I plan to offer several more advanced courses until the summer 2025. More specifically, my plans are the following:
Format: 4 SWS lecture + 2 SWS exercises
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.
Format: 4 SWS lecture, maybe add 2 SWS exercises if teaching capacity and student demand justify this
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.
Format: 4 SWS
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.
Format: 2 SWS Seminar or lecture
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.
This class assumes my C*-algebra class from the winter term, including the theory of C*-correspondences and their Cuntz-Pimsner algebras.
If you want to write a bachelor thesis or master thesis under my supervision, you may approach me at any time. If you do it earlier, I may advise you about courses to take before you actually start to work on the thesis. I try to take into account the courses that you have already taken to suggest a topic that fits your current knowledge. My research interests are rather broad. In particular, I am interested in (bi)categories, Lie groupoids, bornological functional analysis, especially over the p-adic numbers, and homological algebra. Therefore, it is possible that we can find a suitable topic even if you have not taken so many classes with me. And the classes by Chenchang Zhu could also give you useful knowledge for a bachelor thesis with me. Apart from this general flexibility, a standard plan to write a Bachelor's thesis with me would be to do this after the C*-algebra class in August and September 2024. I would recommend to approach me already during the summer term 2024 in order to find a suitable topic, so that you can start work on it soon after the end of lectures. The modelling topological phases and the groupoids and C*-algebras classes are both close enough to my current research, so they form an ideal basis for writing a master's thesis during the summer term 2025. It may, however, be possible to write a master's thesis with me after taking the C*-algebra class or the harmonic analysis class, provided that you have already taken some other relevant courses that allow us to choose a suitable topic. Write me an email if you want to talk about such options for you.
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.