Schedule for: 19w5137 - Multivariable Spectral Theory and Representation Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

This talk is based on joint works with Z. Cuckovic, T. Peebles, A. Tchernev and J. Weyman. Given a tuple ${A_1, ..., A_n}$ of N × N matrices, determinantal hypersurface $σ(A_1, ..., A_n)$ is an algebraic set in the complex projective space $CP^{n−1}$ given by $σ(A_1, ..., A_n) = \{[x_1, ..., x_n] \in CP^{n−1} : det(x_1A_1 + ...x_nA_n) = 0\}$. In the infinite dimensional case of operators acting on a Hilbert space, the corresponding set is called the projective joint spectrum and is defined by $σ(A_1, ..., A_n) =\{[x_1, ..., x_n] \in CP^{n−1}: x_1A_1 + ... + x_nA_n \textrm{ is not invertible}\}.$ The main topic of the talk is how the geometry of the joint spectrum reveals a mutual behavior of the operators in the tuple. We’ll see that this investigation leads to a characterization of representations of finite Coxeter groups in terms of determinantal hypersurfaces. This is the first part of a two talk presentation. The second one will be given by A. Tchernev.

This is the second in a series of two talks. Part I was presented by M. Stessin.

The connection between the geometry of joint spectra and representations of Coxeter groups that was discussed in Part I raises a framework of fundamental structural questions, special cases of which go all the way back to the origins of representation theory. I will formulate these, and describe how joint works with Z. Cuckovic, T. Peebles, R. Schiffler, M. Stessin, and J. Weyman fit into this framework.

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 the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

The nc state space of an operator system $A$ is the nc convex set of u.c.p. maps into Hilbert spaces. The operator system is identified with the nc affine functions on the state space. The space of nc continuous functions on the state space is the maximal C*-algebra generated by $A$, and the space of bounded nc functions corresponds to the double dual. Measures on $C(K)$ are replaced by u.c.p maps, and they represent the restriction to $A$. We identify the states of $A$ with unique representing maps, and show that the nc extreme points correspond to the boundary representations. The convex envelope of a continuous nc function is a multivalued nc function. This function has an equivalent form in terms of u.c.p. maps. This leads to two natural orders on u.c.p. maps, the nc Choquet order and the dilation order, which turn out to coincide. There is a noncommutative Choquet-Bishop-de Leeuw theorem that every state of $A$ has a representation as a u.c.p. map on the C*-algebra which is in a certain sense supported on the extreme boundary. This is joint work with Matthew Kennedy.

In this talk I will begin with a review of the Frobenius operator $\Phi$ on the $\ell$-adic cohomology of a curve $C$ over a finite field of characteristic $p$ different from $\ell$. The spectrum of $\Phi$ determines the zeta function of $C$, and is a basic invariant of $C$. I will show how, under certain hypotheses, a derivative of $\Phi$ can be used to compute certain natural cup products in the cohomology of $C$. It is an open problem with applications in cryptography to determine the computational complexity of finding such cup products. This is joint work with Ted Chinburg.

Finitely generated structures are important subjects of study in various mathematical disciplines. Examples include finitely generated groups, finitely generated Lie algebras and $C^*$-algebras, tuples of several linear operators on Banach spaces, etc. It is thus a fundamental question whether there exists a universal mechanism in the study of these vastly different entities. In 2009, the notion of projective spectrum for several elements $A_1, A_2, ..., A_n$ in a unital Banach algebra ${\mathcal B}$ was defined through the multiparameter pencil $A(z)=A_1+z_2A_2+\cdots +z_nA_n$, where the coefficients $z_j$ are complex numbers. This conspicuously simple definition turned out to have a surprisingly rich content. In this talk we will review some results related to group theory, complex geometry, Lie algebras, operator theory and complex dynamics.

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

The focus of my talk will be on the representation theory of a finite group over a field whose characteristic divides the order of the group. Typically, in this case the indecomposable representations cannot be classified. Over the past four decades, various techniques have been developed to study these modular representations. One such is the theory of rank varieties for elementary abelian groups introduced by Jon Carlson, and developed for general finite groups, and even finite group schemes, by Friedlander and Pevtsova. My plan is to give an introduction to some of these ideas.

I will talk about recent progress on Sarason’s Ha-plitz product problems: characterize entire functions $f$ and $g$ such that the Toeplitz product $T_fT_{\bar g}$ is bounded, or the Hankel product $H^*_{\bar f}H_{\bar g}$ is bounded, or the mixed product $H_{\bar f}T_{\bar g}$ is bounded on the Fock space. The Weyl unitary representation of the Heisenberg group plays a key role in this line of research.

Given two arbitrary complex selfadjoint matrices of size n, work of Tau and Knutson shows that there exists an associated combinatorial object called a hive of size n (or, dually, a honeycomb). Conversely, every hive of size n is associated to some pair of selfadjoint matrices. This association is neither injective nor surjective and it is not known how to effectively realize it. Danilov and Koshevoy conjectured an explicit formula for (one of) the hive(s) associated to a pair of selfadjoint matrices. I will explain some progress made in verifying this conjecture (in a somewhat sharper form) for a number of hives that are known to be rigid. The hope is that this work will be useful in understanding arbitrary pairs of selfadjoint operators that belong to a finite von Neumann algebra. This is joint work (in progress) with W. S. Li and D. Timotin.

Let $H_m(\mathbb B)$ be the analytic functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ with reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. Examples are the Drury-Arveson space, the Hardy space and (weighted) Bergman spaces. We use algebraic operator identities to characterize those commuting row contractions $T \in L(H)^n$ that decompose into the direct sum of a spherical coisometry and a shift $M_z \in L(H_m(\mathbb B,D))^n$. For $m = 1 = n$, this recaptures the classical Wold decomposition, for the case of the unit disc $\mathbb D \subset \mathbb C$ we obtain results of Giselsson and Olofsson from 2012. The same methods can be used to give a Brown-Halmos type characterization of Toeplitz operators with pluriharmonic symbol on the spaces $H_m(\mathbb B)$. The talk is based on joint work with Sebastian Langend\"orfer.

Cluster algebras are commutative algebras with a special combinatorial structure. They originated in Lie theory in the context of canonical bases and total positivity, but are now connected to a number of areas including combinatorics, representation theory, knot theory, integrable systems and algebraic geometry. In this talk, we give an overview of the theory and highlight a recent connection to the joint spectrum of representations of the symmetric group.

A rank-one perturbation $A+K$ of an operator $A$ is one where the range of $K$ is just one-dimensional. Being rather restrictive, they form a small class of perturbations. Yet, rank-one perturbations are related to many deep questions. Here we focus on a relation with Anderson-type Hamiltonians. These are random perturbations which are obtained by taking a countable sum of rank-one perturbation, each weighted by a randomly chosen coupling constant. Such perturbations are non-compact almost surely. Under mild conditions, the essential parts of two realizations of an Anderson-type Hamiltonian are almost surely related by a rank one perturbation.

We present an introduction to the truncated moment problem, based on joint work with L.A. Fialkow, S.H. Lee, H.M. Moeller, S. Yoo and J. Yoon. Inverse problems naturally occur in many branches of science and mathematics. An inverse problem entails finding the values of one or more parameters using the values obtained from observed data. A typical example of an inverse problem is the inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This problem is intimately connected with image reconstruction for X-ray computerized tomography. Moment problems are a special class of inverse problems. While the classical theory of moments dates back to the beginning of the 20th century, the systematic study of \textit{truncated} moment problems began only a few years ago. In this talk we will first survey the elementary theory of truncated moment problems, and then focus on those problems with cubic column relations. Our lecture is organized around the following topics: the classical Fibonacci sequence, truncated moment problems (TMP), the basic positivity condition, the algebraic variety of a TMP, the First Existence Criterion for TMP, the Flat Extension Theorem, localizing matrices, Riesz-Haviland for TMP, the quartic TMP, the extremal TMP, the Division Algorithm in TMP and the Sextic TMP.

Abstract. Non-abelian C*-algebras can be understood better from the examination of their maximal abelian subalgebras. In particular, Renault showed that in the presence of a Cartan subalgebra, a C*-algebra can be associated in a canonical way with a topological twisted groupoid. In joint work with Jon Brown, Adam Fuller, and David Pitts, we extend Renault's result by identifying Cartan pairs revealed by gradings by a group.

Entrywise operations which preserve positive definite (structured) matrices will be analyzed from a historical perspective, starting with classical results of von Neumann, Schoenberg, Helson, Kahane, Katznelson, Rudin. The recent advances on the subject are related to modifications of large correlations matrices in statistics. At this crossroad, Schur polynomials and similar invariants of the symmetric group enter into the picture. Positivity preservers of totally positive matrices will also be touched in the lecture.

Determinantal representations of projective hypersurfaces in the $n$-dimensional projective space ${\mathbb P}^n$ (or of homogeneous polynomials in $n+1$ variables) appear naturally in a variety of areas. The case $n=1$ is trivial. For the case $n=2$ (plane curves) determinantal representations always exist and can be reconstructed from their kernel sheaves. For the case $n>2$ determinantal representations in general do not exist, but there is a large number of both known results and open problems, especially in the real symmetric setting where one assumes that the polynomial has some hyperbolicity (in the sense of Garding) or stability properties. I will survey some of these results, as well as the crucial role of determinantal representations of plane curves in the spectral theory of pairs of commuting nonselfadjoint (especially dissipative) operators (the theory of vessels) that was discovered by M.S. Liv\v sic in the 1980s. This can be generalized to the case of $n$ commuting nonselfadjoint operators, yielding both interesting conditions for the existence of commuting unitary dilations of the corresponding commuting semigroups of contractions and an interesting class of determinantal representations of higher codimensional subvarieties in ${\mathbb P}^n$. In particular, one can show that an algebraic curve in ${\mathbb P}^n$ always admits such a determinantal representation. The proof is based on the generalization of the classical notion of the Bezoutian of two univariate polynomials to the setting of compact Riemann surfaces of a higher genus. This talk is based on a joint work with Eli Shamovich.

The talk will present some of the recent developments in free analysis and geometry of matrix convex sets described by linear matrix inequalities (LMIs). LMIs are ubiquitous in applications such as semidefinite programming, control theory and linear systems engineering, while free LMIs (= LMIs with dimension-free matrix unknowns) are central to the theories of operator systems and spaces, and serve as the paradigm for matrix convex sets. Talk will be delivered by Dr. Jurij Volcic.

We present recent results regarding the characterization of the joint invariant subspaces under the universal model $(B_1,\ldots, B_n)$ associated with a noncommutative variety $\mathcal{V}$ in a regular domain in $B(\mathcal{H})^n$. The main result is a Beurling-Lax-Halmos type representation which is used to parameterize the corresponding wandering subspaces of the joint invariant subspaces of $(B_1\otimes I_{\mathcal{E}},\ldots, B_n\otimes I_{\mathcal{E}})$. We also present a characterization of the elements in the noncommutative variety $\mathcal{V}$ which admit characteristic functions. This leads to an operator model theory for completely non-coisometric elements which allows us to show that the characteristic function is a complete unitary invariant for this class of elements. Our results apply, in particular, in the commutative case.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.