Schedule for: 23w5033 - Joint Spectra and related Topics in Complex Dynamics and Representation Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Random walks on self-similar groups induce random walks on the (Schreier) graphs of their action on the levels of the tree. If the group is contracting, then the graphs converge in a certain sense to the limit space of the group. We will discuss how the geometry of the limit space can be used to study the random walks on the groups. The conformal dimension of the limit space and the critical exponent of contraction play important role in this study. Since the original approach to these questions involved a map equivalent to the Schur complement, there must be a non-trivial connection of the geometry of the limit space and its dimension with the spectral properties and this workshop is a good opportunity to explore it. The talk is joint work with N. Mate Bon and T. Zheng.

We solve the two-sided version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples using Kronecker's tensor product by $L(X_1,\dots,X_m):=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m$. We will show that ranks of linear pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, $m$-tuples $A$ and $B$ of $n\times n$ matrices are simultaneously similar if and only if the ranks of $L(A)$ and $L(B)$ are equal for all linear matrix pencils $L$ of size $mn$. Variants of this property exist for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Finally, if time permits, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups will be presented. The talk is based on joint work with Harm Derksen, Visu Makam and Jurij Vol\^v{c}i\^v{c}.

In a classical work, Lee and Yang proved that zeros of certain polynomials (partition functions of Ising models) always lie on the unit circle. Distribution of these zeros control phase transitions in the model. We study this distribution for a special “Migdal-Kadanoff hierarchical lattice”. In this case, it can be described in terms of the dynamics of an explicit rational function in two variables. More specifically, we prove that the renormalization operator is partially hyperbolic and has a unique central foliation. The limiting distribution of Lee-Yang zeros is described by a holonomy invariant measure on this foliation. These results follow from a general principal of expressing the Lee-Yang zeros for a hierarchical lattice in terms of expanding Blaschke products allowing for generalization to many other hierarchical lattices. This is joint work with Pavel Bleher and Mikhail Lyubich

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A (non-selfadjoint) operator algebra is said to be residually finite dimensional (RFD) if it embeds into a product of matrix algebras. This notion is well studied in the context of C*-algebras. In particular, a theorem of Exel and Loring characterizes RFD C*-algebras in terms of the state space and in terms of a finite-dimensional approximation property for representations. I will talk about a non-selfadjoint version of the Exel-Loring theorem. In particular, I will discuss questions of Clouatre, Dor-On and Ramsay about approximation by finite dimensional representations.

Abstract: We will discuss the foundational aspects of multivariable spectral theory, and provide some applications. We will begin with a description of the algebraic and spatial spectral theory for several commuting operators, with an emphasis on the axiomatic approach to spatial spectra. We will prove the Spectral Mapping Theorem for spatial spectra, assuming that the projection property holds. Next, we will define the analytic functional calculus for the Taylor spectrum, and showed the connections with the Bochner-Martinelli kernel in the case when the operators belong to a $C^*$-algebra. We will also consider the associated Fredholm theory, and present some applications to subnormal $n$–tuples, Bergman $n$–tuples, and the $n$–tuple $M_z$ of multiplications by the coordinate functions acting on the Bergman space of a Reinhardt domain in ${\mathbb C}^n$.

This minicourse aims to survey some recent work on the projective joint spectrum for linear operators. In finite dimension, the notion in part motivated the definition of joint characteristic polynomial for several matrices. The first talk will present some examples and describe its application to the classification of finite dimensional Lie algebras.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $\mu = (1-|z|^2)^{2\lambda-2}dA)$ be the weighted area measure on the unit disc $\mathbb D$ and let $\mathcal H^{(\lambda)}$ be the Hilbert space $L^2(\mathbb D, d\mu)$, $\lambda > 1/2$. Suppose that $\mathcal M \subseteq \mathcal H^{(\lambda)}$ is a closed subspace invariant under the unitary representations $\left (U_{\varphi^{-1}} f \right ) \big (z \big ) := {\varphi^\prime}^{\lambda}\big (z \big ) \big (f \circ \varphi\big ) \big (z \big )$, $\varphi \in \mbox{M\"{o}b}$, of the M\"{o}bius group M\"{o}b, and the operator $M$ of multiplication by the coordinate function on $\mathcal H^{(\lambda)}$. There is one such, namely, the subspace of holomorphic functions in $\mathcal H^{(\lambda)}$. A description of these simultaneous invariant subspaces remain mysterious in general. Never the less, one may go one more step and consider unitary representations of the M\"{o}bius group acting on the Hilbert space $\mathcal H^{(\lambda)} \otimes V$, where $V$ is a finite dimensional Hilbert space by replacing the multiplier ${\varphi^\prime}^\lambda$ by a suitable multiplier taking values in $\text{GL}(V)$. These subspaces are in one to one correspondence with subnormal homogeneous operators, namely, those contractive linear operators $T$ such that $\varphi(T)$ is unitarily equivalent to $T$ for all $\varphi$ in M\"{o}b. We will discuss some partial answers that we have obtained recently to these questions.

We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and $C^*$-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra, which we assume to be generated by a vector space endowed with a Hilbertian seminorm. This allows us in turn to extend these existence results to the case when the generating vector space is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space. Joint work with M. Infusino, T. Kuna, P. Michalski.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Self-similar groups constitute an important class of groups with rare and unusual properties. They can be defined in numerous ways, including by actions on the interval, Cantor set, regular tree or via Mealy automata. The class includes the groups of intermediate growth (between polynomial and exponential), amenable but not elementary amenable groups, groups of Burnside type (i.e. infinite finitely generated torsion groups), iterated monodromy groups associated with polynomials (or rational functions) etc . Very interesting is the spectral theory around self-similar groups which includes spectra and spectral functions associated with Laplace (and more generally Schroedinger ) operator on graphs of algebraic origin (i.e. Cayley and Schreier graphs). The idea of the inclusion of Laplace operator on groups and graphs in the multiparametric pencil of operators (i.e. of a Joint Spectrum) invented in the above context by the L.Bartholdi and speaker in 2000 lead to the numerous results in spectral theory of groups and graphs. In my two lectures I will introduce a general framework of the self-similar groups, explain how self-similarity and a classical tool known as a Schur Complement lead in some cases to complete solution of the spectral problem or to its reduction to the problems of multidimensional dynamics: iteration of a rational function in $C^n$ (or $R^n, n\geq 2$), finding invariant subsets, including the attractor type subsets etc. This will be demonstrated by examples, including the first group of intermediate growth, Lamplighter and Hanoi Towers groups. The lectures will give a ground for understanding the lectures of N-B. Dang and some talks of the workshop.

Certain problems related to the computation of spectrum and joint spectrum of specific self-similar groups can be translated into a study of the dynamics of very particular transformations. In this mini course, I will introduce the main techniques from complex geometry in order to study the iterates of rational transformations. I will first explain how and when one can define some invariant currents by a given rational transformation and then explain in which situation iterative preimages of a hypersurface/subvarieties can converge to these currents.

In this talk we investigate under which conditions a linear functional $L$ defined on a linear proper subspace $B$ of a unital commutative real algebra $A$ admits an integral representation with respect to a nonnegative Radon measure supported on a prescribed closed subset $K$ of the space of homomorphisms of $A$ endowed with the weak topology. This is a generalization of the classical truncated moment problem and it has the advantage of encompassing also infinite dimensional instances, e.g. when $A$ is not finitely generated and $B$ is finite dimensional or when $A$ is finitely generated and $B$ infinite dimensional $A$. We first provide a criterion for the existence of such an integral representation for $L$ in the case when $A$ is equipped with a submultiplicative seminorm. Then we build on this result to prove a Riesz-Haviland type theorem, which also holds when $A$ is not equipped with a topology. Beside infinite dimensional applications, this theorem extends some classical results for the truncated moment problem for polynomials in finitely many variables to situations when the monomial diagram associated to $B$ contains infinitely many monomials in one of the variables, e.g. for rectangular or sparse truncated moment problems. This is a joint work with Raul Curto, Mehdi Ghasemi, and Salma Kuhlmann.

The talk is devoted to a discussion of the formalism of "coloured" random walks on groups (also known as covering random walks, random walks with hidden degrees of freedom, etc. etc.) and their applications.

Given a tuple of square $N\times N$ matrices, $A_1,...,A_n$, the \textit{determinantal hypersurface}, or \textit{determinantal manifold}, of the tuple is $$\sigma(A_1,...,A_n)=\Big\{[x_1:x_2:\cdots :x_n]\in \mathbb{C}\mathbb{P} ^{n-1}: \ det(x_1A_1+\cdots +x_nA_n)=0\Big\}.$$ Determinantal hypersurfaces have been under investigation in the framework of algebraic geometry for more than a century with the main focus on the question when an algebraic set of codimension 1 in the projective space admits a determinantal representation. In 2009 R. Yang \cite{Y} introduced \textit{projective joint spectra} of operator tuples acting on a Hilbert space, which generalized the concept of determinantal hypersurfaces to the infinite dimensional case. The definition is $$\sigma(A_1,...,A_n)=\Big\{[x_1:x_2:\cdots :x_n]\in \mathbb{C}\mathbb{P} ^{n-1}: \ x_1A_1+\cdots +x_nA_n \ \mbox{is not invertible}\Big\}.$$ This initiated a study of spectral properties of operator tuples from the point of view of connection between the geometry of projective joint spectra and relations between the operators in the tuple. Such approach lead to new and meaningful results even for classical investigation of determinantal manifolds of matrix tuples. In particular, an interesting connection to representation theory was found (cf. \cite{GY} and \cite{CST} among others). It was established that for some groups joint spectra of images under linear representations of certain generating sets characterized representations up to an equivalence. In course of this research a sort of inverse theorems which we call \textit{spectral rigidity theorems} were found. We say that a tuple $(A_1, \dots, A_n)$ of operators (matrices) is \textit{spectrally rigid} if any other tuple (or sometimes any tuple satisfying certain additional conditions) that has the same projective joint spectrum is equivalent to $(A_1, \dots, A_n)$. %These are results of a type: %if two tuples have the same projective joint spectra, they are equivalent. We call such phenomenon \textit{spectral rigidity}. In this talk we will discuss spectral rigidity theorems related to representations of\\ a) Coxeter groups; \\ b) certain subgroups of permutation groups which are related to Hadamard matrices of Fourier type; \\ c) Lie algebra $\mathfrak{sl}(2)$; \\ 4) infinitesimal generators of quantum $SU(2)$ groups representations. \vspace{.5cm} The talk will include some results from \cite{PS}, \cite{St}, and \cite{S}. $$\begin{thebibliography}{99} \bibitem{CST} Z.Cuckovic, M. Stessin, A. Tchernev, Deterninantal hypersurfaces and representations of Coxeter groups, \textit{Pacific J. Math}, \textbf{313} (2021), No 1, 103-135. \bibitem{GY} R. Grigorchuk, R. Yang, Joint spectrum and infinite dihedral group, \textit{Proc. Steklov Inst.Math.}, \textbf{297} (2017), 145-178. \bibitem{PS} T. Peebles, M.Stessin, Spectral hypersurfaces for operator pairs and Hadamard matrices of $F$-type, {\it Journal Adv. Oper. Theory}, {\textbf 6} (2021) issue 1, $\# 13$. \bibitem {St} M. I. Stessin, Spectral analysis near regular point of reducibility and representations of Coxeter groups, {\it J. Complex Anal. Oper. Theory}, \textbf{16} (2022), $\# 70$. \bibitem{S} M. Stessin, Spectral characterization of representations of quantum SU(2), \textit{in preparation} \bibitem{Y} R. Yang, Projective spectrum in Banach algebras, \textit{J. Topol. Anal.}, \textbf{1} (2009), 289-306. \end{thebibliography}$$

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss the foundational aspects of multivariable spectral theory, and provide some applications. We will begin with a description of the algebraic and spatial spectral theory for several commuting operators, with an emphasis on the axiomatic approach to spatial spectra. We will prove the Spectral Mapping Theorem for spatial spectra, assuming that the projection property holds. Next, we will define the analytic functional calculus for the Taylor spectrum, and showed the connections with the Bochner-Martinelli kernel in the case when the operators belong to a $C^*$-algebra. We will also consider the associated Fredholm theory, and present some applications to subnormal $n$–tuples, Bergman $n$–tuples, and the $n$–tuple $M_z$ of multiplications by the coordinate functions acting on the Bergman space of a Reinhardt domain in ${\mathbb C}^n$.

A group representation $\pi: G\to U(\mathcal{H})$ is said to be self-similar if it can be lifted intrinsically to a representation $\hat{\pi}: G\to U(\mathcal{H}^d)$. It was discovered by Grigorchuk and his collaborators that the spectral properties of $\pi$ and $\hat{\pi}$ naturally gives rise to a rational map on the projective space $\mathbb{P}^n$. Remarkably, it was shown recently that the Julia set of such map can be explicitly determined in some cases, and they are found to be directly connected with the projective spectrum of linear operators.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The distribution of the zeros of partition functions are intimately related to the analyticity of physical quantities, and to the hardness of approximating the partition function in polynomial time. In the 1950's Lee and Yang proved that the free energy per site of the cubic lattice is analytic at a give positive real parameter when the complex zeros of the partition functions for a sequence of finite graphs converging to the lattice avoid a neighborhood of this parameter. Moreover, under very mild conditions on the sequence of graphs, the limiting free energy per site is independent of the chosen sequence of graphs. Because of these results one might assume that the limit distribution of the zeros is also independent of the sequence of graphs, but in fact they depend very sensitively on the chosen graphs. Even deciding whether the zeros remain bounded or not turns out to be a subtle question, a question that I will address in this talk. This is based on ongoing work with David de Boer, Pjotr Buys and Guus Regts. I hope that by the time of this conference I will be able to give a complete answer for graphs that are tori of different dimensions.

Non-Archimedean local fields and their rings of integers can be naturally identified with the boundaries of locally finite trees. We develop a method of embedding of some HNN extensions of self-similar groups into the group $D(\mathbb Q_p)$ of dilations of the field $\mathbb Q_p$ of $p$-adic numbers. In particular, we embed the finitely presented Grigorchuk group, and HNN extensions of the lamplighter group $\mathbb Z_2\wr\mathbb Z$ and of the Baumslag-Solitar group $BS(1,3)$ into $D(\mathbb Q_p)$. We also discuss a connection to scale groups introduced by Willis and show that the closure of the finitely presented Grigorchuk group is a scale group that acts 2-transitively on the relative boundary of the ternary regular tree (which is a punctured hyperbolic boundary of such a tree). This is a joint work with Rostislav Grigorchuk.

In this talk we will discuss some questions from spectral theory of infinite graphs and how to solve them by studying self-similar groups and their actions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Self-similar groups constitute an important class of groups with rare and unusual properties. They can be defined in numerous ways, including by actions on the interval, Cantor set, regular tree or via Mealy automata. The class includes the groups of intermediate growth (between polynomial and exponential), amenable but not elementary amenable groups, groups of Burnside type (i.e. infinite finitely generated torsion groups), iterated monodromy groups associated with polynomials (or rational functions) etc . Very interesting is the spectral theory around self-similar groups which includes spectra and spectral functions associated with Laplace (and more generally Schroedinger ) operator on graphs of algebraic origin (i.e. Cayley and Schreier graphs). The idea of the inclusion of Laplace operator on groups and graphs in the multiparametric pencil of operators (i.e. of a Joint Spectrum) invented in the above context by the L.Bartholdi and speaker in 2000 lead to the numerous results in spectral theory of groups and graphs. In my two lectures I will introduce a general framework of the self-similar groups, explain how self-similarity and a classical tool known as a Schur Complement lead in some cases to complete solution of the spectral problem or to its reduction to the problems of multidimensional dynamics: iteration of a rational function in $C^n$ (or $R^n, n\geq 2$), finding invariant subsets, including the attractor type subsets etc. This will be demonstrated by examples, including the first group of intermediate growth, Lamplighter and Hanoi Towers groups. The lectures will give a ground for understanding the lectures of N-B. Dang and some talks of the workshop.

This will be an expository introduction to quiver representations and their associated rank schemes. Like joint spectra, they are also defined by determinantal equations. One setting where rank schemes are better understood is for Dynkin quivers. I will survey recent results on this.

Full groups originated from the theory of measurable (and later Cantor) dynamical systems and their von Neumann-algebra ($C^*$-algebra) crossed-products. For a given topological dynamical system $(X,G)$, the full group [G] can be broadly defined as the set of all homeomorphisms of $X$ that act within the $G$-orbits. Thus, the full groups can be viewed as a generalized symmetric group of the orbit equivalence relation of $(X,G)$. In a series of papers by Giordano-Putnam-Skau, Matui, Medynets, Nekrashevych, and others, it was shown that full groups (as abstract groups) encode complete information about the underlying dynamical systems up to (topological) orbit equivalence. In recent years, the development of the theory of full groups for Cantor minimal systems has been having considerable impact on geometric group theory driven primarily by the fact that by tweaking dynamical properties of the underlying dynamical system $(X,G)$, we can produce a (countable) full group $[G]$ with new and unusual properties, which has been successfully used to solve some open problems in geometric group theory. In this talk, we will focus on Vershik’s conjecture dealing with the theory of characters for full groups of minimal $Z$-actions and its interplay with the underlying dynamics.

This talk about a possible connection between between cluster algebras and the joint spectrum. Cluster algebras are commutative algebras with a combinatorial structure. They come with a special set of generators called cluster variables. These cluster variables are Laurent polynomials in several variables. For cluster algebras associated to a Dynkin diagram, the cluster variables are in bijection with the (almost) positive roots of the Dynkin diagram. On the other hand, we consider the joint spectrum of k operators on a finite dimensional complex vector space. In fact, we are interested in the joint spectrum of the regular representation acting on the generators of a finite Coxeter group. This is a determinantal hypersurface. Čučković, Stessin and Tchernev showed that if two finite Coxeter groups have the same joint spectrum on the regular representation, then the groups are isomorphic. The connection to cluster algebras comes from the observation that for the Coxeter group of type $A_{n-1}$ the joint spectrum is the zero locus of the cluster variable of the highest root in type $A_{n-2}$ under a certain specialization.

Every self-similar group G of d-ary tree automorphisms induces a sequence of finite Schreier graphs X_n of the action of G on the level n of the tree, along with a sequence of d-to-1 coverings X_{n+1} -> X_n. There are interesting examples of self-similar groups for which the spectra of the corresponding Schreier graphs are described by backward iterations of polynomials of degree 2 (the first Grigorchuk group, the Hanoi Towers group, the IMG of the first Julia set, ...). In this talk, for every r>1, we provide examples of self-similar groups for which the spectra of the Schreier graphs are described by backward iterations of polynomials of degree r. The examples come from the world of iterated monodromy groups of critically-fixed polynomials. A critically-fixed polynomial is a polynomial that fixes all of its critical points, and such polynomials are clearly post-critically finite. In general, if we start with any post-critically finite rational map f of degree d on the Riemann sphere, the iterated monodromy group of f (due to Nekrashevych) is a self-similar group acting on the d-ary rooted tree by automorphisms in such a way that the corresponding sequence of Schreier graphs approximates the Julia set of f and the coverings approximate the action of f on the Julia set. In our examples, the degree r of the polynomial that describes the spectra of the Schreier graphs coincides with the maximal local degree of f at the critical points.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Arithmetic dynamics is a field at the intersection of complex dynamics and algebraic number theory. The main questions in arithmetic dynamics are motivated by analogous classical problems in arithmetic geometry, especially the theory of elliptic curves. We study one such question, which is an analogue of Serre's open image theorem regarding $\ell$-adic Galois representations arising from elliptic curves. We consider the action of the absolute Galois group of a field on pre-images of a point $\alpha$ under iterates of a rational map $f$ (points that eventually map to $\alpha$ as we apply $f$ repeatedly). These points can be given the structure of a rooted tree in a natural way. This determines a homomorphism from the absolute Galois group of the field to the automorphism group of this tree, called an arboreal Galois representation.

Emergence is way to describe the richness of possible statistical behaviors of orbits of a dynamical system. For example, topological emergence quantifies the size of the set of invariant ergodic measures. It is closely related to topological entropy, but it is possible for a system to have zero entropy and high emergence. While it is known that high emergence is topologically generic in certain settings, it is less clear for "random" systems. We provide a model for random automorphisms of spherically homogeneous rooted trees such that the action on the ends has high emergence almost surely.