# Project C: Non-commutative geometry and probability

In this part of our research program, geometry, groups and their actions
are studied with methods related to analysis. Actions are studied
using concepts from functional analysis, non-commutative geometry,
ergodic theory or probability. Groups are studied using their representations
on Hilbert spaces and by associating algebras of operators.
Differential and algebraic geometry is generalized to the noncommutative
setting using generalized "algebras of functions". Dynamical
systems and processes are studied from the point of view of
probabilistic or ergodic properties. Many of the projects in this research
area have close connections to problems and structures studied
in number theory or in topology and thus to the research areas A and
B.

One of the starting points for projects C1, C2, C3 is the fact that one
can associate algebras with geometric objects, groups, group actions
or groupoids. These can be algebras of functions or convolution algebras
or a mixture of both. They are usually C*-algebras but in some
cases also locally convex algebras and encode important properties of
the original objects. On the other hand, the study of the algebras often
allows to perform the investigation in a framework which includes
generalized versions of geometry, groups etc., like non-commutative
geometry or quantum groups.

Project C1 studies actions on spaces constructed algebraically, such
as adele spaces or group duals. The ergodic properties of such actions
and the algebras associated with them have intriguing structures. For
instance, the algebra constructed from an affine action on the space of
finite adeles over a global field can also be described using generators
labeled by the elements of the ring of integers in the field. An important
role in this project is also played by groupoids and duality theory.
There are close connections to number theory and to algebra and thus
to research area A.

In projects C2 and C3, generalized homology/cohomology theories
such as K-theory and its bivariant forms, but also cyclic homology and
dimension theory play a leading role. These projects have many points
in common with the topology projects in area B (K-theory, isomorphism
conjectures, dimension functions). Important objects of study
here are group algebras or crossed products. An important aspect is the
problem of determining the K-theory of such algebras (Baum-Connes
conjecture). This particular problem is parallel to part of the program
in B6.

Project C4 is related to C1, C2, C3 in that it studies quantum field theory
in the setting of non-commutative geometry. It uses concepts from
this theory such as spectral triples and studies quantum field theories
defined on a non-commutative version of space-time. These can often
be mapped to matrix models with particular properties. The main
example is a candidate for a non-perturbative renormalization of an
interacting quantum field theory in four dimensions. The project aims
at completing the constructive renormalization proof, which requires
hard analysis and combinatorics, but also number theory through iterated
integrals which evaluate to polylogarithms and zeta functions.
The use of combinatorics, of random matrices and of an iteratively
generated ring of functions establishes close links to the probability
projects C5 and C6. The spectral triple underlying the above quantum
field theory is based on supersymmetric quantum mechanics. This
connects to topological quantum field theory, which plays a role in area
B and can also be linked to nets of von Neumann algebras on the circle,
as well as to elliptic cohomology.

Markov chains are an excellent tool from probability theory to investigate
the geometric structure of state spaces. They are the common
theme of the projects in C5. In particular, we will focus on such situations
where the state space of the random walk is itself subject to a
random choice. Such situations arise naturally in the theory of random
walks in random environments, random walks on random trees, or as
well in the study of Markov chain Monte Carlo methods for random
measures. This project is, of course, intimately related with the other
probability project C6. However, via its structures it is connected to
projects focusing on the theory of buildings (part A), while its methods
are also used in and partially derived from differential geometry
(part B).

C6 is an interdisciplinary project between pure and applied mathematics.
The topic is to study certain aspects of sequences of random
functions with a focus on the spectrum of random matrices of increasing
dimension with stochastically dependent entries and the convergence
and stationary regime of iterated function systems (IFS). High
dimensional random matrices are connected to free probability and
hence to von Neumann algebras as well as to number theory, but also
get much attention from theoretical physics. We will try to attack the
problems from the ergodic point of view as well as from a probabilistic
perspective. Concerning IFS, the goal is to understand their asymptotic
behaviour under appropriate contraction conditions on the chosen
functions. The project is related to C5 by the use of probabilistic
methods, but also to C4, via a combinatorial approach to random matrix
theory and to C1 by its connections to von Neumann algebras and
ergodic theory.

## Subprojects

- C1: Dynamical systems from a noncommutative point of view
- C2: The fine structure of nuclear C*-algebras
- C3: (Semi-)group C*-algebras and their invariants
- C4: Mathematical aspects of quantum field theory
- C5: Random walks, branching, random media
- C6: Random matrices and iterated function systems

## Principal investigators

- Prof. Dr. Gerold Alsmeyer
- Prof. Dr. Dr. h.c. Joachim Cuntz
- Prof. Dr. Christopher Deninger
- Prof. Dr. Steffen Dereich
- Prof. Dr. Siegfried Echterhoff
- Prof. Dr. Nina Gantert
- Prof. Dr. Matthias Löwe
- Dr. Christian Voigt
- Prof. Dr. Wilhelm Winter
- Prof. Dr. Raimar Wulkenhaar

## Participating scientists

- Selcuk Barlak
- Prof. Dr. Arthur Bartels
- Johannes Blank
- Christian Boenicke
- Dennis Bohle
- Fabian Buckmann
- Dr. Martijn Caspers
- Sayan Chakraborty
- Dr. Lucio Cirio
- Jins de Jong
- Dr. habil. Steven Duplij
- Dr. Dominic Enders
- Martin Engbers
- Hendrik Flasche
- Dr. Eusebio Gardella
- Dr. Elisabeth Gillaspy
- Alexander Hock
- Prof. Dr. Zakhar Kabluchko
- Andrea Kasprowski
- Dr. Soren Knudby
- Michael Kochler
- Sebastian Krusekamp
- Dr. Xin Li
- Kang Li
- Dr. Snigdhayan Mahanta
- Dr. Matthias Meiners
- Sebastian Mentemeier
- Hamed Nikpey
- Dr. Walther Paravicini
- Dr. Dmitri Pavlov
- Dr. Ulrich Pennig
- Dr. Henning Petzka
- Oliver Pfante
- Dr. Stephan Rave
- Dr. Jan Schlemmer
- Matti Schneider
- Dr. Ansgar Schneider
- Timo Siebenand
- Dr. Jan Spakula
- Nicolai Stammeier
- Dr. Michael Stolz
- Dr. Karen Strung
- Gabor Szabo
- Dr. Hannes Thiel
- Dr. Aaron Tikuisis
- Dr. Thomas Timmermann
- Dr. Felipe Alexis Torres Tapia
- Dr. Stefan Wagner
- Dr. Moritz Weber
- apl. Prof. Dr. Wend Werner

## Publications and preprints of the SFB

- Deninger, Cuntz, Marcelo Laca: C*-algebras of Toeplitz type associated with algebraic number fields
- Li, : Semigroup C*-Algebras and Amenability of Semigroups
- Li, : K-Theory for Ring C*-Algebras attached to Function Fields with only one Infinite Place
- Tikuisis, : The Cuntz semigroup of continuous functions into certain simple C*-algebras
- Tikuisis, Greg Maloney: A classification of finite rank dimension groups by their representations in ordered real vector spaces
- Tikuisis, Leonel Robert: Hilbert C*-modules over a commutative C*-algebra
- Tikuisis, Bruce Blackadar, Leonel Robert, Andrew S. Toms, Wilhelm Winter: An algebraic approach to the radius of comparison
- Tikuisis, : Regularity for stably projectionless, simple C*-algebras
- Cuntz, : Quillen's work on the foundations of cyclic cohomology
- Echterhoff, Marcelo Laca: The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers
- Cuntz, : C*-Algebras associated with endomorphisms and polymorphisms of compact abelian groups
- Echterhoff, Buss, Alcides: Imprimitivity theorems for weakly proper actions of locally compact groups
- Winter, : Classifying crossed product C*-algebras
- Winter, Barlak, Szabo, Dominic Enders, Hiroki Matui: The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras
- Echterhoff, Alcides Buss: Rieffel proper actions
- Echterhoff, Buss, Alcides: Maximality of dual coactions on sectional C*-algebras of Fell bundles and applications
- Timmermann, : On duality of algebraic quantum groupoids
- Timmermann, Enock, Michel: Measured quantum transformation groupoids
- Timmermann, : Integration on algebraic quantum groupoids
- de Laat, Mikael de la Salle: Banach space actions and L2-spectral gap
- de Laat, Yuki Arano, Jonas Wahl: The Fourier algebra of a rigid C*-tensor category
- Paravicini, : The Spectral Radius in Co(X)-Banach Algebras
- Tikuisis, : Finite dimensional ordered vector spaces with Riesz interpolation and Effros-Shen's unimodularity conjecture
- Jacelon, : A Simple, Self-Absorbing, Stable Projectionless C*-Algebra
- Winter, F. Perera, A.S. Toms, S. White: The Cuntz semigroup and stability of close C*-algebras
- Tikuisis, Winter, : Decomposition rank of Z-stable C*-algebras
- Winter, A.S. Toms, S. White: Z-stability and finite dimensional tracial boundaries
- Jacelon, Winter, : Z is universal
- Winter, H. Thiel: The generator problem for Z-stable C*-algebras
- Winter, Yasuhiko Sato, Stuart White: Nuclear dimension and Z-stability
- Winter, Karen Strung: UHF slicing and classification of nuclear C*-algebras
- Thiel, Gardella, : Group algebras acting on Lp-spaces
- Echterhoff, : Crossed Products, the Mackey-Rieffel-Green Machine and Applications
- Echterhoff, Emerson, Heath: STRUCTURE AND K-THEORY OF CROSSED PRODUCTS BY PROPER ACTIONS
- Echterhoff, : A general Kirillov Theory for locally compact nilpotent groups
- Spakula, Goulnara Arzhantseva and Erik Guentner: Coarse non-amenability and coarse embeddings
- Spakula, Rufus Willett: Maximal and reduced Roe algebras of coarsely embeddable spaces
- Spakula, Rufus Willett: On Rigidity of Roe algebras
- Paravicini, : A Generalised Green-Julg Theorem for Proper Groupoids and Banach Algebras
- Paravicini, : The Bost Conjecture and Proper Banach Algebras
- Echterhoff, Alcides Buss: Weakly proper group actions, Mansfield's imprimitivity and twisted Landstad duality
- Echterhoff, Schneider, : Non-commutative T-duality
- Echterhoff, Alcides Buss and Rufus Willett: Exotic crossed products and the Baum-Connes conjecture
- Echterhoff, Buss, Alcides and Willett, Rufus: Exotic Crossed Products
- Echterhoff, : Bivariant KK-Theory and the Baum-Connes conjecure
- Echterhoff, Li, Kang and Nest, Ryszard: The orbit method for the Baum-Connes Conjecture for algebraic groups over local function fields
- : Witt Vector Rings and the Relative de Rham Witt Complex
- de Laat, Federico Vigolo: Superexpanders from group actions on compact manifolds
- Werner, Bohle, : A K-theoretic approach to the classification of symmetric spaces
- Werner, Bohle, : The universal enveloping TRO of a JB*-triple system
- Werner, Hendrik Schlieter: Unbounded multipliers on operator spaces
- Wulkenhaar, Harald Grosse: 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory
- Wulkenhaar, Gayral, Victor: Spectral geometry of the Moyal plane with harmonic propagation
- Wulkenhaar, Grosse, Harald: Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
- Wulkenhaar, Grosse, Harald: Construction of the \Phi^4_4-quantum field theory on noncommutative Moyal space
- Wulkenhaar, Grosse, Harald: Solvable limits of a 4D noncommutative QFT
- Bartels, Christopher Douglas, Andre Henriques: Dualizability and index of subfactors
- Bartels, Christopher Douglas, Andre Henriques: Conformal nets I: coordinate-free nets
- Bartels, Christopher Douglas, Andre Henriques: Conformal nets III: fusion of defects
- Pavlov, Scholbach, : Symmetric operads in abstract symmetric spectra
- Wulkenhaar, de Jong, : The asymptotic volume of diagonal subpolytopes of symmetric stochastic matrices
- Alsmeyer, Meiners, : Fixed points of the smoothing transforms: Two-sided solutions
- Alsmeyer, Meiners, : Fixed Points of Inhomogeneous Smoothing Transforms
- Alsmeyer, Meiners, J.D. Biggins: The Functional Equation of the Smoothing Transform
- Löwe, Torres Tapia, : On hitting times for simple random walk on dense Erdos-Renyi graphs
- Löwe, Torres Tapia, : A note on hitting times for simple random walk on rooted, subcritical Galton-Watson trees
- Löwe, Meiners, : Moderate deviations for random field Curie-Weiss models
- Meiners, Anselm Reichenbach: On the accuracy of the normal approximation for the free energy in the REM
- Gantert, Thomas Kochler: Cutoff and mixing time for transient random walks in random environments
- Meiners, Iksanov, Alexander Marynych, Alexander: Moment convergence in renewal theory
- Alsmeyer, Soeren Groettrup: A host-parasite model for a two-type cell population
- Alsmeyer, Meiners, Iksanov, Alexander: Power and exponential moments of the number of visits and related quantities for perturbed random walks
- Gantert, Mathieu, P. Piatnitski, A.: Einstein relation for reversible diffusions in random environment
- Alsmeyer, : The smoothing transform: a review of contraction results
- Alsmeyer, Groettrup, Soeren: Branching within branching I: The extinction problem
- Alsmeyer, Groettrup, Soeren: Branching within branching II: Limit theorems
- Alsmeyer, Buckmann, : Fluctuation theory for Markov random walks
- Alsmeyer, Dyszewski, Piotr: Thin tails of fixed points of the nonhomogeneous smoothing transform
- Alsmeyer, Iksanov, Alexander, Marynych, Alexander: Functional limit theorems for the number of occupied boxes in the Bernoulli sieve
- Alsmeyer, : Ladder epochs and ladder chain of a Markov random walk with discrete driving chain
- Alsmeyer, Buckmann, : An arcsine law for Markov random walks
- Alsmeyer, Kabluchko, Alexander Marynych: A leader-election procedure using records
- Alsmeyer, Kabluchko, Alexander Marynych: Leader election using random walks
- Kabluchko, Alexander Iksanov: Functional limit theorems for Galton-Watson processes with very active immigration
- Kabluchko, Vladislav Vysotsky, Dmitry Zaporozhets: Convex hulls of random walks: Expected number of faces and face probabilities
- Kabluchko, Vladislav Vysotsky, Dmitry Zaporozhets: A multidimensional analogue of the arcsine law for the number of positive terms in a random walk
- Kabluchko, Alexander Marynych, Henning Sulzbach: General Edgeworth expansions with applications to profiles of random trees
- Kabluchko, Alexander Marynych, Henning Sulzbach: Mode and Edgeworth expansion for the Ewens distribution and the Stirling numbers
- Kabluchko, Vladislav Vysotsky, Dmitry Zaporozhets: Convex hulls of random walks, hyperplane arrangements, and Weyl chambers
- Kabluchko, Rudolf Grübel: Asymptotic expansions for profiles of branching random walks
- Kabluchko, Alexander Iksanov: A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk
- Kabluchko, : Weak convergence of renewal shot noise processes in the case of slowly varying normalization
- Kabluchko, Alexander Iksanov, Alexander Marynych, Georgiy Shevchenko: Fractionally integrated inverse stable subordinators
- Kabluchko, Alexander Marynych: Renewal shot noise processes in the case of slowly varying tails
- Dereich, Cecile Mailler, Peter Mörters: Non-extensive condensation in reinforced branching processes
- Dereich, Leif Döring, Andreas E. Kyprianou: Real self-similar processes started from the origin
- Dereich, : Preferential attachment with fitness: unfolding the condensate
- Dereich, Christian Mönch, Peter Mörters: Distances in scale free networks at criticality
- Dereich, Leif Döring: Random interlacements via Kuznetsov measures
- Alsmeyer, Mentemeier, : Tail behavior of stationary solutions of random difference equations: the case of regular matrices
- Mentemeier, Ewa Damek, Mariusz Mirek, Jacek Zienkiewicz : Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices
- Löwe, Friesen, : The Semicircle Law for Matrices with Independent Diagonals
- Mentemeier, Dariusz Buraczewski, Ewa Damek, Mariusz Mirek : Heavy tailed solutions of multivariate smoothing transforms
- Alsmeyer, Mentemeier, Ewa Damek: Precise tail index of fixed points of the two-sided smoothing transform
- Löwe, Friesen, Stolz, : Gaussian Fluctuations for Sample Covariance Matrices with Dependent Data
- Löwe, Friesen, : A phase transition for the limiting spectral density of random matrices
- Thomas Kriecherbauer, Kristina Schubert: Spacings - an example for Universality in Random Matrix Theory
- Thomas Kriecherbauer, Kristina Schubert, Katharina Schuler, Martin Venker: Global Asymptotics for the Christoffel-Darboux Kernel of Random Matrix Theory
- Torres Tapia, Jiri Lember, Heinrich Matzinger: General approach to the fluctuations problem in random sequence comparison
- Löwe, Friesen, : On the Spectral Density of Large Sample Covariance Matrices with Markov Dependent Columns
- Löwe, Franck Vermet: Capacity of an associative memory model on random graph architectures
- Löwe, Franck Vermet: The Hopfield model on a spare Erdos-Renyi graph
- Löwe, Peter Eichelsbacher: DIE ENTWICKLUNG DER LINDEBERG-METHODE DER VERGANGENEN 90 JAHRE
- Alsmeyer, Winkler, Andrea: Metabasins - a state space aggregation for highly disordered energy landscapes
- Stolz, Christian Doebler: A quantitative central limit theorem for linear statistics of random matrix eigenvalues
- Stolz, Christian Doebler: Stein's method and the multivariate CLT for traces of powers on the classical compact groups
- Stolz, Christian Doebler, Peter Eichelsbacher: Large deviations for disordered bosons and multiple orthogonal polynomial ensembles
- Stolz, Andreas Lodwig, Hermann Schulz-Baldes: Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions
- Kristina Schubert: SPACINGS IN ORTHOGONAL AND SYMPLECTIC RANDOM MATRIX ENSEMBLES
- Stolz, : STEIN'S METHOD AND CENTRAL LIMIT THEOREMS FOR HAAR DISTRIBUTED ORTHOGONAL MATRICES: SOME RECENT DEVELOPMENTS
- Alsmeyer, : Quasi-stochastic matrices and Markov renewal theory
- Alsmeyer, : On the stationary tail index of iterated random Lipschitz functions
- Alsmeyer, Marynych, Alexander: Renewal approximation for the absorption time of a decreasing Markov chain
- Alsmeyer, Buckmann, : Stability of perpetuities in Markovian environment
- Alsmeyer, Buraczewski, Dariusz, Iksanov, Alexander: Null-recurrence and transience of random difference equations in the contractive case
- Kabluchko, Alexander Iksanov, Alexander Marynych: Local universality for real roots of random trigonometric polynomials
- Flasche, : Expected number of real roots of random trigonometric polynomials