Lectures WS 2024/2025

Speaker: Michael Preeg, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by Karl-Hermann Neeb

Abstract: In accordance with a projection P on a Hilbert space H, such that H = PH + (1-P)H, an irreducible representation of the CAR algebra CAR(H) can be constructed on the tensor product Gamma_P(H) of the Fock spaces F(PH) and the complex conjugate of F((1-P)H). Unitary operators U on H for which [U,P] is a Hilbert-Schmidt operator form the restricted unitary operators group U_(res)(H). For such a restricted operator, the Bogoliubov transformation Bog(u) of CAR(H) is implementable. This talk examines unitary one-parameter subgroups of restricted operators, with a particular focus on multiplication operators of bounded functions on square-integrable functions with the unit circle as the domain in the case of projection onto the space H_+ of positive Fourier modes. This leads to the bounded functions of the Sobolev space of fractional order 1/2. In general, restricted operators can be expressed as an operator-valued 2×2 matrix with respect to the splitting of H into PH and (1-P)H. Their norm is defined by the operator norm on the diagonal entries and the Hilbert-Schmidt norm on the off-diagonal entries. This norm enables us to conclude that the unitary restricted multiplication operators constitute a Banach-Lie group. To investigate the generators of one-parameter subgroups, we draw upon tools from infinite-dimensional Lie theory, demonstrating that their generators are also restricted and exhibiting some properties of the exponential function. The appropriate topology on these unitary restricted operators is the so-called P-strong topology, which replaces the operator norm topology on the diagonal entries with the strong operator topology. In the case of general restricted operators, we present an examples of a unbounded non-restricted generator and a characterisation of (possibly unbounded) restricted generators. In the case of multiplication operators, we demonstrate that every (possibly unbounded) real-valued function of the Sobolev space of fractional order 1/2 generates a P-strong continuous unitary one-parameter subgroup.

Speaker: Latévi M. Lawson, Max Planck Institute for the Science of Light, Erlangen – Invited by Catherine Meusburger

Speaker: Karl-Hermann Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg

Abstract: We are interested in obtaining local nets in the sense of Haag–Kastler from unitary representations of a connected Lie group G. A natural set of axioms leads to a causal structure on M. We focus on the case where M = G/P is a flag manifold of a simple Lie group G, or a covering space thereof. Then G must be a hermitian Lie group and M a conformal compactification of a euclidean Jordan algebra V. Its simply connected covering is a simple space-time manifold in the sense of Mack–de Riese.

We show that the unitary representations permitting non-trivial nets are the positive energy representations (direct integrals of lowest weight representations). These nets have several interesting features. One is that locality properties of the net can be specified in terms of open G-orbits in the space of pairs, which is most interesting for covering spaces because the number of these orbits corresponds to the number of sheets in the covering.

Speaker: Valerio Proietti, University of Oslo – Invited by Kang Li

Abstract: Given a class of topological dynamical systems, we study the associated mapping torus from the point of view of foliated spaces. By studying the interaction between the leafwise Dirac operator and the invariant transverse measures, we reframe in a geometric fashion the Elliott invariant for the crossed product of the dynamical system, and prove a rigidity result for the mapping torus, lifting leafwise homotopy equivalences to isomorphism of the noncommutative leaf space. Joint work with Hao Guo and Hang Wang.

Speaker: Maximilian Ludwig, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by Catherine Meusburger

Abstract: In dem Vortrag stelle ich Ergebnisse aus meiner Masterarbeit vor. Diese ist thematisch in den Bereich der topologischen Quantenfeldtheorien (TQFT) einzuordnen. Wir betrachten (ungetwistete) Dijkgraaf-Witten Theorie mit Defekten, welche als ein Spezialfall der Turaev-Viro-Barrett-Westbury (TVBW) TQFT mit Defekten betrachtet werden kann. Letztere ist eine Verallgemeinerung der TVBW TQFT ohne Defekte und wurde von Turaev, Viro und Barrett, Westbury als Zustandssummenmodel für 3-MFKen eingeführt. Diese verwendet als algebraisches Datum eine sphärische Fusionskategorie C.

C. Meusburger verallgemeinerte diese zu dem Fall mit Defekten, wobei Defektflächen durch eingebettete 2d MFKen und Defektlinien und Defektpunkte durch eingebettete Graphen in der Defektfläche realisiert werden. Defektdaten werden jeweils zu 3d Regionen, 2d Defektbereichen, 1d Defektlinien und 0d Defektpunkten zugeordnet. Diese sind systematisch durch sphärische Fusionskategorien C,D, (C,D)-Bimodulkategorien mit Bimodulspuren, (C,D)-Bimodulfunktoren und (C,D)-bimodulnatürliche Transformationen gegeben und formen für feste C,D eine pivotale 2-Kategorie Bimodθ(C,D). Die Zustandssumme für Defekt-3-MFKen wird mit Hilfe eines 2d diagrammatischen Kalküls für Bimodθ(C,D) definiert, der an die betrachteten Defekte angepasst ist.

Ungetwistete Dijkgraaf-Witten Theorie mit Defekten ist der Spezialfall, dass alle sphärischen Fusionskategorien C,D Kategorien VecG von G-graduierten endlichdimensionalen C-Vektorräumen für endliche Gruppen G sind und dass alle Kozykeln, die als Teil der Defekt-Daten auftreten können, trivial sind. Dies führt dazu, dass wir für die Defekt-Daten nur eine pivotale Unter-2-Kategorie Bimodθtriv(C,D) ⊂ Bimodθ(C,D) betrachten. In diesem Fall vereinfachen sich alle Defekt-Daten und bilden ebenfalls eine pivotale Bikategorie Rep(G × Hop−Set). Das erste Resultat ist, dass die pivotalen Bikategorien Bimodθtriv(C,D) und Rep(G × Hop−Set) pivotal 2-äquivalent sind. Darauf aufbauend habe ich einen vereinfachten diagrammatischen Kalkül für Rep(G × Hop−Set) entwickelt und gezeigt, dass dieser äquivalent zu dem diagrammatischen Kalkül für Bimodθtriv(C,D) ist. Abschließend habe ich noch Beispiele für Defekt-3-MFKen betrachten, die aus Flächen vom Geschlecht g entstehen, und deren Zustandssumme berechnet.

Speaker: Tobias Simon, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by Karl-Hermann Neeb

Abstract: In this talk, we explain the connection of globalizations of Harish-Chandra modules and the complex crown domain, which allows holomorphic extension of orbit maps. This allows one to apply methods from complex analysis to obtain sharp polynomial estimates from above and below of the derivatives of these orbit maps since the geometry of the domain determines the asymptotic at the boundary.

Speaker: Alistair Miller, Odense University – Invited by Kang Li

Abstract: Self-similar groups are groups of automorphisms of infinite rooted trees obeying a simple but powerful rule. Under this rule, groups with exotic properties can be generated from very basic starting data, most famously the Grigorchuk group which was the first example of a group with intermediate growth.

Nekrashevych introduced a groupoid and a C*-algebra for a self-similar group action on a tree as models for some underlying noncommutative space for the system. Our goal is to compute the K-theory of the C*-algebra and the homology of the groupoid. Our main theorem provides long exact sequences which reduce the problems to group theory. I will demonstrate how to apply this theorem to fully compute homology and K-theory through the example of the Grigorchuk group.

This is joint work with Benjamin Steinberg.

Speaker: Lucas Seco, Universidade de Brasília on postdoc leave at Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by Karl-Hermann Neeb

Abstract: The fundamental alcove of a reduced root system encodes essential features of the root system, e.g. its affine Weyl group and its fundamental group.

I will talk about work on the geometry of the fundamental alcove A. Our first main result shows that the full isometry group Aut(A) of the alcove is naturally isomorphic to the automorphism group of the Dynkin diagram of the corresponding affine root system. Building on this connection, we establish that Aut(A) is an abstract Coxeter group, with generators corresponding to affine isometric involutions of the ambient space. Although these involutions may not be reflections, our second main result leverages them to construct, by slicing the Komrakov-Premet fundamental polytope K for the action of the fundamental group, a family of fundamental polytopes for the action of Aut(A) on A, whose vertices are contained in the vertices of K, and faces parametrized by the so-called balanced minuscule roots.

This is joint work with Arthur Garnier, LAMFA-Université de Picardie Jules Verne, France, Karl-Hermann Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg.

Speaker: Alexander Shapiro, University of Edinburgh – Invited by Catherine Meusburger