Im Schulunterricht wird seit einiger Zeit mehr Wert auf darauf gelegt, Mathematik auf die Wirklichkeit anzuwenden.  Eine
mathematische Modellierung ist jedoch ein komplexer Prozess mit mehreren Schritten, die ineinander greifen.  Einzelne Teile dieses Prozesses für sich liefern kaum einen Erkenntnisgewinn. Insbesondere bringen die mathematischen Handlungen, losgelöst von der mathematischen Modellierung, kaum Einblick in das Anwendungsproblem.  Viel kritisiert werden daher Pseudoanwendungen, die weder zum Verständnis der Wirklichkeit noch zum mathematischen Verständnis etwas beitragen.  In diesem Seminar soll der Prozess der mathematischen Modellierung daher als Ganzes betrachtet und an einer Reihe von Beispielen eingeübt werden.  Daneben bietet das Seminar auch die Gelegenheit, wichtige Konzepte aus der Differenzial- und Integralrechnung zu vertiefen.  Oft möchte man qualitative Aussagen über das Verhalten der mathematischen Modelle herleiten und verwendet dabei oft mehr oder weniger tiefliegende mathematische Sätze.

Es gibt interessante mathematische Modellierungsprobleme in den verschiedensten Bereichen, ob Physik, Biologie, Verkehrsplanung oder die Preisfindung in der Wirtschaft.  In diesem Seminar werde ich mich auf biologische Probleme konzentrieren.  Eine Rechtfertigung hierfür ist der Wunsch, mit dem Seminar auch einen kleinen Beitrag zum Nachhaltigkeitsziel der Biodiversität zu leisten.  Einige überraschende Vorgänge in komplexen Ökosystemen lassen sich auch schon in einfachen mathematischen Modellen beobachten.  Mit Hilfe dieser Modelle kann man die Vorgänge in Ökosystemen daher besser verstehen und erklären.  Ein typisches Beispiel hierfür ist das Umkippen von Gewässern.  Wird zu viel Phosphat in ein Gewässer eingeleitet, so kippt es um in einen lebensfeindlichen Zustand. Danach erholt es sich erst wieder, wenn die Phosphateinleitung unter eine andere Schwelle gesenkt wird, die deutlich niedriger ist als die Schwelle, ab der das Gewässer umkippt.

Die Wirklichkeit ist so komplex, dass „exakte“ mathematische Modelle weder möglich noch nützlich sind.  Denn je genauer ein Modell, desto mehr Parameter braucht es für die Beschreibung.  All diese Parameter müssen aus den vorhandenen Daten geschätzt werden, aber die Auswirkungen dieser Schätzungen auf die Vorhersagen des Modells werden mit der Komplexität des Modells immer undurchsichtiger.  Für bestimmte Probleme wie zum Beispiel die Wettervorhersage braucht es natürlich sehr komplexe Modelle.  Wir werden uns im Seminar allerdings auf einfachere Modelle beschränken, deren Ziel das Verständnis naturwissenschaftlicher Phänomene ist.  Solche Modelle enthalten eigentlich immer vereinfachende Annahmen, die offensichtlich falsch sind.  Zum Beispiel werden Populationsgrößen in vielen Modellen als reelle Zahlen angesehen, die bestimmten Differenzialgleichungen genügen sollen -- obwohl Populationsgrößen offensichtlich ganzzahlig sind.  Darum verdient auch die Übersetzung von der Wirklichkeit zum Mathematischen Modell und zurück besondere Beachtung.  Diese Schritte sind auch konzeptionell schwieriger als die Lösung der mathematischen Probleme, die ein Modell stellt, weil sie sowohl mathematisches als auch fachwissenschaftliches Verständnis benötigen.

Die mathematische Modellierung bietet auch die Gelegenheit, unter Computereinsatz mit Daten zu experimentieren.  Man kann das mathematische Modell programmieren und danach ausprobieren, wie es sich verhält, wenn man Anfangswerte und Parameter verändert.  Solche Simulationen sind eine verbreitete Methode in den Naturwissenschaften, so dass es sich lohnt, wenn dies auch im Mathematikunterricht in der Schule vorkommt.  Deutlich schwieriger ist es, Modelle auch quantitativ auf reale Daten anzuwenden.  Dabei müssen Parameter geschätzt, dann Vorhersagen gemacht und mit der Wirklichkeit verglichen werden.  Hierbei sind jedoch verschiedene Aspekte schwierig, so dass ich in diesem Seminar darauf verzichten werde.  Auf die computergestützte Modellierung möchte ich jedoch nicht verzichten.  In den Vorträgen des Seminars sollten daher die folgenden diversen Aspekte jeweils vorkommen, jedoch in
unterschiedlicher Gewichtung:

• Übersetzen von wirklichen Problemstellungen in mathematische, und Motivation der dabei auftretenden systematischen Modellierungsfehler;
• Mathematische Sätze beweisen;
• Qualitative Aussagen über spezifische Modelle herleiten und ihre Anwendung auf die Wirklichkeit diskutieren;
• Computerexperimente mit den mathematischen Modellen, Parameter  variieren und qualitative Aussagen durch Computerexperimente überprüfen.
   

Der Computereinsatz ist auch deshalb für den Einsatz der Modellierung in Schulen wesentlich, weil er es erlaubt, die in der
mathematischen Biologie verbreiteten Modelle mit Differenzialgleichungen durch Modelle mit Differenzengleichungen zu
ersetzen.  Letztere setzen keine Differenzialrechnung voraus, so dass ihr Einsatz im Unterricht nicht auf die Oberstufe beschränkt bleibt.  Sie sind auch gerade in der Biologie sogar dichter an der Wirklichkeit, weil viele modellierte Größen von Natur aus diskret und nicht kontinuierlich sind.

Gemeinsam mit Sebastian Bauer habe ich ein ähnliches Seminar im Sommer 2022 angeboten.  Sein Buch „Mathematisches Modellieren als fachlicher Hintergrund für die Sekundarstufe I+II“ ist auch die Hauptquelle für dieses Seminar.  Eine wichtige Veränderung gegenüber dem damaligen Durchlauf betrifft Experimente mit echten Daten: Hierauf verzichte ich diesmal.  So wie im damaligen Durchlauf erscheint es mir sinnvoll, nicht einzelne Vorträge zu vergeben, sondern einzelne Themenblöcke an Gruppen von 2–4 Studierende zu vergeben, die diese Themen dann gemeinsam präsentieren.  Ein Themenbereich, der im Lehrbuch von Bauer nicht vorkommt, ist die Epidemiemodellierung.  Diese wurde aus Anlass der Covid-Pandemie in den letzten Jahren intensiv beforscht, und es erscheint mir weiterhin lohnend, Epidemiemodelle als Themenblock zu betrachten. Hierbei sollen die klassischen Modelle zusammen mit neueren Modellen betrachtet werden, die bestimmte Aspekte der Covid-Pandemie erhellen sollen.
 

Which topics do I offer?

My research interests are quite diverse, so that I may accommodate many different directions for bachelor's and master's theses. This includes, among others:

•  C*-Algebras
•  K-Theory
• some parts of Algebraic Topology
•  some parts of Functional Analysis, especially involving bornologies
• Homological Algebra
• bicategories
• groupoids.

The list of courses below and the previous bachelor's and master's theses listed below give you more concrete ideas of possible topics.

Zwei-Fächer-Bachelor und Master of Education?

Ich habe bisher eine Bachelorarbeit im 2-FBA betreut und einige ko-betreut, aber noch keine Masterarbeit im Master of Education.  Ich bin mir bewusst, dass Studierende in diesen Studiengängen nur wenige Mathematikvorlesungen belegen können, so dass ich andere Themen wähle, die weniger voraussetzen.  Ein dankbares Gebiet hierfür ist die mathematische Biologie, zum Beispiel Modellierung von Epidemien, ich kann mir aber auch andere Themen vorstellen.

Should I take some particular courses before?

As a result, there is not one course that you must have taken to write a thesis with me, but you should have taken courses that allow you to get quickly into some research question in the areas listed above.  At the moment, there is a cycle of lectures in noncommutative geometry, with courses about

• Abstract Harmonic Analysis
• C*-algebras, especially Toeplitz and Cuntz-Pimsner algebras
• Mathematical modelling of topological materials through K-theory
• C*-algebras and groupoids.

More details are listed below.  These courses would, of course, be an ideal preparation to write a thesis with me.  At the same time, I already supervised a number of master's theses that did not involve any of these topics.  In particular, category theory would be a possible entry point for theses in the direction of bicategories.  I have recorded a number of courses in recent years, which are available for self-study.  They may also be very useful preparation for writing a thesis with me. 

When and how best to contact me?

Please send me an email to ask for an appointment.  It is useful for me if you write also about the kind of mathematics that you particularly like and/or some of the subjects that you have already learnt or are learning right now, especially those that you enjoy doing and would like to figure in your thesis project.  If you approach me well in advance, there would be some time for you to take courses suitable to prepare you for a thesis topic in a particular direction.  That is why I recommend to approach me early if you are interested in a thesis with me.   If you come to a week before you want to start working on your thesis, it may work out nicely or it might not.  This depends on whether you have already taken enough relevant courses for me to choose an interesting and doable topic for you.

Previous Bachelor's Theses

• Examples of covariance rings (2022).  This thesis studies a construction of rings from generalised dynamical systems that is inspired by crossed products for group actions and Cuntz–Pimsner algebras for C*-correspondences. It shows that purely algebraic analogues of them such as twisted group rings and Leavitt path algebras fit into this language, and examines to what extent partial and twisted partial actions of groups on rings may be covered as well.
• Convolution of measures on locally compact groupoids (2021).  This thesis explores a possible generalisation of the Haar system on a locally compact groupoid where invariant measures are replaced by quasi-invariant measures. The somewhat surprising outcome is that if such a generalised Haar system exists, then there must be a Haar system in the usual sense.
• Epidemienmodellierung mit Differenzengleichungen (2021, 2-Fächer-Bachelor).  This thesis grew out of a seminar for students in the Master of Education about mathematical modelling of the Covid pandemic. It studies difference equations that are analogous to the differential equations that are used in the literature. These difference equations are much more accessible for school children.
• Continuity of joint spectra (2021).  This thesis studies the continuity of spectral for families of operators on Hilbert spaces. The starting point is the dissertation of S.~Beckus, who showed that for bounded normal operators, the spectra vary continuously if and only if the operators form the sections of a continuous field of C*-algebras. This is extended to several commuting normal operators. The unbounded case is also explored, but the results are more messy in that case.
• Classifying spaces over topological groupoids (2021).  This thesis studies the classifying space of a topological groupoid, with an emphasis on imposing no normality or Hausdorffness assumptions on the groupoids and spaces. Instead of restricting to paracompact spaces, it restricts to numerable covers in the definition of a topological groupoid and a principal bundle.
• On the K-theory and KK-theory of Cuntz-Pimsner algebras (2020).  This thesis studied Pimsner's proof that the Toeplitz C*-algebra of a C*-correspondence is KK-equivalent to the underlying C*-algebra of the C*-correspondence. This proof is rewritten using quasi-homomorphisms instead of KK-theory.
• Exakte Moduln über dem von Manuel Köhler beschriebenen Ring (2018).  This thesis studies modules over a certain rather complicated ring that occurs in the classification of group actions on certain C*-algebras. The Bachelor's thesis helped me to write the article On the classification of group actions on C*-algebras up to equivariant KK-equivalence.

Previous Master's Theses

• Stammeier's C*-algebras for several injective group endomorphisms as C*-algebras of diagrams of étale groupoid correspondences (2023).  Nicolai Stammeier has studied a class of C*-algebras, which are associated to irreversible algebraic dynamical sytems. This thesis shows that these C*-algebras are special cases of C*-algebras defined by diagrams of groupoid correspondences, where the groupoid correspondences are associated to injective endomorphisms of a group. In particular, it follows that these C*-algebras are groupoid C*-algebras in case the endomorphisms have finite index. In this case, the properties of this groupoid that are related to simplicity and pure infiniteness of the groupoid C*-algebra are studied.
• A classification of 2-groups (2022).  This thesis reproves the classical Whitehead classification of crossed modules by reducing it to the classification of skeletal bigroups, which turns out to be rather easy. To reduce to this case, the Whitehead Theorem that characterises when a homomorphism of bicategories is an equivalences of bicategories is shown directly, without using the MacLane Coherence Theorem.
• Invariants for topological insulators coming from decompositions of coarse spaces (2021). This thesis continues the study of topological insulators using the K-theory of Roe C*-algebras. It simplifies an argument in an article by Ludewig and Thiang using the coarse Mayer–Vietoris sequence and then studies the universality of the boundary map in van Daele's K-theory, following previous work by George Elliott and myself.
• On groupoid models for diagrams of groupoid correspondences (2021).  This thesis proves that any diagram of proper correspondences between locally compact étale groupoids has a groupoid model. The issue is mainly to prove that a certain category of actions contains a final object, and this is shown by verifying that any action extends to a certain relative Stone–Čech compactification.  Some results of this thesis are published in the joint articles Existence of groupoid models for diagrams of groupoid correspondences and The bicategory of groupoid correspondences
• Leavitt path algebras as Cohn localisations and their Hochschild homology (2021).  This thesis proves that Leavitt path algebras of directed graphs are quasi-free and computes their Hochschild homology in complete generality. This extends previous results for row-finite graphs. The main method is to realise the Leavitt path algebras as a Cohn localisation of a variant of the path algebra. Issues that require some care are to define this variant carefully for irregular graphs and to handle nonunital rings in case the graph has infinitely many vertices.
• Geometric construction of Hamiltonians (2022).  This thesis starts from some computations by Prodan and Schulz-Baldes for a certain Hamiltonian for topological materials. These Hamiltonians are defined in a uniform manner in all dimensions and depend on a mass parameter. The question is which topological phase they have depending on the mass. This is computed using explicit generators of van Daele's K-theory for spheres and geometric bivariant K-theory to relate K-theory groups for spheres and tori. The computations are also done in the case of “real” K-theory. This is a cohomology theory, which was not treated in the earlier work on bivariant K-theory. It is checked in the thesis that geometric bivariant K-theory also works for this cohomology theory.  Some results of this thesis are published in the joint article Geometric construction of classes in van Daele's K-theory.
• Partial actions on C*-algebras (2022).  This thesis explores partial actions of groups and inverse semigroups on C*-algebras from the point of view of morphisms to the bicategory of C*-correspondences. It is already known that homomorphisms correspond to ordinary actions of groups and inverse semigroups. Both morphisms and partial actions are more general than that, but not the same. It turns out that extra conditions are needed on a morphism to correspond to a partial action and that extra conditions for cones over a morphism are needed to get the correct notion of covariant representation, which is used to define the crossed product for a partial action. As a class of examples, the article studies Toeplitz algebras of automorphisms and correspondences.

Ich habe in den Semestern Winter 2020/1 bis Winter 2021/2 und Winter 2022/3 bis Winter 2023/4 jeweils die Vorlesungen Mathematik für Studierende der Physik 1-3 (MaPhy) angeboten.  Bei der MaPhy3-Vorlesung im Winter 2023/4 habe ich ein Skript erstellt, siehe https://www.studip.uni-goettingen.de/sendfile.php?force_download=1&type=0&file_id=212157ea4fd453638fb075134159be8b&file_name=main.pdf

Die Vorlesungen wurden jeweils aufgezeichnet, und diese Vorlesungen sind in Stud.IP weiterhin verfügbar und grundsätzlich zum Selbststudium geeignet.

My main area of research is noncommutative geometry, a generalisation of our idea of what space and geometry are. This theory is inspired by quantum mechanics. The Heisenberg commutation relation predicts that the position and momentum observables in quantum mechanics form a noncommutative algebra. This noncommutative algebra describes the phase space of a quantum mechanical system. Thus some noncommutative algebras seem to describe geometric objects. Which geometric ideas carry over to such spaces, and how?

Algebraic topology provides many tools to distinguish different spaces by some homological invariants. Most of these invariants do not carry over to noncommutative spaces. The main invariant in that setting is K-theory, which for ordinary spaces classifies the vector bundles over a space. In noncommutative geometry, K-theory is part of a bivariant theory, which provides powerful tools to compute K-theory and a rich categorical structure. One important aspect of my work is extra structure in bivariant K-theory and how to use it, among others, for the classification of C*-algebras.

Another important homological invariant for noncommutative algebras are cyclic cohomology theories. Periodic cyclic cohomology generalises the de Rham homology of smooth manifolds, but only makes sense for algebras that behave like smooth functions. Variants of it like analytic and local cyclic cohomology also give good results for larger algebras like C*-algebras, which behave more like continuous functions. The original periodic cyclic cohomology is computable through its relationship to Hochschild cohomology. One important aspect of my work has been to extend various tools of homological algebra to algebras that carry some functional analytic structure, such as algebras of smooth or holomorphic functions and their noncommutative analogues. I found that bornologies are better behaved than topologies when functional analysis meets algebra.  In recent research of mine about analytic cyclic homology for algebras over nonarchimedean fields, this turned out to be so in a nice way, giving a good analytic interpretation for the weak completions that are used in the context of Monsky-Washnitzer and rigid cohomology for varieties over finite fields.  We may also want to compare the homological algebra for two different algebras that describe the same noncommutative space with more or less extra structure. One may be an algebra of polynomial functions, the other an algebra of smooth functions on that noncommutative space. We would expect the homological algebra for these to be related, and my notion of an isocohomological embedding gives one way to make precise how.

Ordinary symmetries of spaces are usually described through group actions. Local symmetries may lead to groupoid actions. In noncommutative geometry, however, the more general notion of space should also lead to drastically different types of geometry. One Ansatz to describe noncommutative symmetries are quantum groups, which are noncommutative C*-algebras with extra structure that resembles that in a group.  Another Ansatz which I found even more interesting is higher category theory.  Several constructions in the study of C*-algebras become very clear if we treat C*-algebras as a bicategory with C*-correspondences as arrows and isomorphisms of correspondences as 2-arrows.

For instance, the noncommutative torus may be thought of as the algebra of functions on the quotient of the circle group by a dense subgroup. This quotient is a rather badly behaved non-Hausdorff space, which is replaced in noncommutative geometry first by a groupoid and then by its groupoid C*-algebra. The group structure on this quotient does not correspond to a quantum group structure on noncommutative tori. The philosophy of noncommutative geometry suggests to replace a non-Hausdorff quotient group by a bigroup or crossed module. There is indeed an action of this crossed module on the noncommutative torus that describes the translation action of the underlying non-Hausdorff quotient group on itself.

Besides providing more general notions of symmetry, bicategories also allow clarify the role of Fell bundles: a saturated Fell bundle over a group or groupoid is equivalent to an action by equivalences. For non-invertible dynamics, we also get a common treatment for crossed products and Cuntz-Pimsner algebras. An action of a monoid by C*-correspondences on a C*-algebra is equivalent to a product system over that monoid. If the product system is proper, then the Cuntz-Pimsner algebra of this product system is characterised by a universal property that is analogous to that of crossed products. This universal property is a bicategorical analogue of the definition of a limit or colimit.

Such limit constructions make sense also in suitable bicategories of rings and groupoids, as opposed to C*-algebras. Ongoing work by my students and me interprets various constructions of rings and groupoids as such bicategorical limits.

I plan to offer several more advanced courses until the summer 2025. More specifically, my plans are the following:

Winter term 2023/4: Harmonic Analysis

Format: 4 SWS lecture + 2 SWS exercises

Keywords: Representation theory for locally compact groups, especially for finite and compact groups, including compact Lie groups such as SU(2), Abelian groups, and some examples of nilpotent groups such as the Heisenberg group

This class deals with the unitary representations of groups, especially compact and Abelian groups, and some examples like the Heisenberg group. The starting point of Harmonic Analysis is the Fourier transform for functions on the circle. This may be viewed as the decomposition of the regular representation of the circle group into irreducible representations. More generally, for compact groups it is true that any representation decomposes as a direct sum of irreducible representations, and the latter are finite-dimensional and may be classified for concrete examples of compact groups. We will carry this out at least for the group SU(2). For all Abelian locally compact groups, we will introduce an analogue of the Fourier transform on the real numbers or the circle group. Then we will study the irreducible representations of the Heisenberg group. If time permits, we may consider the more general case of nilpotent Lie groups, where the results are similar to the Heisenberg group, and look into induced representations as a tool to describe the representation theory of more complicated groups. This lecture assumes functional analysis and basic mathematical competence. In particular, students should already be familar with Hilbert spaces, and some form of spectral theory for selfadjoint operators on Hilbert spaces.

The harmonic analysis lecture is intended not just as the first part of my own cycle of lectures. Advanced lectures in the direction of mathematical physics or PDE may also use such this course as a prerequisite. Some structural changes to our study programme are currently being discussed, which would include offering harmonic analysis courses regularly each year.

This class was offered by Christopher Wulff in my place.  He followed the book Abstract Harmonic Analysis by Folland and covered the following topics:

I. Locally compact groups 
I.1 Topological groups (2.1 in Folland) 
I.2 Haar measure (2.2 in Folland, ohne die komplizierten Beweise) 
I.3 The modular function (2.4 in Folland) 
I.4 Convolutions (2.5 in Folland) 

II. Basic representation theory 
II.1 Unitary representations (3.1 in Folland) 
II.2 Representations of G and L^{^1}(G) (3.2 in Folland) 

III. Analysis on locally compact abelian groups 
III.1 The dual group (4.1 in Folland) 
III.2 Spectral theory & the Gelfand transform (Hier ging es um einige grundlegende Aussagen aus Kapitel 1 in Follands Buch, die ich aber anders aufgezogen habe, weil ich Follands Argumentation in Teilen nicht nachvollziehbar fand.) 
III.3 Representations of locally compact abelian groups (4.4 in Folland) 
III.4 The Fourier transform (Auszüge aus 4.2 in Folland.) 

IV. Analysis on compact groups 
IV.1 Representations of compact groups (5.1 in Folland) 
IV.2 The Peter-Weyl theorem (5.2 in Folland) 
IV.3 Example SU(2) (aus Kapitel 5.4 in Folland) 

V. The Heisenberg groups

He is going to offer a seminar in the summer term 2024 covering some further topics in representation theory. 

Summer term 2024: C*-Algebras

Schedule: Monday and Thursday 10-12 (planned)
Assistance: Akshara Prasad and/or Yuetong Luo
Language: English
Audience: M.Sc. and last year B.Sc. students
Exam requirements: 50% in homework assignments, show own solutions at least twice during exercise sessions

Format: 4 SWS lecture, maybe add 2 SWS exercises if teaching capacity and student demand justify this

Keywords: General theory of C*-algebras, up to the Gelfand-Naimark Theorem; Hilbert modules, C*-correspondences and their Cuntz-Pimsner algebras; graph C*-algebras and C*-algebras of self-similar groups

This course begins with the general theory of C*-algebras, up to the Gelfand-Naimark Theorem, which realises any C*-algebra as a C*-subalgebra of bounded operators on a Hilbert space. Then I will move on to study Hilbert modules over C*-algebras. These generalise Hilbert spaces by allowing a module over a C*-algebra instead of a vector space, equipped with a C*-algebra valued inner product. Such a Hilbert module over a C*-algebra together with a representation of the C*-algebra on it is called a C*-correspondence. This is the initial data for the construction of Cuntz-Pimsner algebras. This is an important method to define interesting C*-algebras. In particular, graph C*-algebras or C*-algebras of self-similar groups are defined in this way. Cuntz-Pimsner algebras also figure prominently in my own research. One important aspect in my research is that C*-correspondences form a bicategory and that bicategory theory offers a useful perspective on constructions of C*-algebras such as Cuntz-Pimsner algebras. This would be a good direction for Bachelor and Master thesis under my direction. I plan, however, to focus on the analytical aspects of the Cuntz-Pimsner algebra construction, leaving the bicategorical links to individual reading or a separate class, which may be offered depending on demand and capacity. Group representations may be studied using group C*-algebras and crossed products by group actions on C*-algebras. This links this course to the Harmonic Analysis course in the previous term. Nevertheless, students who missed the Harmonic Analysis course may still do fine in this class, except for a few lectures. I do assume knowledge of functional analysis, however.

Winter term 2024/5: Modelling topological phases of matter

Format: 4 SWS

Keywords: quantum mechanics background, some homotopy theory, classification of topological phases of physical systems through van Daele's K-theory, anti-unitary symmetries and Clifford algebras, van Daele K-theory classes for spheres

Prerequisites: This lecture requires functional analysis, but not the lectures on harmonic analysis and C*-algebras, see below for more details. 

A recent discovery in quantum physics is that certain materials are insulators, in principle, but conduct electricity on the boundary of a finite chunk of them. Even more, this conductivity on the boundary is forced to be present by topological properties of the underlying physical system. These topological properties may be understood using C*-algebra K-theory. The course will begin with a crash course in quantum mechanics, to clarify how the mathematics that we are going to do may be applied to physical systems. Then we define topological phases through homotopy classes of Hamiltonians in a given C*-algebra that describes the physical system. Usually, this C*-algebra consists simply of matrix-valued functions on a torus. More complicated C*-algebras appear when one adds magnetic fields or disorder to the setup. The topological classification of Hamiltonians may be formulated using basic homotopy theory. This even allows for some computations, but I plan to focus on C*-algebra K-theory as a more powerful machinery to encode the relevant topological information. There are different ways to introduce K-theory for Banach algebras or C*-algebras, and the approach most convenient for topological phases is not the one most commonly used by C*-algebraists. Many interesting topological materials have extra symmetries, and only show topological properties if the symmetries are taken into account. To describe these extra symmetries conveniently and in a way that interacts nicely with K-theory, we will use C*-algebras over the field of real numbers and equipped with a Z/2-grading, where van Daele has given a definition of K-theory that is very close to physical applications. Clifford algebras play a crucial role in the K-theory of real C*-algebras, and they may also be used to describe the various physical symmetry types.

This class also requires functional analysis, and you should know what a C*-algebra is and be ready to take a few facts about C*-algebras for granted. In this sense, you may succeed in this course without taking the earlier harmonic analysis or C*-algebras classes, if you are willing to spend a few days to learn the most basic things about C*-algebras. You need some physics background to fully appreciate this course. I do give a short crash course in quantum mechanics, but this may not be enough if it is your first contact with it. The mathematics in this course may be considered interesting in its own right. In particular, this may be the only course about the rich subject of K-theory of C*-algebras in the near future. If there is a lot of interest by students with no physics background, I could split the course into two strands, one about K-theory for C*-algebras and one that is more about the physical applications. This may be difficult to do properly, however, so that I will only do this if there is sufficient interest in this option. So please write to me if you are interested in such an arrangement.

I have offered a 2SWS course on modelling topological phases in the past, which is still available as a reading course, that is, online materials are available that may suffice for you to learn this subject. My plan for the Winter 2024/5 course differs in that I use the more powerful machinery of C*-algebra K-theory instead of just basic homotopy theory. I usually would recommend the 4SWS course. The smaller course, however, has the advantage of lighter prerequisites. If you prefer to take it instead of the more substantial 4SWS course, please write me an email. If you have already taken the 2SWS reading course, it would still make sense to take the more substantial 4SWS course in addition to that.

Winter term 2024/5: Groupoids and C*-algebras

Format: 2 SWS Seminar or lecture

Keywords: Étale groupoids and their C*-algebras, Cartan subalgebras in C*-algebras, constructing C*-algebras from groupoid correspondences.

Prerequisites: This class assumes my C*-algebra class from the winter term, including the theory of C*-correspondences and their Cuntz-Pimsner algebras.

This class will introduce another important construction of C*-algebras, starting with étale groupoids. (C*-algebras of more general locally compact groups shall not be covered.) Renault has shown that a C*-algebra has this form if and only if it has a C*-subalgebra with certain properties, called a Cartan subalgebra. This result has received a lot of attention recently because large classes of C*-algebras may be shown to be groupoid C*-algebras in this way. My own recent research deals with groupoid correspondences and how to associate C*-algebras to them. This allows to prove that many Cuntz-Pimsner algebras are groupoid C*-algebras, by describing the underlying groupoid directly from the groupoid correspondence.

An ideal form for this class would be a seminar. This, however, requires at least 10 students, while a lecture may already make sense with a handful of students. Therefore, it may turn out that the class will be a lecture and not a seminar, or it may be integrated into the regular noncommutative geometry research seminar.

Research seminar

My research seminar (Oberseminar Nichtkommutative Geometrie) mostly has talks by my doctoral and master students about their ongoing work, or articles that they are reading. Sometimes, the seminar follows a programme to learn about a certain topic.

During the winter term 2022–23, the seminar was about representations of *-algebras by possibly unbounded operators and C*-algebras attached to them.

Synergies with Chenchang Zhu's cycle

One of my research specialities is bicategories in noncommutative geometry. Bicategories are close in spirit to 2-categories, which is the first step beyond categories in the generalisation to infinity-categories. The relevant bicategories in noncommutative geometry have groupoids, C*-algebras, or just rings as objects, and groupoid correspondences, C*-correspondences, or suitable bimodules over rings as arrows. This is one reason why I have already offered courses on category theory based on Emily Riehl's book. In addition, I can offer master thesis topics that bring together advanced category theory with C*-algebras or étale groupoids. I also have doctoral students working with Lie groupoids and Lie algebroids, though I have not yet thought of master's thesis projects in this direction.

I recorded several more or less advanced courses in the past, and some of them are suitable for self-study:

• Bicategories in noncommutative geometry (Summer Term 2022)
• Noncommutative differential geometry (Summer Term 2020)
• Modelling topological phases of matter (Winter Term 2018–19)
• Category theory in context (Winter Term 2021–22)

If you are a student at Göttingen University and intend to take one of these courses for credit, please write me an email about the possibility of offering them as a reading course.

Mathematics is already difficult enough.  Write concisely.  Use simple grammar to make your writing easier to read.

When you construct overly complicated, such as this one, which is an intentionally bad example designed to show what can go wrong, the reader will almost certainly have already forgotten what it was that you were discussing at the very beginning of the sentence, when he reaches the end of the sentence, forcing her to go back to the beginning and read it again, maybe several times, until eventually after many false attempts understanding what you want to say, or giving up and turning to another author, who is capable of writing sentences that are easier to follow, being more concise.

Grammar also subtly changes the meaning of a sentence.  In a sentence, there is more emphasis on the subject and the verb, and also on the very first words in it, and less on subordinate clauses such as relative clauses.  For instance:

Continuous functions on compact spaces are bounded.
Good sentence, everything is said straight, without subtle shades of emphasis or room for misunderstanding.

It is clear that continuous functions on compact spaces are bounded.
This is a sentence about “it”, which “is clear”.  Whatever is clear is less important, hardly worth mentioning.  I see no use for this construction.

Clearly, continuous functions  on compact spaces are bounded.
This is a sentence about continuous functions being bounded, but it is made very clear that this is a trivial statement.  Use this if you really want to emphasise the triviality of the statement, but nevertheless want to mention the statement.  So this is rarely useful.

It is proved in Lemma 1.5 that continuous functions on compact spaces are bounded.
Once again, this is a statement about “it”, which “is proved”.  The second-most important thing in the sentence is Lemma 1.5, which is where the action takes place.  The continuity of bounded functions is the least important thing in this sentence.  Once again, not recommended.

Lemma 1.5 proves that continuous functions on compact spaces are bounded.
Now Lemma 1.5 is the main player, and it proves something.  Once again, what is being proved is marked as less important.  Actually, the sentence does not directly assert that continuous functions are bounded.  To deduce this, the reader has to combine it with the common knowledge that statements that are proved are also true.   This sentence may be useful if you are discussing the structure of an article, to lead the reader to interesting results.

We prove in Lemma 1.5 that continuous functions on compact spaces are bounded.
Now “We proving” something is the main action.  Use this to emphasise your own contribution to this proof.

Continuous functions on compact spaces are bounded by Lemma 1.5.
Now the continuity of bounded functions is the main statement; that it is proved in Lemma 1.5 is only a sideremark, for the reader wanting to check this fact.  This is probably how you want to refer to this fact during a proof.

By Lemma 1.5, continuous functions on compact spaces are bounded.
Now Lemma 1.5 is emphasised through the unusual word order.  The continuity of bounded functions is asserted neutrally.  I can imagine this construction being useful sometimes, but rarely.

From the Stone-Weierstraß Theorem, we have the following result.
This is a result about us having something.  I see no situation where this formulation is good style.  The comma is needed here, by the way, to excuse the unusual word order.  Better write “The Stone-Weierstraß Theorem implies/gives/yields the following result.” or “The following result follows from the Stone-Weierstraß Theorem.”, depending on whether you consider the Stone-Weierstraß Theorem or the following result to be slightly more important.  I often see the phrase “we have that”, but I rarely see a use for it because its does not add to the meaning and complicates the grammar by turning a main clause into a subordinate clause. 

Well, a general style guideline says that a sentence should not begin with a mathematical formula.  So the sentence “We have that c²=a²+b².” fulfils this rule, unlike the simpler “c²=a²+b².”  But the meaning of the rule is that such a formula should be put into context, by saying, say, “Hence c²=a²+b².” or “Now we use the Pythagorean Theoremc²=a²+b².”

Note that continuous functions on compact spaces are bounded.
Here “Note that” turns the main statement into a subordinate clause, hinting that this statement is not so important.  If you really want to mark statements as being less important, rather say clearly why they are not so important, but still important enough that you need to write them.

“Such that” or “satisfying” versus “with”
What is the difference between “We seek x with f(x)=0.” and “We seek x such that f(x)=0.” or “We seek x satisfying f(x)=0.”?  The second and third formulations create a subordinate clause, namely, “f(x)=0”, which is treated here as an assertion, not as an equation.  An equation may have two roles: it is a statement, f(x) is equal to 0, and a mathematical formula and thus an object in its own right.  The assertion “f(x)=0” is so simple that I prefer the first formulation with its simpler grammar, treating the equation as an object in its own right.  I would use “such that” if what is coming afterwards is so complicated that a new subordinate clause is needed or helpful to understand what we require of x.  For instance: “We seek x such that the sign of f changes at x.”


Punctuation marks have a function.
The punctuation marks “.!?;:” all end gramatically complete sentences.  For instance, “Given a C*-algebra A.” is not correct because it is not a sentence.  “Let A be a C*-algebra.” is correct.  An experienced reader expects a full stop as a sign to pause and digest the sentence, before going on.  Over-long sentences make this hard, so avoid them.

It is perfectly OK if every sentence expresses only one idea.  Sometimes, relationships between ideas are easier to express by combining them in one sentence together with appropriate links, such as “We will show that the function is continuous in order to prove that it is bounded.”; this points out that you only care about the function being bounded, but prove continuity instead.  The alternative formulation “We will show that the function is continuous; hence it is bounded.” says almost the same, but without commenting on whether continuity or boundedness is what you really care about.

The colon “:” differs from the the full stop “.” in asking the reader to read further, indicating that this is needed to understand the last sentence, as in “The following are equivalent:”.  I use the semicolon “;” to highlight a close relationship between the assertions before and after it.

For commas, there are many rules where they must or must not appear.  Some words are always separated from the rest of the text by commas.  This includes furthermore, moreover, however, therefore, of course, namely, that is, for example, say, and many more.  A comma must not be put around a necessary relative clause, that is, if the meaning of the sentence is changed or lost by leaving out the relative clause; example: “The function which we construct is continuous.”.  In contrast, a comma must be put around unnecessary relative clauses; example: “This function, which is very important for the following construction, is continuous.”

When no rule requires or forbids a comma, I write a comma if it helps the reader to structure a complicated sentence.

Ich bin Sprecher des Chapters Mathematik und Informatik bei Alumni Göttingen.

• Ich bin Vorsitzender des Allgemeinen Fakultätentags.
• Ich bin Mitglied im Beirat der Konferenz Mathematischer Fachbereiche (KMathF).
• Ich bin Mitglied bei den Grünen.
• Ich singe im Göttinger Vokalensemble.
• Ich singe auch im Vokalensemble Lenglern.

Die Möglichkeit, das Studium durch ein Stipendium zu finanzieren, ist vielen entweder unbekannt, oder sie lassen sich von dem Wort „Begabtenförderungswerke“ abschrecken. Dabei werden die Stipendien nicht nur nach Schul- und Studiennoten vergeben, sondern es kommt besonders auf Persönlichkeit und Engagement an. Sind Sie im Sportverein, in einer Kirchengemeinde oder einem Chor aktiv? Diese und ähnliche Aktivitäten gelten als „ehrenamtliches Engagement“. Wenn Ihre Noten nun noch mindestens im Zweier-Bereich liegen, lohnt eine Bewerbung auf jeden Fall. Sogar, wenn beim Abischnitt eine drei vorm Komma steht, Sie aber beispielsweise im Studium viel besser sind, sich besonders sozial engagieren, kann eine Bewerbung chancenreich sein.

An staatsfinanzierten Stiftungen gibt es neben der Studienstiftung (bei der man sich nur eingeschränkt selbst bewerben kann) Stiftungen verschiedener Parteien, Kirchen und Gewerkschaften. Ein gemeinsames Internetportal informiert über diese Stiftungen. Außerdem gibt es eine sehr große Zahl meist kleinerer privater Stiftungen. Häufig sind deren Stipendien auf bestimmte Zielgruppen, Studienfächer oder Orte ausgerichtet. Auch hier lohnt sich aber die Suche, denn wenn Sie das Glück haben, zur Zielgruppe einer dieser Stiftungen zu gehören, ist eine Bewerbung dort besonders chancenreich, weil weniger Bewerber(innen) um diese Stipendien konkurrieren. Eine Datenbank über Stipendien privater Stiftungen

Die Initiative Arbeiterkind bietet neben Links auf Stipendiengeber und -datenbanken auch Interviews mit erfolgreichen Bewerber(inne)n und Bewerbungstipps.

Einige Bewerbungsfristen, die ich vor langer Zeit im Internet recherchiert habe, die aber eventuell nicht mehr aktuell sind: 

Konrad-Adenauer-Stiftung: 15. Januar und 1. Juli
Friedrich-Ebert-Stiftung: keine Fristen
Friedrich-Naumann-Stiftung: 31. Mai und 30. November
Hans-Böckler-Stiftung: einige Zeit vor dem 1. Februar und 1. September (mehrstufiges Verfahren: Stipendiat(inn)en vor Ort müssen Bewerbungen prüfen und bis zu diesem Datum weiterleiten)
Heinrich-Böll-Stiftung: 1. März 2009
Evangelisches Studienwerk e.V. Villigst: 1. März und 1. September