Past Functional Analysis Seminar

When we look at a graph, one of the first questions that might naturally come to mind is "Does this graph have any symmetry?" In fancier language, is the automorphism group trivial? The quantum automorphism group is a compact matrix quantum group which is meant to capture the quantum symmetries of the graph. Although we've ramped up the jargon, we still have the same question: Given a graph, does it have any quantum symmetries? In this talk, I'll explain why this question is undecidable, thanks to an Adian-Rabin theorem for quantum automorphism groups. Consequently, there are graphs where the quantum automorphism group is non-trivial but has no non-trivial finite-dimensional representations (or in other words, the discrete quantum automorphism group is trivial).

Joint work with Josse van Dobben de Bruyn. 
 

Any countable group G comes equipped with a canonical dynamical system, namely its conjugation action on its space of subgroups Sub(G). This 0-dimensional compact space is a central object in measured and geometric group theory, especially because it supports invariant and stationary random subgroups.

In general, describing this space is hard. In 2024, Carderi, Gaboriau, Le Maître, and Stalder initiated the study of the space of subgroups of non-amenable Baumslag-Solitar groups BS(m,n). They provided an explicit description of the perfect kernel of Sub(BS(m,n)). This is the largest closed subspace without isolated points, i.e. the space that remains after performing successive derivations that remove the isolated points.

In this talk, I will provide a complete classification of the spaces Sub(BS(m,n)) up to homeomorphism. I will prove that there exist exactly four homeomorphism types among the non-amenable ones. This relies on a detailed study of the Cantor-Bendixson erasing process, which depends on the arithmetical properties of the parameters m,n. This is based on a joint work with Damien Gaboriau, François Le Maître, and Yves Stalder.

In 1989, Benjamin Weiss proved that for a countable group G, there exists a global minimal G-flow which is a topological model for every essentially free p.m.p. Borel G-action. In this talk, I will present our vast generalization, asserting that any essentially free Borel measured action (without the p.m.p. assumption), admits a minimal compact model. Moreover, we show that given a random walk on the group, there exists a global minimal topological model for all essentially free stationary Borel actions.

Based on joint work with Yair Hartman.

Let $(M,\varphi)=(M_1,\varphi_1)*(M_2,\varphi_2)$ be a free product of non-trivial von Neumann algebras equipped with arbitrary faithful normal states. In 2011, Ueda showed that the diffuse summand of $M$ is a factor if and only if the maximum vector space dimension of $M_1$ and $M_2$ is at least 3, and in this case one further has that the diffuse summand has no asymptotically central sequences. Now consider the von Neumann algebra $N$ generated by non-trivial subalgebras $N_i\leq M_i$ with $\varphi_i$-preserving expectations for $i=1,2$, and let $z \in N$ be the central projection supporting the diffuse summand of $N$. Ueda's result characterizes when $Nz$ is a (full) factor, and, in fact, several cases treated in his proof yield the stronger property that $zMz$ lacks asymptotically $(Nz)$-central sequences. Thus, is natural to wonder: is this stronger property equivalent to the maximum vector space dimension of $N_1$ and $N_2$ being at least 3? In this talk, I will make use of amalgamated free products to answer this question in the affirmative. These same techniques can also be used to analyze the centralizer $M^\varphi = \{ x \in M \colon \varphi(xy) = \varphi(yx)\ \forall y\in M\}$ of the free product state, generalizing a separate 2011 result of Ueda from almost periodic states to arbitrary states. This is based on joint work with Aldo Garcia Guinto, Fehmi Ekin Giritlioglu, Yoonkyeong Lee, and Rahul K. Ramachandran.

(Joint work with Brent Nelson) In this talk we investigate the anticoarse space of the von Neumann algebras generated by the kernel and the domain of a closable derivation. We show that when a tuple (x_i)_{i\in I} admits a conjugate system, then for any proper subset J \subset I  with |J| \geq 2 the inclusion W*(x_j :j \in j) \subset W*(x_i: i \in I) is irreducible, infinite index and non-regular.

In this talk, I will start by presenting what KK-rigidity means and how I together with Chris Schafhauser have modified it to the dynamical setting for strongly outer actions on classifiable C*-algebras. This includes an application to the uniqueness of the Bernoulli shift on the CAR-algebra and Jiang-Su algebra for torsion-free amenable countable discrete groups. Then I will move towards the machinery leading up to the rigidity theorem, which includes some exposition on the cocycle category and classification of equivariant KK-lifts

This talk by Peter Huston gives an overview of the motivation for and classification of fusion 1-categories and 2-categories. In particular, we will review how fusion 1-categories naturally arise in operator algebras from the subfactor classification programme, which furnishes exotic examples of fusion category, such as the Haagerup subfactor, which are inaccessible by other approaches. Fusion 2-categories are a categorification of fusion 1-category, arising naturally from the study of TQFT in 4D, or as quantum symmetries of fusion 1-categories. We will outline the classification of fusion 2-categories. In particular, we will see that, while fusion 1-categories are wild in the sense that they cannot be constructed from lower dimensional data like finite groups, fusion 2-categories are comparatively tame, expressible in terms of braided fusion 1-categories and extension theory.

The boundedness of Fourier multipliers on non-commutative $L_p$-spaces ($1 < p < \infty$) is a fundamental problem in non-commutative analysis. Building on the non-commutative Cotlar identity introduced by Mei and Ricard (2017), which yields $L_p$-boundedness ($1 < p < \infty$) of Hilbert transforms on amalgamated free products of finite von Neumann algebras, their approach relies heavily on freeness in the underlying free product structure.

In this talk, Xiaoqi Lu introduces a new strategy that overcomes this limitation. Our approach combines a generalized Cotlar identity, which holds on suitable subspaces and captures non-freeness information, with an additional condition related to the property of Rapid Decay to control the remaining components. From this framework, we establish the $L_p$-boundedness ($1 < p < \infty$) of Rademacher-type Hilbert transforms on graph products of finite von Neumann algebras. This unified framework extends earlier results for free products of finite von Neumann algebras and for graph products of groups acting on right-angled buildings. This is a joint work with Runlian Xia.

In the context of Fourier analysis on the real line, a \textit{one-sided problem} involves deducing properties of a function $f$ from some information about the restriction of its Fourier transform $\widehat{f}$ to a half-line, for instance to $\mathbb{R}_- := (-\infty, 0)$. A prototypical result, which is foundational to the theory of Hardy spaces on $\mathbb{R}$, asserts that if $f \in L^2(\mathbb{R})$ is non-zero and $\widehat{f}$ vanishes on a half-line, then $f$ satisfies the \textit{Szeg\H{o} condition} $\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty$. 

Various problems in operator theory involve the study of functions $f$ satisfying a weaker condition of decay of $\widehat{f}$ on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local Szeg\H{o}-type conditions hold: if the decay of $\widehat{f}$ is strong enough on a half-line, then the mass of the function $f \in L^2(\mathbb{R})$ must concentrate enough for the integral $\int_E \log |f(x)| dx$ to converge on a "massive" set $E$. 

In his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.

The Poisson boundary of a probability measure on a countable group is a probability space endowed with a stationary group action that captures the asymptotic behaviour of the associated random walk. Since its introduction by Furstenberg in the 1960s, the study of Poisson boundaries and stationary actions has become a powerful tool for understanding geometric and algebraic properties of groups.

In this talk, I will discuss connections between stabilizers of stationary actions, in particular, those arising from the Poisson boundary, and the C*-simplicity of the associated reduced group C*-algebra. I will also address the (seemingly unrelated) problem of realizing different Poisson boundaries on a common underlying topological model. The talk is based on joint work with Anna Cascioli and Martín Gilabert Vio, and with Josh Frisch.

It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.

The notion of invariant random subgroups (IRS) is a fruitful, well-studied concept in dynamics on groups. In this talk, Hanna Oppelmayer will explain what it is and how to extend this notion to group von Neumann algebras LG, where G is a discrete countable group. We call it invariant random sub-von Neumann algebra (IRA). As an application, Hanna will provide a result concerning amenable IRAs, which generalises (in the discrete setup) a theorem of Bader-Duchesne-Lécureux about amenable IRSs. This is joint work with Tattwamasi Amrutam and Yair Hartman.

The problem of stability of approximate homomorphisms was first posed by S. Ulam in the context of groups equipped with a metric. If $G$ and $H$ are groups and $H$ is equipped with a metric $d$, then $\varphi\colon G\to H$ is an $\varepsilon$-homomorphism if $d(\varphi(xy), \varphi(x)\varphi(y))\leq \varepsilon$ for all $x,y\in G$. Ulam’s well-studied problem asks how closely such a map can be approximated by a true homomorphism.
Analogous questions have been investigated in many algebraic and analytic settings. For C*-algebras, the notion of an $\varepsilon$-*-homomorphism admits several possible formalizations. The variant I will discuss, while perhaps not the most immediate, turns out to be particularly interesting, because its associated Ulam stability problem is closely related to rigidity for corona C*-algebras. Namely, Ulam stability of $\varepsilon$-*-isomorphisms between C*-algebras in a certain class (e.g., AF algebras) is equivalent to the rigidity question for coronas of direct sums of C*-algebras in this class.

 

We will introduce a class of countable homeomorphism groups that share many properties with Thompson's group V, known as FSS* groups. This talk from Patrick Henry DeBonis will focus on some of the group constructions and deformation/rigidity arguments needed to prove FSS* group von Neumann algebras are prime - and have potential for wider applications.

We introduce equivariant bivariant K-theory for bornological algebras by taking a presentable refinement of the bivariant K-theory of Lafforgue and Paravicini. An upshot of this refinement is that we may purely formally define a Bost-Connes assembly map via localisation in the sense of Meyer-Nest. Another feature built into the refinement is a large UCT-class; on this UCT-class, we show that the rationalised Chern-Connes character from KK-theory to local cyclic homology is an equivalence. This is joint work with Anupam Datta.

Graph products of groups were introduced by Green as a construction that encompasses both direct products and free products. Likewise, the notion of graph product of von Neumann algebras, introduced by Caspers and Fima, recovers both tensor products and free products. Camille Horbez will present rigidity theorems for graph products of tracial von Neumann algebras, and discuss the computation of their symmetries, drawing parallels with the case of groups. This is a joint work with Adrian Ioana. 

AF-embeddability, i.e., the question whether a given C*-algebra can be realised as a subalgebra of an AF-algebra, has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1980. Since then, many AF-embeddings have been constructed for concrete examples but also many non-constructive AF-embeddability results have been obtained for classes of algebras typically assuming the UCT. 

In this talk by Joachim Zacharias, we will consider a separable unital C*-algebra A of decomposition rank at most 1 and construct from a suitable system of 1-decomposable cpc-approximations an AF-algebra E together with an embedding of A into E and a conditional expectation of E onto A without assuming the UCT. We also consider some extensions of this inclusion and indicate some applications.

Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, Kasia Wyczesany will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. Whilst Mahler’s influential 1939 conjecture regarding the minimiser of the volume product will be mentioned, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions. Main new results are joint work with S. Artstein-Avidan and S. Sadovsky, and S. Artstein-Avidan and M. Fradelizi. 

In this talk from Alexander Ravnanger, he provides a dynamical description of the largest AF-ideal in certain crossed products by the integers. In the case of the uniform Roe algebra of the integers, this reveals an interesting connection to a well-studied object in topological semigroup theory. On the way, he gives an overview of what is known about the abundance of projections in such crossed products, the structure of the simple quotients, and concepts of low-dimensionality for uniform Roe algebras.

In 1962, Horn raised the following problem: Let A and B be n-by-n Hermitian matrices with respective eigenvalues a_1,...,a_n and b_1,...,b_n. What can we say about the possible eigenvalues c_1,...,c_n of A + B?

The deterministic perspective is that the set of possible values for c_1,...,c_n are described by a collection of inequalities known as the Horn inequalities.

Free probability offers the following alternative perspective on the problem: if (A_n) and (B_n) are independent sequences of n-by-n random matrices with empirical spectra converging to probability measures mu and nu respectively, then the random empirical spectrum of A_n + B_n converges to the free convolution of mu and nu.

But how are these two perspectives related?

In this talk Samuel Johnston will discuss approaches to free probability that bridge between the two perspectives. More broadly, Samuel will discuss how the fundamental operations of free probability (such as free convolution and free compression) arise out of statistical physics mechanics of corresponding finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials, Horn inequalities, permutations etc).

This talk is based on joint work with Octavio Arizmendi (CIMAT, Mexico), Colin McSWiggen (Academia Sinica, Taiwan) and Joscha Prochno (Passau, Germany).

This talk by Tim Austin, at the University of Warwick, will be an introduction to "almost periodic entropy".  This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra.  It is an analog of Lewis Bowen's "sofic entropy" from ergodic theory.  This analogy extends to many of its properties, but some important differences also emerge.  Tim will not assume any prior knowledge about sofic entropy.

After setting up the basic definition, Tim will focus on the special case of finitely generated free groups, about which the most is known.  For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations.  This, in turn, offers a new proof of the Collins—Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.

In this talk, Konstantin Recke, University of Oxford,  will report on some results pertaining to the interplay between geometric group theory, operator algebras and probability theory. Konstantin will introduce so-called invariant percolation models from probability theory and discuss their relation to geometric and analytic properties of groups such as amenability, the Haagerup property (a-T-menability), $L^p$-compression and Kazhdan's property (T). Based on joint work with Chiranjib Mukherjee (Münster).

The simplex of traces of a unital C*-algebra has long been regarded as a central invariant in the theory. Likewise, from the group-theoretic perspective, the simplex of traces of a discrete group (namely, the simplex of traces of its maximal C*-algebra) is a fundamental object in harmonic analysis, and the study of this simplex led to many applications in recent years.

Itamar Vigdorovich , UCSD, will discuss several results describing the simplex of traces in concrete and significant cases. These include Property (T) groups and especially higher rank lattices, for which the simplex of traces is as tame as possible. In contrast, for free products, the simplex is typically as wild as possible, yet still admits a canonical and universal structure—the Poulsen simplex. In ongoing work, an analogous result is obtained for the space of traces on the fundamental group of a closed surface of genus g≥2.

Itamar presents these results, outlines the main ideas behind the proofs, and gives an overview of the central concepts. The talk is based on joint works with Gao, Ioana, Levit, Orovitz, Slutsky, and Spaas.

With the classification theory of simple and nuclear C*-algebras of real rank zero advanced to a level which may very well be final, it is natural to wonder what happens when one allows ideals, but not too many of them. Contrasting the simple case, the K-theoretical classification theory for real rank zero C*-algebras with finitely many ideals is only satisfactorily developed in subcases, and in many settings it is even unclear and/or disputed which flavor of K-theory to use.

Restricting throughout to the setting of real rank zero, Søren Eilers will compare what is known of the classification of graph C*-algebras and of approximately subhomogeneous C*-algebras, with an emphasis on what kind of conclusion can be extracted from restrictions on the complexity of the ideal lattice. The results presented are either more than a decade old or joint with An, Liu and Gong.

An epsilon-representation of a discrete group G is a map from G to the unitary group U(n) that is epsilon-multiplicative in norm uniformly across the group. In the 1980s, Kazhdan showed that surface groups of genus at least 2 are not uniform-to-local stable in the sense that they admit epsilon-representations that cannot be perturbed, even locally (on the generators), to genuine representations.
 

In this talk, Marius Dadarlat of Purdue University will discuss the role of bounded 2-cohomology in Kazhdan's construction and explain why many rank-one lattices in semisimple Lie groups are not uniform-to-local stable, using certain K-theory properties reminiscent of bounded cohomology.

A group is said to be matricially stable if every function from the group to unitary matrices that is "almost multiplicative" in the point-operator norm topology is "close," in the same topology, to a genuine representation. A result of Dadarlat shows that even cohomology obstructs matricial stability. The obstruction in his proof can be realized as follows. To each almost-representation,  a vector bundle may be associated. This vector bundle has topological invariants, called Chern characters, which lie in the even cohomology of the group. If any of these invariants are nonzero, the almost-representation is far from a genuine representation. The first Chern character can be computed with the "winding number argument" of Kazhdan, Exel, and Loring, but the other invariants are harder to compute explicitly. In this talk, Professor Forrest Glebe will discuss results that allow the computation of higher invariants in specific cases: when the failure to be multiplicative is scalar (joint work with Marius Dadarlat) and when the failure to be multiplicative is small in a Schatten p-norm.

We study the fixed points of the Berezin transform on the Fock-type spaces F^{2}_{m} with the weight e^{-|z|^{m}}, m > 0. It is known that the Berezin transform is well-defined on the polynomials in z and \bar{z}. In this talk from Ghazaleh Asghari from Reading University, we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many m \in (0,\infty). We also show that, in some particular cases, the fixed point polynomials are harmonic for all m.

In this talk, Aaron Kettner, Institute of Mathematics, Czech Academy of Sciences, will show how to construct a C*-correspondence from a vector bundle together with a (partial) homeomorphism on the bundle's base space. The associated Cuntz-Pimsner algebras provide a class of examples that is both tractable and potentially quite large. Under reasonable assumptions, these algebras are classifiable in the sense of the Elliott program. If time permits, Aaron will sketch some K-theory calculations, which are work in progress.

Given a von Neumann algebra M, we can consider the set of values of p such that Lp(M) has the approximation property: the identity on it is a limit of finite rank operators for a suitable topology. Apart from the case when p is infinite, which has been the subject of a lot of work initiated by Haagerup in the late 70s, this invariant has not been very much exploited so far. But ancient works in collaboration with Vincent Lafforgue and Tim de Laat suggest that, maybe, it can distinguish the factors of SL(n,Z) for different values of n. I will explain something that I realized only recently, and that explains why this is a difficult question: it implies some form of the classical Kakeya conjecture, which predicts the shape of sets in the Euclidean space in which a needle can be turned upside down. This talk from Mikael de la Salle will be an opportunity to discuss other connections between classical Fourier analysis and analysis in group von Neumann algebras, including in collaboration with Javier Parcet and Eduardo Tablate

Groupoids provide a rich supply of C*-algebras, and there are many results describing the structure of these C*-algebras using properties of the underlying groupoid. For non-Hausdorff groupoids, less is known, largely due to the existence of 'singular' functions in the reduced C*-algebra. This talk will discuss two approaches to studying ideals in non-Hausdorff groupoid C*-algebras. The first uses Timmermann's Hausdorff cover to reduce certain problems to the setting of Hausdorff groupoids. The second will restrict to isotropy groups. For amenable second-countable étale groupoids, these techniques allow us to characterise when the ideal of singular functions has dense intersection with the underlying groupoid *-algebra. This is based on joint work with K. A. Brix, J. B. Hume, and X. Li, as well as upcoming work with J. B. Hume.

The Cuntz semigroup of a C*-algebra A consists of equivalence classes of positive elements, where equivalence means roughly that two positive elements have the same rank relative to A.  It can be thought of as a generalization of the Murray von Neumann semigroup to positive elements and is an incredibly sensitive invariant. We present a calculation of the homotopy groups of these Cuntz classes as topological subspaces of A when A is classifiable in the sense of Elliott.  Remarkably, outside the case of compact classes, these spaces turn out to be contractible.  

I will describe recent developments in information geometry (the study of optimal transport and entropy) for the setting of free probability.  One of the main goals of free probability is to model the large-n behavior of several $n \times n$ matrices $(X_1^{(n)},\dots,X_m^{(n)})$ chosen according to a sufficiently nice joint distribution that has a similar formula for each n (for instance, a density of the form constant times $e^{-n^2 \tr_n(p(x))}$ where $p$ is a non-commutative polynomial).  The limiting object is a tuple $(X_1,\dots,X_m)$ of operators from a von Neumann algebra.  We want the entropy and the optimal transportation distance of the probability distributions on $n \times n$ matrix tuples converge in some sense to their free probabilistic analogs, and so to obtain a theory of Wasserstein information geometry for the free setting.  I will present both negative results showing unavoidable difficulties in the free setting, and positive results showing that nonetheless several crucial aspects of information geometry do adapt.

Directed graph algebras have long been studied as tractable examples of C*-algebras, but they are limited by their inability to have torsion in their K_1 group. Graphs of groups, which are famed in geometric group theory because of their intimate connection with group actions on trees, are a more recent addition to the C*-algebra scene. In this talk, I will introduce the child of these two concepts – directed graphs of groups – and describe how their algebras inherit the best properties of its parents’, with a view to outlining how we can use these algebras to model a class of C*-algebras (stable UCT Kirchberg algebras) which is classified completely by K-theory.

Roe algebras were introduced in the late 1990's in the study of indices of elliptic operators on (locally compact) Riemannian manifolds. Roe was particularly interested in coarse equivalences of metric spaces, which is a weaker notion than that of quasi-isometry. In fact, soon thereafter it was realized that the isomorphism class of these class of C*-algebras did not depend on the coarse equivalence class of the manifold. The converse, that is, whether this class is a complete invariant, became known as the 'Rigidity Problem for Roe algebras'. In this talk we will discuss an affirmative answer to this question, and how to approach its proof. This is based on joint work with Federico Vigolo.

A landmark result in the study of locally compact, abelian groups is the Pontryagin duality. In simple terms, it says that for a given locally compact, abelian group G, one can uniquely associate another locally compact, abelian group called the Pontryagin dual of G. In the realm of C*-algebras, whenever such an abelian group G acts on a C*-algebra A, there is a canonical action of the dual group of G on the crossed product of A by G. In particular, it is natural to ask to what extent one can relate properties of the given G-action to those of the dual action. 

In this talk, I will first introduce a property for actions of locally compact abelian groups called the abelian Rokhlin property and then state a duality type result for this property. While the abelian Rokhlin property is in general weaker than the known Rokhlin property, these two properties coincide in the case of the acting group being the real numbers. Using the duality result mentioned above, I will give new examples of continuous actions of the real numbers which satisfy the Rokhlin property. Part of this talk is based on joint work with Johannes Christensen and Gábor Szabó.

A G-kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C*-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, G-kernels on the hyperfinite II_1-factor have been completely understood via deep results of Connes, Jones and Ocneanu. 

In the talk I will explain how G-kernels on C*-algebras and the lifting obstructions can be interpreted in terms cohomology with coefficients in crossed modules. G-kernels, group actions and cocycle actions then give rise to induced maps on classifying spaces. For strongly self-absorbing C*-algebras these classifying spaces turn out to be infinite loop spaces creating a bridge to stable homotopy theory.

The talk is based on joint work with S. Giron Pacheco and M. Izumi, and with my PhD student V. Bianchi.
 

Let G be an infinite discrete group; e.g. free group, surface groups, or hyperbolic 3-manifold group.

Finite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps. 

The talk is a broadly accessible discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas, R. van Handel.

Quantized universal enveloping algebras admit an intriguing class of (unbounded) Hilbert space representations obtained via their cluster structure. In these so-called positive representations the standard generators act by (essentially self-adjoint) positive operators. 

The aim of this talk is to discuss some analytical questions arising in this context, and in particular to what extent these representations can be understood using the theory of locally compact quantum groups in the sense of Kustermans and Vaes. I will focus on the simplest case in rank 1, where many of the key features (and difficulties) are already visible. (Based on work in progress with Kenny De Commer, Gus Schrader and Alexander Shapiro). 

Building on the concept of diagonal dimension introduced by Li, Liao, and Winter in 2023, we propose a topological dimension for an inclusion pair of C*-algebras. This new framework allows for finite values in cases of Cartan inclusions that are not diagonal. In this talk, we present calculations for both upper and lower bounds concerning the inclusion of the unitization of c_0(\mathbb{N}) into the Toeplitz algebra. This work is a collaboration with W. Winter.

The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.

 The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.

Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension for an étale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra.  To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite.  I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph,  if the C*-algebra is stably finite, then its nuclear dimension is at most 1.  This is joint work with Dana Williams. 

Rota’s Alternierende Verfahren theorem in classical probability theory, which examines the convergence of iterates of measure preserving Markov operators, relies on a dilation technique. In the noncommutative setting of von Neumann algebras, this idea leads to the notion of absolute dilation.  

In this talk, we explore when a Fourier multiplier on a group von Neumann algebra is absolutely dilatable. We discuss conditions that guarantee absolute dilatability and present an explicit counterexample—a Fourier multiplier that does not satisfy this property. This talk is based on a joint work with Christian Le Merdy.

We propose a natural version of Connes' Rigidity Conjecture (1982) that involves property (T) groups with infinite centre. Using methods at the rich intersection between von Neumann algebras and geometric group theory, we identify several instances where this conjecture holds. This is joint work with Ionut Chifan, Denis Osin, and Hui Tan.