dem letzten Semester.

Preliminary Program

The aim of this lecture is to find mathematical models for the observable algebras of quantum systems. We will take the necessary requirements from physics as orientation to pass from general algebraic structures, *-algebras and their states, to more analytic versions. Analysis will be needed to arrive at meaningful and manageable algebras. This will include among others the class of C*-algebras. Having found this class as physically very appealing algebras we will study their properties in detail. Here the spectral calculus plays a dominant role. While having already very nice properties, an abstract C*-algebra is not yet enough for physical models: one needs to implement the abstract algebras as an algebra of operators on a Hilbert space. Thus we will investigate the bounded operators on a Hilbert space from a more conceptual point of view to establish the bounded-measurable functional calculus for Hermitian operators. If time permits, we will extend our discussion then to unbounded operators and investigate the spectral theory of possibly unbounded self-adjoint operators in Hilbert spaces.

Prerequisites

Elementary knowledge on functional analysis (Banach spaces, Hilbert
space etc) will be assumed and can, if wanted, be briefly recalled. A
familiarity with basic concepts of quantum mechanics may be useful but
are not necessarily required.

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Tuesday 10-12 and Friday 8-10
Tutorial: Friday 12-14

WueCampus course

Preliminary Program

Topological and differential manifolds

Tangent structures

Immersions and submersions

Glimpse of "classical differential geometry"

Short Intro. to Lie groups and Lie algebras

Fiber bundles

Tensor analysis

Connections and covariant derivatives

Prerequisites

Solid knowledge in

1)

Calculus I-IV, i.e., Real Analysis 1-2, Geometric Analysis (Differential Forms, Stokes Thm.), Course on Integral Calculus (Gauß, Green, div, rot, grad, etc...)

2)

Linear Algebra I-II

3)

Ordinary Differential Equations (Linear Theory, Ex. & Uniqueness of solutions, etc...)

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Monday 14:15 - 15:45, venue: S1.101 and Thursday 8:30-10:00, venue: SE30, Start Thursday 19.10.23
Exercises: Thursday 14:00 - 15:30, venue: SE30, Start: details on Thursday 19.10.23 8:30

wuecampus

Die Lineare Algebra 2 ist einer der beiden grundlegenden Vorlesungen des Mathematikstudiums und behandelt die Theorie der linearen Vektorräume und Abbildungen mit einigen ihrer Anwendungen. In diesem Semester bildet sie die Fortsetzung Lineare Algebra 1 aus dem vergangenen Sommersemester 2023. Inhaltlich schließt sie unmittelbar daran an. Alle Details zu dieser Vorlesung finden Sie im WueCampus Kurs, in den Sie automatisch eingeschrieben werden, wenn Sie sich in WueStudy fuer diese Vorlesung angemeldet haben. Dies geschieht kurz nach Vorlesungsbeginn Die interessierten Studierenden können sich aber auch unabhängig davon selbständig dort einschreiben.

 WueCampus Kurs

In this course we will find geometric formulations for various field theories from mathematical physics:
the most important ones will be the Yang-Mills field theories from particle physics, but also other
field theories like the more phenomenological ones from fluid dynamics or solid state physics are in
principle part of this geometric approach. The main idea is that fields will be described as sections
of certain fiber bundles over a spacetime manifold. In many situations the fiber bundle is actually a
vector bundle but we will see situations where we have to go beyond this point of view. Closely related
to the geometric approach to field theory is the role of symmetries played by Lie group actions. In
fact, one can turn the accidental presence of a symmetry into a principle and construct field theories
based on the symmetry directly. This will essentially be the idea in the Yang-Mills field theories. In
the lecture we will mainly consider the kinematic aspects of field theory, i.e. the geometric description
of fields. Understanding the dynamical aspects, i.e. the study of actual field equations, is then a second and typically
much more difficult step.

WueCampus course

Die Analysis 1 ist einer der beiden grundlegenden Vorlesungen des Mathematikstudiums und behandelt die Analysis von Funktionen einer Variablen. Im folgenden Semester gibt es dann eine Fortsetzung Analysis 2. Alle Details zu dieser Vorlesung finden Sie im WueCampus Kurs, in den Sie automatisch eingeschrieben werden, wenn Sie sich in WueStudy fuer diese Vorlesung angemeldet haben. Dies geschieht kurz vor Vorlesungsbeginn.

WueCampus Kurs

Lectures:

Monday 16:00, SE 40; Tusday 14:00, SE 40

Exercises:

Friday 8:30, SE 40

Start:

18. April 2023 14:00, SE 40

Prerequisites:

This course builds on the master course in differential geometry from WS 22/23.

Contents:

This master course covers important topics in Riemannian and pseudo-Riemannian geometry.

In particular, definite and indefinite metrics, curvature, geodesics, connections, covariant derivatives.

Moreover, we study certain aspects of symmetric and natural reductive homogeneous spaces.

Literature:

Suggestions to literature will be published on Wuecampus in due course.

https://wuecampus.uni-wuerzburg.de/moodle/course/view.php?id=60058

Die Lineare Algebra 1 ist einer der beiden grundlegenden Vorlesungen des Mathematikstudiums und behandelt die Theorie der linearen Vektorräume und Abbildungen mit einigen ihrer Anwendungen. Im folgenden Semester gibt es dann eine Fortsetzung Lineare Algebra 2. Alle Details zu dieser Vorlesung finden Sie im WueCampus Kurs, in den Sie automatisch eingeschrieben werden, wenn Sie sich in WueStudy fuer diese Vorlesung angemeldet haben. Dies geschieht kurz vor Vorlesungsbeginn.

 WueCampus Kurs

Dieser Bachelorkurs behandelt Kurven und Flächen in der Sprache der modernen Differentialgeometrie.
Er bietet somit eine Brücke zwischen intuitiv anschaulichen geometrischen Objekten und der eher abstrakten Begriffswelt,
die in vielen weiterführenden Veranstaltungen benötigt wird.

Wesentliche Konzepte wie Tangentialvektoren, Metrik, Krümmung, etc werden anschaulich motiviert und viele Beispiele betrachtet.

WueCampus Kurs

Prerequisites:

For this RiG you will need the knowledge/contents of the course Riemannian and pseudo-Riemannian geometry from the last semester, summer 2022.

The RiG will serve as an entry ticket for a potential master thesis in differential geometry.

Registering exclusively per email (stud-mail account!) before Monday 17.10.22, 18:00

Diese Vorlesung baut auf Analysis 1+2 auf. 

Inhalte sind u.A.:

• Fortführung der Analysis von Funktionen mehrerer Veränderlicher;
• Lebesgue-Maß und Lesbegue-Integral im Rⁿ,
• Integralsätze.

Vorlesung:

Di.12:15 HS 2 , Mi. 14:00 HS 2, Start ist 18.10.22

Übungen:

Siehe Vorlesungsverzeichnis sowie Details auf WüCampus (ab 10.10. online)

Literatur, Klausurtermin, etc siehe WüCampus.

Aufbauend auf LA 1-2 werden weiterführende Themen der LA behandelt.

Die Teilnehmer halten einen ca. 60 min Vortrag incl. Diskussion, ev. Handout,

sowie geben eine 4-seitige PDF-Zusammenfassung ab.

Es herrscht Anwesenheitspflicht.

Anmeldung:

per email (einzeln, via stud-mail-account!) bis 17.10.22, 18:00, zwingend erforderlich.

Vorbesprechung incl. Themenvergabe:

Donnerstag 20.10.22, 10:15, S1.101.

Notenvergabe am Semesterende.

Preliminary Program

In mathematical models of physical systems, the physical observables are typically described by C*-algebras or von Neumann algebras. While this gives a very elegant and powerful spectral calculus, many situations will not directly yield a such nice classes of algebras. In various quantization theories the construction of C*-algebras is difficult or unclear. One way out is to focus on the algebraic features before taking into account the analytic issues as well. This is the main motivation for considering *-algebras without any analysis involved. Beyond quantization theories, other important examples are group algebras or universal enveloping algebras of Lie algebras but also algebras of differential operators. Here one typically has by no means a C*-norm available.

The aim of this RiG is to find a common algebraic framework for a reasonable representation theory of such algebras. It turns out that aspects of positivity can be formulated in a entirely algebraic way yielding interesting structures for the representation theory. In general, it will be difficult if not impossible to understand the representation theory of a given algebra completely. However, and this is quite  surprising, it might be possible to compare it to the representation theory of a different algebra and determine whether or not the two algebras have the same representation theory. This is the main task of  Morita theory, which we will present both in a purely ring-theoretic context and in an adapted version for *-algebras. We will develop the necessary category-theoretic notions to put the question of Morita equivalence in the right perspective.

Literature

A detailed list of references will be available in the WueCampus course.

Prerequisites

For this RiG you will not need much prerequisites. In fact, a good knowledge in (multi-) linear algebra will be sufficient for most things. Depending on the examples you want to discuss, some ideas about Lie algebras, differential geometry, algebra, functional analysis might be useful but certainly not necessary. We will meet some category-theoretic notions on the way, they will be developed in the lecture part of the RiG.

WueCampus

Preliminary Program

This master course is a first introduction to the theory of Lie groups and Lie algebras. It should be seen as a companion course of the differential geometry course. In the beginning we will discuss matrix Lie groups and their Lie algebras to obtain the important classes of examples. On the algebraic side we will investigate Lie algebras in quite some details, explaining their structures. In a second part we will then use techniques of differential geometry to define and study Lie groups in general. Depending on time and interests we then can discuss representation theory with some functional-analytic aspects.

Literature

A detailed list of references will be available in the WueCampus course.

Prerequisites

This is an introductory master class which requires only solid knowlwdge of bachelor mathematics. However, it will be advantageous to have some familiarity with differential geometry as we will need notions like manifolds and their Cartan calculus in the second half of the lecture.

WueCampus

In this master course we provide a first contact with the vast subject of algebraic topology. The main theme of the course is the deep connection between algebra and topology, that we begin to investigate via the study of algebraic invariants for topological spaces. However, it is not only algebra used in topology, but we are also led to introduce advanced algebraic techniques that are motivated from topology. The algebraic invariants are naturally formulated using the language of categories. We will consider homology and cohomology functors, homotopy groups and develop on the topological side the language of simplex categories and CW-complexes, on the algebraic side chain complexes and their homotopies and category related concepts.

Preliminary Program

This master course is a first introduction to geometric mechanics. We will discuss classical Newtonian mechanics using differential geometric methods within the context of symplectic geometry. The Hamiltonian approach to mechanics based on cotangent bundles as phase spaces will be central for this approach. The Lagrangean formulation of mechanics will be formulated on the tangent bundle of the configuration space. The comparison between the two will give, among many other aspects, a new look on Riemannian geometry through the symplectic looking glass. But geometric mechanics will not stop at these two (equivalent) versions but requires more general symplectic manifolds as phase spaces. In particular, the presence of symmetries and conserved quantities will lead to phase space reductions which yields quite generic symplectic manifolds as result. Finally, Poisson manifolds provide a yet more general framework of great importance when it comes to questions like quantization.

Literature

A detailed list of references will be available in the WueCampus course.

Prerequisites

The course can be see as a continuation of the lecture Differential Geometry. Some good knowledge in basic notions of differential geometry (manifolds, Cartan calculus) is required. If in doubt, please get in contact early, then we can arrange the details.

Vorläufiges Programm

Die zweisemestrige Vorlesung Lineare Algebra I und II ist neben der parallelen Analysisvorlesung die zentrale Grundlage eines jeden Mathematikstudiums. Ihre Bedeutung kann daher kaum überschätzt werden.

Im zweiten Teil werden nun einige Details zur Spektraltheorie nachgereicht, dann liegt der Schwerpunkt auf spezielleren Themen der linearen Algebra. Wir werden lineare Differentialgleichungen durch Matrix-Exponentation lösen, Quotienten und Tensorprodukte eingehend studieren und schließlich Bilinearformen betrachten.

Literatur

Im WueCampus Kurs gibt es eine detaillierte Literaturliste.

Prerequisites:

This course builds on the master course in differential geometry from WS 21/22.

Würzburg students are invited to download the slides from the corresponding Wuecampus course.

Contents:

This master course covers important topics in Riemannian and pseudo-Riemannian geometry.

In particular, definite and indefinite metrics, curvature, geodesics, connections, covariant derivatives.

Moreover, we study certain aspects of symmetric and natural reductive homogeneous spaces.

Literature:

Suggestions to literature will be published on Wuecampus in due course.

https://wuecampus.uni-wuerzburg.de/moodle/course/view.php?id=52003

Dieser Bachelorkurs ist zweigeteilt aufgebaut. Der erste Teil beschäftigt sich mit den grundlegenden Definition von topologischen Räumen mittels offener und abgeschlossener Mengen sowie dem Begriff der Umgebung, von stetigen Abbildungen, Abzählbarkeits- und Trennungsaxiomen und Kompaktheitsbegriffen. Universelle Konstruktionen wie direkte Summe, direktes Produkt, pullback, intiale, terminale Topologien, werden nach einer kurze Einführung in die Sprache der Kategorien einheitlich behandelt. Beispiele aus der Analysis und Geometrie werden diskutiert. Im zweiten Teil beschäftigen wir uns mit dem Begriff der Homotopie und betrachten die Fundamentalgruppe eines topologischen Raums.

Preliminary Program

Topological and differential manifolds

Tangent structures

Immersions and submersions

Glimpse of "classical differential geometry"

Short Intro. to Lie groups and Lie algebras

Fiber bundles

Tensor analysis

Connections and covariant derivatives

Prerequisites

Solid knowledge in

1)

Calculus I-IV, i.e., Real Analysis 1-2, Geometric Analysis (Differential Forms, Stokes Thm.), Course on Integral Calculus (Gauß, Green, div, rot, grad, etc...)

2)

Linear Algebra I-II

3)

Ordinary Differential Equations (Linear Theory, Ex. & Uniqueness of solutions, etc...)

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Tuesday 14-16 and Wednesday 16-18
Exercises: Thursday 8-10

wuecampus

Vorläufiges Programm

Die zweisemestrige Vorlesung Lineare Algebra I und II ist neben der parallelen Analysisvorlesung die zentrale Grundlage eines jeden Mathematikstudiums. Ihre Bedeutung kann daher kaum überschätzt werden.

Zum einen werden die Begriffe der linearen Algebra, Vektorräume und linearen Abbildungen, in jeder weiteren Mathematikveranstaltung benötigt und benutzt werden. Dies gilt sowohl für die Entwicklung der theoretischen und angewandten Mathematik als auch für die zahlreichen Anwendungen, nicht zuletzt in der (mathematischen) Physik und der Wirtschaftsmathematik, aber auch in der Schulmathematik.

Zum anderen wird in der linearen Algebra Mathematik das erste Mal, im Vergleich zur Schulmathematik, auf wissenschaftlichem Niveau betrachtet. Insbesondere werden in der Vorlesung die grundlegenden Techniken der axiomatischen Herangehensweise der Mathematik, die Notwendigkeit einer stringenten Beweisführung und die zugehörige Abstraktion erlernt. Dies ist erfahrungsgemäß am Anfang nicht immer einfach.

Literatur

Im WueCampus Kurs gibt es eine detaillierte Literaturliste.

Termine

Vorlesung: Montag und Freitag 8 -- 10 Uhr

Übungen: Termine werden auf WueStudy/WueCampus bekanntgegeben und verteilt.

WueCampus Kursraum

Preliminary Program

In this RiG we discuss more advanced topics in Riemannian and pseudo-Riemannian geometry and further generalizations from sub-Riemannian geometry. 

Prerequisites

The course can be see as a continuation of the lecture Riemannian and pseudo-Riemannian Geometry (SS 2020 and/or SS 2021). 

Literature

A detailed list of references will be available in the WueCampus course.

Dates

An introductory zoom session will be held Tuesday 19.10.2021 16:00-16:30, for more details see

 wuecampus

Preliminary Program

In this RiG we discuss deformation quantization of classical Poisson algebras.  Starting with canonical Poisson brackets for the functions on a phase space M of a mechanical system one tries to find a description of the corresponding quantum system by deforming the classical Poisson algebra of observables C^\infty(M) into a new, non-commutative algebra. This is done by a (formal) star product *, an associative product for C^\infty(M)[[\hbar]] which in zeroth order coincides with the classical product and which in first order of the commutator gives the Poisson bracket.

In the lecture part of the RiG we study such star products, both from the conceptual point of view as well as for various classes of examples. Depending on the preferences of the participants, we can either take a fairly geometrical point of view or focus more on the algebraic and analytic aspects of the theory.

The second component will be a seminar by the students on more particular topics. We expect the participants to write a small proceeding-like summary of their seminar talks. The precise topics for the student talks will be communicated once we decided on the direction of the RiG: fully geometric or more algebraic and analytic. We can also have talks in both directions. Many of the topics can also be shared by two or more students. In this case, the students are expected to work in teams, both for the preparation of the talks as well as for the proceedings.

Prerequisites

The course can be see as a continuation of the lecture Geometric Mechanics. However, we will try to make this RiG as self-contained as possible: depending on the prerequisites of the interested audience we either formulate things geometrically for Poisson manifolds or more locally for phase spaces being open subsets of R^n. Of course, a good knowledge in the bachelor courses of geometric analysis, functional analysis, linear algebra etc. are expected.

For motivation some background in quantum mechanics might be helpful though not necessary. Here the parallel lecture Algebra and Dynamics of Quantum Systems is a perfect companion in this semester.

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture:

WueCampus course

Vorläufiges Programm

In dem Bachelorseminar Matrix-Lie-Gruppen studieren wir eine wichtige Klasse von Gruppen mittels der linearen Algebra und Analysis aus den Grundvorlesungen. 

Lie-Gruppen und ihre Wirkungen sind ein zentraler Bestandteil in vielen Gebieten der Mathematik
und Physik, und dienen meist zur Beschreibung von Symmetrien.
Lie-Gruppen bilden aber auch für sich genommen interessante mathematische Objekte mit vielen
Verbindungen zu Geometrie, Topologie und Analysis. Die große Klasse von Matrix-Lie-Gruppen lassen sich nur mit Hilfe der linearen Algebra definieren lassen, und viele allgemeinere Aussagen über Lie-Gruppen recht direkt zugänglich machen.

Literatur

Im WueCampus Kurs gibt es eine detaillierte Literaturliste.

Termine

Vorbesprechung: Montag, 18.10 um 14.15 (Online)

Seminar: Montag 14 – 16 Uhr (Online)

WueCampus Kursraum

Preliminary Program

Category theory is central to modern mathematics. When studying a mathematical object like a group, a ring or a topological space,
it is often necessary to investigate its relation to other objects of the same type, that is to study a certain category.
Also in mathematical physics such relations between objects are becoming more and more important, like studying a certain quantum field  theory not on one space-time, but on all space-times of a certain type. Another common situation is the investigation of a class of theories that depends on additional parameters and the attempt to "integrate" the theory over the parameter space. This leads to the theory of limits and colimits in categories.  

As usual for a RiG the course consists of two parts. In the first part we have a weekly lecture and in the second part at the end of the term we will have one or two days with the student seminars, where you give a talk on a subject that we distribute during the first lectures.

In the lecture part I give a rather short introduction to the language of categories and functors. And then we proceed to discuss

• universal properties, limits, colimits,
• adjoint functors,
• monoidal categories, duals and the diagrammatic calculus,
• topological categories, simplicial sets and related constructions,
• derived functors,
• the structure of quantum observables in mathematical physics with an eye towards factorization algebras.

  The subjects for your talks depend on your interest, mainly they are related to applications to mathematical physics.  Possibilities include

• ribbon categories and Hopf algebras,
• applications to knot theory,
• Categories in algebraic topology (e.g. the fundamental groupoid functor, homotopy theory),
•  Categories in field theories, principal bundles,
•  Introduction to topological field theories,
• Lie groupoids in geometric mechanics,
• Hochschild homology and derived functors.

Prerequisits

Apart from the Bachelor lectures we require a basic familiarity with algebra, such as modules, tensor products, etc. If needed, these concepts can be also gained during the lecture. No knowlegde of category theory is required, but some self-reading at the beginning is intended. Some knowledge of topology is useful but not strictly required.

Dates

The dates and more information you will find on the wuecampus site.

Riemannian and Pseudo-Riemannian Geometry (Prof. Dr. Knut Hüper)

Prerequisites

This course builds on the master course in differential geometry from WS 20/21.

Würzburg students are invited to download the slides from the corresponding Wuecampus course.

Contents

This master course covers basics in Riemannian and pseudo-Riemannian geometry.

In particular, definite and indefinite metrics, curvature, geodesics, connections, covariant derivatives, etc...

Literature

Suggestions to literature will be published on Wuecampus in due course.

https://wuecampus.uni-wuerzburg.de/moodle/course/view.php?id=45073

Voraussichtliche Inhalte

Siehe Wuecampus-Kurs

Literatur

Siehe Wuecampus-Kurs

Vorlesungs- und Übungszeiten

Siehe Wuecampus-Kurs

https://wuecampus.uni-wuerzburg.de/moodle/course/view.php?id=45072

Geometric Mechanics (Dr. Gregor Schaumann)

Preliminary Program

This master course introduces geometric mechanics, which is perhaps the most successful interplay of mathematics and physics. Classical Newtonian mechanics has its natural home in the context of symplectic geometry and the geometric point clarifies the kinematical and dynamical aspects of physical theories.
We start with the Hamiltonian approach to mechanics, which corresponds geometrically to cotangent bundles as phase spaces. In the course we discuss symmetries and the corresponding reduced phase spaces, which lead to more general symplectic manifolds. The Lagrangian formulation of classical mechanics is formulated on tangent bundles and we put the Legendre transformation into its geometric context. Furthermore we discuss  the even more general Poisson manifolds and algebraic aspects of multivector fields.

Literature

A detailed list of references will be available in the WueCampus course.

Preliminary Program

In this course we define and study Hopf algebras and their relation with topological field theories,
a relation which goes back to the Yang-Baxter equation in statistical mechanics.
The focus of the RiG is on algebraic structures and their use for topological invariants. The RiG consists of a lecture part and a seminar part at the end of the semester.
We first discuss Hopf algebras in particular the examples coming from the theory of Lie algebras (Quantum groups).
From Hopf algebras we define so-called quantum codes, particular models for topological field theories and consider the relation with the Yang-Baxter equation.
 

Prerequisites

The course is meant to be introductory and suitable for all master students.
A certain knowledge of topology and algebra is useful but can be obtained during the course.

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Wednesday 10-12, SE40.

WueCampus course

Vorläufiges Programm

Vorkenntnisse

Literatur

Im WueCampus Kurs gibt es eine detaillierte Literaturliste.

Termine

WueCampus course

Voraussichtliche Inhalte

Siehe Wuecampus-Kurs

Literatur

Siehe Wuecampus-Kurs

Vorlesungs- und Übungszeiten

Siehe Wuecampus-Kurs

wuecampus

Preliminary Program

Topological and differential manifolds

Tangent structures

Immersions and submersions

Glimpse of "classical differential geometry"

Lie groups and Lie algebras

Fiber bundles

Tensor analysis

Connections and covariant derivatives

Prerequisites

Solid knowledge in

1)

Calculus I-IV, i.e., Real Analysis 1-2, Geometric Analysis (Differential Forms, Stokes Thm), Course on Integral Calculus (Gauß, Green, div, rot, grad, etc...)

2)

Linear Algebra I-II

3)

Ordinary Differential Equations (Linear Theory, Ex. & Uniqueness of solutions, etc...)

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Wednesday 16-18 and Thursday 14-16
Exercises: Thursday 8-10

wuecampus

Preliminary Program

The aim of this lecture is to find mathematical models for the observable algebras of quantum systems. We will take the necessary requirements from physics as orientation to pass from general algebraic structures, *-algebras and their states, to more analytic versions. Analysis will be needed to arrive at meaningful and manageable algebras. This will include among others the class of C*-algebras. Having found this class as physically very appealing algebras we will study their properties in detail. Here the spectral calculus plays a dominant role. While having already very nice properties, an abstract C*-algebra is not yet enough for physical models: one needs to implement the abstract algebras as an algebra of operators on a Hilbert space. Thus we will investigate the bounded operators on a Hilbert space from a more conceptual point of view to establish the bounded-measurable functional calculus for Hermitian operators. If time permits, we will extend our discussion then to unbounded operators and investigate the spectral theory of possibly unbounded self-adjoint operators in Hilbert spaces.

Prerequisites

Elementary knowledge on functional analysis (Banach spaces, Hilbert
space etc) will be assumed and can, if wanted, be briefly recalled. A
familiarity with basic concepts of quantum mechanics may be useful but
are not necessarily required.

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Tuesday 10-12 and Thursday 12-14
Tutorial: Friday 12-14

WueCampus course

Preliminary Program

In this RiG we discuss deformation quantization of classical Poisson algebras.  Starting with canonical Poisson brackets for the functions on a phase space M of a mechanical system one tries to find a description of the corresponding quantum system by deforming the classical Poisson algebra of observables C^\infty(M) into a new, non-commutative algebra. This is done by a (formal) star product *, an associative product for C^\infty(M)[[\hbar]] which in zeroth order coincides with the classical product and which in first order of the commutator gives the Poisson bracket.

In the lecture part of the RiG we study such star products, both from the conceptual point of view as well as for various classes of examples. Depending on the preferences of the participants, we can either take a fairly geometrical point of view or focus more on the algebraic and analytic aspects of the theory.

The second component will be a seminar by the students on more particular topics. We expect the participants to write a small proceeding-like summary of their seminar talks. The precise topics for the student talks will be communicated once we decided on the direction of the RiG: fully geometric or more algebraic and analytic. We can also have talks in both directions. Many of the topics can also be shared by two or more students. In this case, the students are expected to work in teams, both for the preparation of the talks as well as for the proceedings.

Prerequisites

The course can be see as a continuation of the lecture Geometric Mechanics. However, we will try to make this RiG as self-contained as possible: depending on the prerequisites of the interested audience we either formulate things geometrically for Poisson manifolds or more locally for phase spaces being open subsets of R^n. Of course, a good knowledge in the bachelor courses of geometric analysis, functional analysis, linear algebra etc. are expected.

For motivation some background in quantum mechanics might be helpful though not necessary. Here the parallel lecture Algebra and Dynamics of Quantum Systems is a perfect companion in this
semester.

Literature

A detailed list of references will be available in the WueCampus course.

Dates

Lecture: Friday 8 - 10

WueCampus course

Inhalte

In diesem Bachelor-Kurs behandeln wir zunächst die klassische Differentialgeometrie. Das beinhaltet Kurven und Flächen im R³, Frenet-Beine, 

Krümmungsbegriffe, Fundamentalformen. Anschließend Untermannigfaltigkeiten im Rⁿ, Tangentialbündel, Kovariante Ableitung, Geodätische. Falls Zeit bleibt etwas Minimalflächen.

Diese Vorlesung dient auch als "Warming Up" für die Mastervorlesungen zur Differentialgeometrie.

Voraussetzungen

Solide Kenntnisse in Linearer Algebra und Analysis, ferner ist etwas Grundwissenen über gewöhnliche Differentialgleichungen von Vorteil, für Minimalflächen etwas Funktionentheorie.

Literatur

Geeignete Literaturhinweise finden Sie in Kürze auf dem Wuecampus-Kurs.

WueCampus 

Contents

This master course covers basics in Riemannian and pseudo-Riemannian geometry. In particular, definite and indefinite metrics, curvature, geodesics, connections, covariant derivatives, etc...

Prerequisites

This course builds on the master course in differential geometry from WS 19/20.

Literature

Suggestions to literature will be published on Wuecampus in due course.

WueCampus 

Preliminary Program

This master course is a first introduction to geometric mechanics. We will discuss classical Newtonian mechanics using differential geometric methods within the context of symplectic geometry. The Hamiltonian approach to mechanics based on cotangent bundles as phase spaces will be central for this approach. The Lagrangean formulation of mechanics will be formulated on the tangent bundle of the configuration space. The comparison between the two will give, among many other aspects, a new look on Riemannian geometry through the symplectic looking glass. But geometric mechanics will not stop at these two (equivalent) versions but requires more general symplectic manifolds as phase spaces. In particular, the presence of symmetries and conserved quantities will lead to phase space reductions which yields quite generic symplectic manifolds as result. Finally, Poisson manifolds provide a yet more general framework of great importance when it comes to questions like quantization.

Literature

A detailed list of references will be available in the WueCampus course.

WueCampus course

Preliminary Program

In this course we will focus on one of the main motivations for
locally convex analysis: the theory of distributions also known as
generalized functions. The main idea is to define distributions as
elements in the dual space of a suitably chosen space of test
functions. The original approach of Laurent Schwartz was to use as
test functions the space of smooth and compactly supported functions.
But also other test function spaces can be investigated like the
Schwartz space, leading to other types of distributions, like the
tempered distributions etc. Based on the last semester course on
locally convex analysis we will now apply the general machinery to
these particular examples and study distributions, differential
operators, convolutions, fundamental solutions etc. If time permits we
will move into the direction of wave front sets and the kernel
theorem.

Prerequisites

The RiG is meant to be a continuation of the last semester course on
locally convex analysis. However, a solid knowledge on functional
analysis (bachelor courses) together with the ability to learn a
couple of concepts alone should be sufficient to attend this
course. If in doubts, please get in touch early then we can arrange
the details.

Literature

A detailed list of references will be available in the WueCampus course.

WueCampus course

Preliminary Program

The aim of this course is to provide a introduction to the mathmatical theory of knots in 3-dimensional space.
The subject lies at the intersection of topology, algebraic topology and algebra. The RiG consists of a lecture part and a seminar part at the end of the semester. After providing the topological notions that are used to study knots, we want to learn how to distinguish two given knots. To this end we use algebraic invariants of knots which are the main focus of the course. There exists a variety of important invariants ranging from the knot genus, the classical knot polynomials, invariants related to the knot group to the modern, so called quantum invariants starting with the Jones polynomial. In the end we obtain a systematic approach to quantum invariants using string diagram calculus.

Prerequisites

The course is meant to be elementary and also accessible for motivated bachelor students and teaching students (most of the material has aspects of interest for advanced high school classes). 
A certain knowledge of topology and algebra is useful but can be obtained during the course.

Literature

A detailed list of references will be available in the WueCampus course.

WueCampus course

Topological and differential manifolds

Tangent structures

Immersions and submersions

Glimpse of "classical differential geometry"

Lie groups and Lie algebras

Fiber bundles

Tensor analysis

Connections and covariant derivatives

WueCampus (Self registering)

Vorträge zu den Themen

Vektoriterationen, Quaternionen, Schurkomplement, Bilinearformen, Quadriken, lineare gewöhnliche Differentialgleichungen, verallgemeinerte Inverse, indefinite Innenprodukträume, symplektische und Hamiltonmatrizen

In this master course we provide a first contact with the vast subject of algebraic topology. The main theme of the course is the deep connection between algebra and topology, that we begin to investigate via the study of algebraic invariants for topological spaces. However, it is not only algebra used in topology, but we are also led to introduce advanced algebraic techniques that are motivated from topology. The algebraic invariants are naturally formulated using the language of categories. We will consider homology and cohomology functors, homotopy groups and develop on the topological side the language of simplex categories and CW-complexes, on the algebraic side chain complexes and their homotopies and category related concepts.

WueCampus course

In this research in groups we investigate various oid-geometries. Of particupar interest will be Lie algebroids and Lie groupoids but also Courant algebroids will be discussed.

WueCampus course

This master course will provide a first approach to locally convex spaces. The aim is two-fold: on the one hand, we will investigate locally convex spaces and continuous linear maps, dual spaces, tensor products, and their properties on a fairly general basis including detailed proofs of the main theorems of locally convex analysis like Hahn-Banach, Banach-Alaoglu, Bipolar theorem, Krein-Milman etc. On the other hand, we discuss the very explicit example of test functions and distributions for open subsets in Euclidean space or, perhaps, on manifolds. This will allow to have a more conceptual view on classical topics in functional analysis.

WueCampus course

Ziel dieser Vorlesung ist es den Satz von Stokes auf Untermannigfaltigkeiten zu verstehen und zu beweisen.

Vorlesungsinhalte: Differenzierbare Untermannigfaltigkeiten, Differentialformenkalkül (a la Cartan), Integration auf Untermannigfaltigkeiten.

In der Vergangenheit wurden diese Inhalte oft mit "Vektoranalysis" bezeichnet, allerdings oftmals dann auch nur der R³ betrachtet.
WueCampus-Kurs

This course is partially based on the master course Differential Geometry M=ADGM-1V  (WS 18/19). We will try to cover

• Riemannian and pseudo-Riemannian manifolds
• Connections and Curvature
• Geodesics and exponential mapping
• Jacobi fields
• Comparison theory

There is a vast literature on Riemannian geometry, less exists for differential manifolds with indefinite metrics. Although the emphasis will be on the positive definite case we try to strive indefinite metrics as well. Most of the books listed below are available in our library, some even in electronic form.

In due course there will be a Wuecampus website available for self enrolment for the exercises. Nevertheless, do not forget to enrol via Wuestudy for this course, as well.
Wuecampus

Dieser Bachelorkurs ist zweigeteilt aufgebaut. Der erste Teil beschäftigt sich mit den grundlegenden Definition von topologischen Räumen mittels offener und abgeschlossener Mengen sowie dem Begriff der Umgebung, von stetigen Abbildungen, Abzählbarkeits- und Trennungsaxiomen und Kompaktheitsbegriffen.
Universelle Konstruktionen wie direkte Summe, direktes Produkt, pullback, intiale, terminale Topologien, werden nach einer kurze Einführung in die Sprache der Kategorien einheitlich behandelt.
Beispiele aus der Analysis und Geometrie werden diskutiert.
Im zweiten Teil beschäftigen wir uns mit dem Begriff der Homotopie und betrachten die Fundamentalgruppe eines topologischen Raums.
WueCampus course

Objective of this Research in Groups, Topics in Mathematical Image Processing, is to deepen the students' understanding of the mathematical fundamentals and background of mathematical image processing. Topics covered are discrete and continuous wavelet, Fourier and Radon transformations, filtering, epipolar geometry in computer vision, cartoon and texture image decomposition.

The original motivation to study algebraic structures up to homotopy can be traced back to questions in algebraic topology where one was interested whether the cohomology of a given topological space carries an algebraic structure like a Lie algebra, an associative algebra etc. The question was then what structures on the space itself were responsible for this, leading to the notions of Lie algebras up to homotopy aka L_\infty-algebras etc. Ever since these structures have been studied for their own sake as they show up in various branches of mathematics. Recently, the usage of such L_\infty-algebras proved to be crucial for the understanding of deformation problems, leading in particular to the famous Kontsevich formality theorem. In this RiG we will establish the notions of L_\infty-algebras and their morphisms as the guiding example of algebraic structures up to homotopy. A first difficulty to overcome is that the very definition requires some more sophisticated notions of cofree coalgebras which we will introduce in detail.
WueCampus course

This master course is a first introduction to geometric mechanics. We will discuss classical Newtonian mechanics using differential geometric methods within the context of symplectic geometry. The Hamiltonian approach to mechanics based on cotangent bundles as phase spaces will be central for this approach. The Lagrangean formulation of mechanics will be formulated on the tangent bundle of the configuration space. The comparison between the two will give, among many other aspects, a new look on Riemannian geometry through the symplectic looking glass. But geometric mechanics will not stop at these two (equivalent) versions but requires more general symplectic manifolds as phase spaces. In particular, the presence of symmetries and conserved quantities will lead to phase space reductions which yields quite generic symplectic manifolds as result. Finally, Poisson manifolds provide a yet more general framework of great importance when it comes to questions like quantization.
WueCampus course

Vorlesung mit Übungen.
WueCampus Kurs

This master course is a first introduction to the topics of differential geometry. We will discuss differentiable manifolds as geometric objects in an intrinsic approach. A particular emphasis will be put on the global calculus on manifolds, showing how the coordinate-based calculations can be minimized as far as possible. After manifolds, we discuss vector bundles as the next important ingredient in differential geometry. Integration on manifolds will be presented in two ways, using an orientation and without orientation. We will see the most important cohomology theories attached to manifolds. If time permits, we will also give a short introduction to Lie groups.
WueCampus course

The course can be seen as a follow-up of the last semester course on Analysis and Geometry of Classical Systems. We will discuss the construction of gauge theories based on the usage of principal fiber bundles and associated bundles. Beside the construction of several relevant models of gauge theories we will see other applications of principal fiber bundles like characteristic classes and resulting invariants. The second component will be a seminar by the students on more particular topics. I expect the participants to write a small proceeding-like summary of their seminar talks. Details on the topics can be found in the WueCampus course.
WueCampus course

Vorlesung mit Übungen
WueCampus Kurs

Vorlesung mit Übungen
WueCampus Kurs

Arbeitsgemeinschaft

In this course we will find geometric formulations for various field theories from mathematical physics:
the most important ones will be the Yang-Mills field theories from particle physics, but also other
field theories like the more phenomenological ones from fluid dynamics or solid state physics are in
principle part of this geometric approach. The main idea is that fields will be described as sections
of certain fiber bundles over a spacetime manifold. In many situations the fiber bundle is actually a
vector bundle but we will see situations where we have to go beyond this point of view. Closely related
to the geometric approach to field theory is the role of symmetries played by Lie group actions. In
fact, one can turn the accidental presence of a symmetry into a principle and construct field theories
based on the symmetry directly. This will essentially be the idea in the Yang-Mills field theories. In
the lecture we will mainly consider the kinematic aspects of field theory, i.e. the geometric description
of fields. Understanding the dynamical aspects, i.e. the study of actual field equations, is then a second and typically
much more difficult step.
WueCampus Kurs

The course can be seen as a follow-up of the last semester course on Algebra and Dynamics of Quantum Systems.
WueCampus Kurs

The program for the lecture component includes the following topics:

• Quantization for R²ⁿ and first star products
• Symbol calculus for differential operators and quantization of cotangent bundles
• Star products on symplectic manifolds
• Fedosov construction of star products
• Fedosov construction on cotangent bundles

WueCampus course

The goal of this lecture is to understand the algebras of operators as they occur in quantum mechanics and in quantum field theory models. The important mathematical structure will be a C*-algebra. We will study such algebras in quite some detail together with their representation theory.
WueCampus Course

Diese Vorlesung zur mengentheoretischen Topologie kann zum einen als Grundlage und Vorbereitung für weiterführende Vorlesungen im Bereich der Analysis (insbesondere der Funktionalanalysis) sowie im Bereich der Geometrie gesehen werden. Zum anderen lässt sich in der Topologie der Weg in die Abstraktion und Allgemeinheit besonders gut illustrieren. Anhand bekannter Phänomene aus der Analysis wird versucht, den Kern dieser Situationen besonders klar herauszustellen und so eine Axiomatisierung der Analysis vorzunehmen. Dies wird am Ende nicht zuletzt für die Analysis der Grundvorlesungen sehr gewinnbringend sein, aber eben auch deutlich darüber hinausführen.

Der zentrale Begriff in der Topologie ist der des topologischen Raumes mit den stetigen Abbildungen zwischen solchen. Wir werden hier verschiedene Aspekte und Beispiele kennenlernen, welche es uns erlauben, völlig neue Gesichtspunkte in den Begriffen Konvergenz und Kompaktheit zu finden. Anwendungen in der Funktionalanalysis erfordern dann ein genaueres Studium verschiedener Funktionenräume von stetigen Funktionen auf topologischen Räumen.
WueCampus Kurs

Lecture with tutorials
WueCampus course

Vorlesung mit Übungen

Vorlesung mit Übungen

Lecture with tutorials

Bachelor and master seminar

Vorlesung

Master seminar

This master course is a first introduction to the topics of differential geometry. We will discuss differentiable manifolds as geometric objects in an intrinsic approach. A particular emphasis will be put on the global calculus on manifolds, showing how the coordinate-based calculations can be minimized as far as possible. After manifolds, we discuss vector bundles as the next important ingredient in differential geometry. Integration on manifolds will be presented in two ways, using an orientation and without orientation. We will see the most important cohomology theories attached to manifolds. If time permits, we will also give a short introduction to Lie groups.
WueCampus course

In mathematical models of physical systems, the physical observables are typically described by C*-algebras or von Neumann algebras. While this gives a very elegant and powerful spectral calculus, many situations will not directly yield such nice classes of algebras. In various quantization theories the construction of C*-algebras is difficult or unclear. One way out is to focus on the algebraic features before taking into account the analytic issues as well. This is the main motivation for considering *-algebras without any analysis involved. Beyond quantization theories, other important examples are group algebras or universal enveloping algebras of Lie algebras but also algebras of differential operators. Here one typically has by no means a C*-norm available.

The aim of this RiG is to find a common algebraic framework for a reasonable representation theory of such algebras. It turns out that aspects of positivity can be formulated in an entirely algebraic way yielding interesting structures for the representation theory. In general, it will be difficult if not impossible to understand the representation theory of a given algebra completely. However, and this is quite surprising, it might be possible to compare it to the representation theory of a different algebra and determine whether or not the two algebras have the same representation theory. This is the main task of Morita theory, which we will present both in a purely ring-theoretic context and in an adapted version for *-algebras. We will develop the necessary category-theoretic notions to put the question of Morita equivalence in the right perspective.
WueCampus course

In diesem Bachelor-Seminar werden wir verschiedene algebraische Strukturen kennenlernen, die an vielen Stellen in der Mathematik auftreten.
WueCampus Kurs

Bachelor and master seminar

Vorlesung mit Übungen

Lecture with tutorials

Masterseminar

Workshop

In this seminar we investigate the basic theory of Riemann surfaces.

Die zweisemestrige Vorlesung Lineare Algebra I und II ist neben der parallelen Analysisvorlesung die zentrale Grundlage eines jeden Mathematikstudiums. Ihre Bedeutung kann daher kaum überschätzt werden.

Zum einen werden die Begriffe der linearen Algebra, Vektorräume und linearen Abbildungen, in jeder weiteren Mathematikveranstaltung benötigt und benutzt werden. Dies gilt sowohl für die Entwicklung der theoretischen und angewandten Mathematik als auch für die zahlreichen Anwendungen, nicht zuletzt in der (mathematischen) Physik und der Wirtschaftsmathematik, aber auch in der Schulmathematik.

Zum anderen wird in der linearen Algebra Mathematik das erste Mal, im Vergleich zur Schulmathematik, auf wissenschaftlichem Niveau betrachtet. Insbesondere werden in der Vorlesung die grundlegenden Techniken der axiomatischen Herangehensweise der Mathematik, die Notwendigkeit einer stringenten Beweisführung und die zugehörige Abstraktion erlernt. Dies ist erfahrungsgemäß am Anfang nicht immer einfach.
WueCampus Kurs

In this Research in Groups, we discuss Lorentzian geometry: every manifold can be equipped with a Riemannian metric. However, it needs some additional features that a manifold can also be equipped with a Lorentzian metric, where the signature is now (+, -, ..., -). Such inner products on every tangent space are now the crucial ingredient for a geometric formulation of general relativity: our spacetime carries a Lorentz metric which encodes not only the propagation of light but the whole causal structure of the spacetime. We will discuss the geometry arising from such a metric in detail. Particular emphasis will be put on the causal structure, Cauchy hypersurfaces and globally hyperbolic spacetimes, and, if time permits, propagation of linear waves.
WueCampus Course

In this Research in Groups we will investigate various recent developments in differential geometry starting from Lie algebroid theory and including generalized geometries like Courant algebroids, generalized complex structures, Dirac structures and their applications in symplectic and Poisson geometry.
WueCampus course

Dieses lehrstuhlübergreifende Seminar behandelt Minimalflächen und harmonische Abbildungen, ein Thema an der Schnittstelle der Forschungsinteressen der drei Lehrstühle Mathematik in den Naturwissenschaften, Mathematische Physik und Funktionentheorie.

Minimalflächen sind Flächen im Raum mit "minimalem" Flächeninhalt und lassen sich mithilfe harmonischer Funktionen beschreiben. Sie spielen eine zentrale und überaus aktuelle Rolle sowohl in der Reinen als auch in der Angewandten Mathematik sowie in der Physik und in den Material- und Ingenieurswissenschaften. Bei der mathematischen Untersuchung von Minimalflächen kommen elegante Methoden aus verschiedenen mathematischen Gebieten wie der Differentialgeometrie, der Variationsrechnung und der komplexen Analysis zur Anwendung. Minimalflächen treten u.a. bei der Untersuchung von Seifenhäuten und der Konstruktion stabiler Objekte (z.B. in der Architektur und Automobilindustrie) in Erscheinung.

Die zweisemestrige Vorlesung Lineare Algebra I und II ist neben der parallelen Analysisvorlesung die zentrale Grundlage eines jeden Mathematikstudiums. Ihre Bedeutung kann daher kaum überschätzt werden.

Zum einen werden die Begriffe der linearen Algebra, Vektorräume und linearen Abbildungen, in jeder weiteren Mathematikveranstaltung benötigt und benutzt werden. Dies gilt sowohl für die Entwicklung der theoretischen und angewandten Mathematik als auch für die zahlreichen Anwendungen, nicht zuletzt in der (mathematischen) Physik und der Wirtschaftsmathematik, aber auch in der Schulmathematik.

Zum anderen wird in der linearen Algebra Mathematik das erste Mal, im Vergleich zur Schulmathematik, auf wissenschaftlichem Niveau betrachtet. Insbesondere werden in der Vorlesung die grundlegenden Techniken der axiomatischen Herangehensweise der Mathematik, die Notwendigkeit einer stringenten Beweisführung und die zugehörige Abstraktion erlernt. Dies ist erfahrungsgemäß am Anfang nicht immer einfach.
WueCampus Kurs

In der algebraischen Deformationstheorie versucht man gewisse algebraische Strukturen wie etwa die Multiplikationsvorschrift in einer Algebra leicht abzuändern und anschließend zu prüfen, ob die neue Struktur zur alten in geeigneter Weise isomorph ist. Dies ist zunächst ein recht abstraktes und allgemeines Konzept, welches aber viele ganz konkrete und in der Anwendung relevante Fragestellungen beinhaltet. So kann man beispielsweise durch Deformation von algebraischen Strukturen folgende Probleme der mathematischen Physik formulieren:

• Quantisierung wird als Deformation der klassischen Observablenalgebra, also der Funktionen auf dem Phasenraum, verstanden, wobei das kommutative Produkt in ein nicht-kommutatives aber nach wie vor assoziatives Produkt deformiert wird.
• Der Übergang von Newtonscher Mechanik mit Galilei-Invarianz zur speziellen Relativitätstheorie mit Lorentz-Invarianz lässt sich als eine Deformation der Lie-Algebra der Galilei-Gruppe in die Lie-Algebra der Lorentz-Gruppe verstehen. Hier ist die algebraische Struktur also eine Lie-Algebra.

In der Arbeitsgemeinschaft wollen wir nun die Grundlagen der Deformationstheorie kennenlernen. Hier gilt es also zunächst zu klären, welche algebraischen Strukturen man deformieren möchte und was man unter einer Deformation genau zu verstehen hat. Da wir einen rein algebraischen Zugang wählen, betrachtet man formale Deformationen, also formale Potenzreihen in einem Deformationsparameter. In den obigen Beispielen ist der Deformationsparameter das Plancksche Wirkungsquantum beziehungsweise die inverse Lichtgeschwindigkeit. Es zeigt sich, dass sich jedes Deformationsproblem mit Hilfe einer geeigneten differentiell gradierten Lie-Algebra und deren Maurer-Cartan-Elementen beschreiben lässt. Existenz und Klassifikationsergebnisse zu Deformationsproblemen beschreibt man dann durch kohomologische Methoden. Wir werden diese neuen algebraischen Konzepte eingehend studieren. Neuere Zugänge zu differentiell gradierten Lie-Algebren benutzen wesentlich koalgebraische Ideen, welche wir im Detail diskutieren wollen.
WueCampus Kurs

In diesem Masterseminar werden wir die Haag-Kastler-Formulierung von Quantenfeldtheorien auf dem Minkowski-Raum kennenlernen.

Diese Mastervorlesung stellt eine Einführung in die geometrische Mechanik dar. Hier wird die klassische Newtonsche Mechanik mit differentialgeometrischen Methoden formuliert und im Kontext der symplektischen Geometrie untersucht. Zentral wird der Hamiltonsche Zugang zur Mechanik sein, der auf dem Kotangentenbündel des Konfigurationsraums zu Hause ist. Der aus der Physik eventuell besser bekannte Lagrangesche Ansatz wird differentialgeometrisch das Tangentenbündel erfordern. Wir werden diese beiden Sichtweisen sowie viele weitere, übergeordnete Aspekte der geometrischen Mechanik im Detail kennenlernen und anwenden. Es wird sich so insbesondere auch ein Hamiltonscher Blick auf die Riemannsche Geometrie ergeben, die den geodätischen Fluss auf einem Konfigurationsraum als einen Hamiltonschen Fluss zur kinetischen Energie auf dem Kotangentenbündel liefert. Eine große Rolle in der Theorie spielen die Symmetrien, welche über das Noether-Theorem zu Erhaltungsgrößen führen. Geometrisch werden Symmetrien durch Gruppenwirkungen von Lie-Gruppen implementiert, Erhaltungsgrößen beschreibt man dann durch Impulsabbildungen. Das Festlegen der Erhaltungsgrößen auf gewisse Werte vereinfacht die Bewegungsgleichungen und entspricht geometrisch einer Phasenraumreduktion. Eine wesentliche Erweiterung der geometrischen Mechanik ergibt sich schließlich, wenn man symplektische Phasenräume durch allgemeine Poisson-Mannigfaltigkeiten ersetzt.
WueCampus Kurs

Poisson manifolds are smooth manifolds with the additional structure of a Poisson bracket for the smooth functions. Equivalently, this is the additional datum of a Poisson tensor, an antisymmetric contravariant tensor field satisfying a particular non-linear PDE, the Jacobi identity.
WueCampus Kurs

Diese Mastervorlesung stellt eine erste Einführung in die Differentialgeometrie dar. Behandelt werden differenzierbare Mannigfaltigkeiten als geometrische Objekte in einem intrinsischen Zugang. In dieser Vorlesung wird ein großer Wert auf den globalen Kalkül auf Mannigfaltigkeiten gelegt und aufgezeigt, wie der koordinatenlastige Kalkül dadurch gewinnbringend ersetzt werden kann. Wir werden allgemeine Vektorbündel kennenlernen. Die Integrationstheorie auf Mannigfaltigkeiten wird in zwei Zugängen vorgestellt: mit und ohne Orientierung. Im orientierten Fall ergeben sich wichtige Verbindungen zu verschiedenen Kohomologie-Theorien. Wenn die Zeit es zulässt, werden wir noch eine Einführung in die Theorie der Lie-Gruppen geben.
WueCampus Kurs

Masterseminar
WueCampus Kurs

Ziel der Vorlesung ist es, die geometrische Interpretation verschiedener Feldgleichungen der mathematischen Physik zu verstehen. Felder werden dabei als Abbildungen einer Mannigfaltigkeit, der Raumzeit, in eine andere Mannigfaltigkeit, die die Werte der Felder beschreibt, verstanden. Etwas allgemeiner und geometrischer interpretiert man Felder als Schnitte von Faserbündeln über der Raumzeit. Die Fasern, also die möglichen Feldwerte, können dabei Vektorräume aber auch allgemeinere Mannigfaltigkeiten sein.

Während dieser sehr allgemeine Feldbegriff durchaus schon interessante Beispiele liefert, etwa die Sigma-Modelle, spielen speziellere Feldtheorien eine besondere Rolle in der Physik. Hier werden wir vor allem Eichtheorien kennenlernen, welche auf mathematischer Seite durch Hauptfaserbündel und ihre assoziierten Bündel gegeben sind. Dazu werden wir die nötigen Symmetrien durch Lie-Gruppen und ihre glatten Wirkungen beschreiben.
WueCampus Kurs

Arbeitsgemeinschaft
WueCampus Kurs

Ziel der Vorlesung ist es, die in der Quantenmechanik und Quantenfeldtheorie auftretenden Algebren von Operatoren systematisch zu studieren. Die entscheidende mathematische Struktur dabei wird die einer C*-Algebra sein. Diese werden wir eingehend kennenlernen und ihre Darstellungen studieren.
WueCampus Kurs

Ziel dieses Bachelor-Seminars ist es, anhand von Beispielen einen ersten Einblick in die Theorie der Matrix-Lie-Gruppen und ihrer Lie-Algebren zu gewinnen. Die auftretenden Gruppen, wie die allgemeine lineare Gruppe, die spezielle lineare Gruppe oder die unitären und orthogonalen Gruppen, sind aus der Linearen Algebra bekannt. Wir werden sehen, wie die Exponentialabbildung für Matrizen dazu benutzt werden kann, viel über die Gruppenstruktur dieser Matrix-Gruppen zu lernen. Die Lie-Algebren werden in gewisser Hinsicht das infinitesimale Gegenstück zu den Gruppen sein.
WueCampus Kurs