2TART Seminar Series and Conference

Welcome to the home page for The Online Operator Theory and Related Topics page, founded due to the total collapse of in-person travel during the 2019-2020 Novel Coronavirus outbreak. Please email me if you want to be added to the list.

The Seminar Series

The time is tentatively set to 11am US Eastern Time, which will hopefully accommodate members in the Americas, Europe, Africa and the Middle East.

2/23 Vaki Nikitopoulos, (UCSD)

  • Speaker: Evangelos “Vaki” Nikitopoulos
  • Title: Noncommutative $C^k$ Functions, Multiple Operator Integrals, and Derivatives of Operator Functions
  • Abstract: Let $\mathcal{A}$ be a $C^*$-algebra, $f \colon \mathbb{R} \to \mathbb{C}$ be a continuous function, and $\tilde{f} \colon \mathcal{A} _{\text{sa}} \to \mathcal{A}$ be the functional calculus map $ \mathcal{A}_{\text{sa}} \ni a \mapsto f(a) \in \mathcal{A}$. It is elementary to show that $\tilde{f}$ is continuous, so it is natural to wonder how the differentiability properties of $f$ relate/transfer to those of $\tilde{f}$. This turns out to be a delicate, complicated problem. In this talk, I introduce a rich class $NC^k(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of \textit{noncommutative} $C^k$ functions $f$ such that $\tilde{f}$ is $k$-times differentiable. I shall also discuss the interesting objects, called \textit{multiple operator integrals}, used to express the derivatives of $\tilde{f}$.

3/9 Joe Ball, (VTech)

3/23 Adam Humeniuk, (Waterloo)

Title:

Jensen’s Inequality for separately convex noncommutative functions

Abstract:

I’ll give a brief introduction to noncommutative (AKA “matrix”) convexity, and a sampling of how classical theorems in convexity seem to always have matricial analogues. Classically, Jensen’s Inequality asserts that if $f$ is a convex function and $\mu$ is a probability measure, then $\int f\; d\mu$ is at least as big as the evaluation of $f$ at the “average point” or barycenter of $\mu$. Noncommutative convex functions satisfy a Jensen Inequality for any unital completely positive (ucp) map, which plays the role of a noncommutative measure. I’ll show how the much broader class of noncommutative multivariable functions which are convex in each variable separately satisfy Jensen’s Inequality for certain “free product” ucp maps. It turns out such ucp maps show up naturally in free probability, and we’ll see a sample application yielding operator inequalities for free semicircular systems.

4/6 Cynthia Vinzant, (NCSU)

4/20 Benoit Sehba (University of Ghana)

5/4 Yi Wang (Vanderbilt)

Talks from previous semesters

4/28/20 Nikhil Srivastava

Title: Quantitative Diagonalizability

Abstract: A diagonalizable matrix has linearly independent
eigenvectors. Since the set of nondiagonalizable matrices has measure
zero, every matrix is a limit of diagonalizable matrices. We prove a
quantitative version of this fact: every n x n complex matrix is
within distance delta in the operator norm of a matrix whose
eigenvectors have condition number poly(n)/delta, confirming a
conjecture of E. B. Davies. The proof is based adding a complex Gaussian
perturbation to the matrix and studying its pseudospectrum.

Finally, we mention a recent application of this result to numerical analysis,
yielding the fastest known provable algorithm for diagonalizing an
arbitrary matrix.

Joint work with J. Banks, A. Kulkarni, S. Mukherjee, J. Garza Vargas.

5/11/20 Alan Sola

Title: One-dimensional scaling limits in a Laplacian random growth model

Abstract:

Conformal maps in the complex plane can be used to encode several types of aggregation processes that arise in mathematical physics. Simulations of these conformal aggregation models produce intriguing pictures hinting at rich and perhaps fractal structures in the resulting clusters, but for the most part these processes have resisted rigorous analysis.
In my talk I will discuss an instance where a rigorous analysis is possible: if the self-reinforcement in the growth local rule is strong enough, then the natural scaling limits of the growing clusters collapse into one-dimensional structures, in contrast to the weak reinforcement regime where growth is uniformly spread out.
This reports on joint work with Amanda Turner (Lancaster University/University of Geneva) and Fredrik Viklund (KTH).

5/25/20 Adam Dor-On

Title: Doob equivalence and non-commutative peaking for Markov chains

Abstract:

The theory of Markov chains has applications in diverse areas of Mathematics and Physics such as group theory, dynamical systems, electrical networks and information theory. Connections with operator algebras however, seem to be mostly in the non-commutative scenario of quantum information.
In this talk we will show how questions about operator algebras of stochastic matrices, studied by Markiewicz and myself, motivate new problems in Markov chain theory. We will show how the study of harmonic functions allows for better classification results of our algebras, and how non-commutative peaking in the sense of Arveson can be completely characterized in terms of the stochastic matrix.
*This is based on joint work with Xinxin Chen, Langwen Hui, Christopher Linden and Yifan Zhang, conducted as part of an undergraduate research project in Illinois Geometry Lab at UIUC.

6/9/20 Andreas Thom

Title: Asymptotics of Cheeger constants and unitarisability of groups
Abstract: Given a group Γ, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of Γ for increasingly large generating sets.
The connection hinges on an analytic invariant Lit(Γ)∈[0,∞] which we call the Littlewood exponent. Finiteness, amenability, unitarisability and the existence of free subgroups are related respectively to the thresholds 0,1,2 and ∞ for Lit(Γ). Using graphical small cancellation theory, we prove that there exist groups Γ for which 1<Lit(Γ)<∞. Further applications, examples and problems are discussed.

6/24/20 Joel Rosenfeld

Title: New Hilbert Space methods in Dynamic Mode Decomposition

Abstract: In this talk we will discuss new methods for performing Dynamic Mode Decompositions (DMD) on time series data. These methods utilize the concept of occupation kernels and their interactions with the Liouville operator  (also known as the Koopman Generator). We introduce a generalization of the Liouville operator, the “scaled” Liouville operator, that allows for operator norm convergence of DMD procedures.

This talk will outline the theoretical benefits from this new perspective, which allows for the analysis of a broader classes of continuous time dynamics than DMD methods based on the Koopman (composition) operators and stronger convergence results.

This talk will serve as a practice run for a talk at a workshop hosted by the American Control Conference that I’m organizing with Dr. Rushikesh Kamalapurkar which will run on June 30th. More details may be found at https://urldefense.proofpoint.com/v2/url?u=https-3A__scc.okstate.edu_ACC2020Workshop&d=DwIGaQ&c=sJ6xIWYx-zLMB3EPkvcnVg&r=lIG1gw2HKsxrEwDXxWVabw&m=L7fThwgvPwYDTUyMooTnLncK3JXKYA5R5wQF4z8WhCA&s=bqopb0Beu7e-ObKGsbBHueX4f4UBDePA2NHpbzdegdc&e=

7/7/20 Malte Gerhold

Title: Dilations and Matrix Ranges of Unitary Tuples

Abstract: We define three distance functions on the space of d-tuples of unitary operators on a Hilbert space related to dilations, matrix ranges, and representations, respectively. For any of those distance functions, two d-tuples U and V have zero distance if and only if there is a *-isomorphism between the generated C*-algebras which maps every entry of U to the corresponding entry of V, in which case we consider U and V as equivalent. We establish estimates between the considered distance functions, and conclude that those distance functions give rise to equivalent metrics on a large subspace of all unitary d-tuples (up to equivalence).

As an example, we consider the rotation C*-algebras, i.e., the universal C*-algebras generated by a pair of unitaries (u, v) subject to the relation vu=quv for a complex number q of modulus 1. Based on estimates for the “dilation distance” between pairs of generators, we find a new and rather simple proof of the fact (and certain refinements thereof) that the rotation C*-algebras form a continuous field.

(based on joint work with Satish Pandey, Orr Shalit, and Baruch Solel)

7/21/20 Monique Laurent

Title: Convergence Analysis of Approximation Hierarchies for Polynomial Optimization

Speaker: Monique Laurent, CWI Amsterdam and Tilburg University

Abstract:
Minimizing a polynomial function f over a compact set K is a computationally hard problem. Upper bounds have been introduced by Lasserre (2011), which are obtained by searching for a degree 2r sum-of-squares density function minimizing the expected value of f over K with respect to a given reference measure supported by K.
We will discuss several techniques that permit to analyse the performance guarantee of these bounds. This includes an eigenvalue reformulation of the bounds, links to extremal roots of orthogonal polynomials and to cubature rules, and reducing to the univariate case by means of push-forward measures.
For simple sets K like the box, the unit ball, the simplex and the unit sphere one can show that the convergence rate to the global minimum is in O(1/r^2) and that this bound is tight for some classes of polynomials and, for general convex bodies, one can show a slightly weaker convergence rate in O((log(r)/r)^2).
Based on joint works with Etienne de Klerk and Lucas Slot.

8/4/20 Kari Eifler

Title: Quantum Symmetries of Quantum Metric Spaces
Abstract: I will first give a light introduction to the theory of compact quantum groups. We will then look at Weaver and Kuperberg’s quantum metric spaces, which are a non-commutative analogue of finite metric spaces. Banica has defined the quantum symmetry group of a finite metric space, and I will talk about how to capture Banica’s definition using the Weaver-Kupperberg framework of quantum metric spaces.

8/18/20 Jamie Mingo

Title: Infinitesimal Freeness and Non-commutative Functions
Speaker: Jamie Mingo, Queen’s University
Abstract: The basic tool in free probability is the $R$-transform. It connects the moments and the free cumulants of a random variable through a functional equation of analytic functions. Infinitesimal freeness is finer type of independence used to analyze spike models in random matrix theory. I will show how the non-commutative functions can be used to find the functional relations for infinitesimal freeness. This is joint work with Pei-Lun Tseng.

9/1/20 Cody Stockdale

Title: A different approach to endpoint weak-type estimates for Calderón-Zygmund operators
Abstract: The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.

9/15/20 Dima Shlyakhtenko

Title: Fractional free convolution powers.

Abstract: (joint work with T. Tao) The extension
k↦μ⊞k of the concept of a free convolution power to the case of non-integer k≥1 was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory. In this paper we give two proofs of the monotonicity of the free entropy and free Fisher information of the (normalized) free convolution power in this continuous setting, and also establish an intriguing variational description of this process.

9/29/20 Jeet Sampat

Title: “Cyclicity preserving operators”
Abstract: For certain spaces of analytic functions in several complex variables, we will show that bounded linear operators that preserve shift-cyclic functions (i.e. Tf is shift-cyclic whenever f is shift-cyclic) are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space, Dirichlet type spaces, Drury-Arveson space etc. In the case of Hardy spaces we will show that this condition is also sufficient. That is, all bounded weighted composition operators preserve cyclic functions in the Hardy spaces.
We will also see that some of the facts required to prove this theorem enable us to prove a version of the Gleason-Kahane-Zelazko theorem for partially multiplicative linear functionals on these spaces.

9/29/20 J. E. Pascoe

> Title: The tracial fundamental group
>
> Abstract: The functional calculus allows us to induce a function on
> matrices with spectrum in some open set from an analytic function on
> that open set. Similarly, one can consider the tracial functional
> calculus, where one then takes the trace of the value of the function
> induced by the functional calculus. Call a germ of an analytic function
> global if it analytically continues along every path in the ambient
> domain. One motivation for homotopy, and thus the fundamental group,
> comes from the theory of analytic continuation. Two paths are homotopic
> if and only if for every global germ, the continuation along both paths
> agree. We therefore formulate the notion of a “fundamental group” in
> the functional calculus and tracial functional calculus essentially by
> considering paths equivalent in terms of global germs, which now need to
> analytically continue along every path in the space of matrices with
> spectrum in the domain. Free universal monodromy implies that in the
> classical functional calculus, the notion is trivial. The tracial
> fundamental group is, perhaps surprising, abelian and isomorphic to a
> direct sum of copies of the rationals.
>
> The results also hold in more than one variable and for more general
> classes of functions. (Such as analytic trace functions in the sense of
> Jekel’s OTWIA talk.)
>
> In this talk, we will introduce the functional calculus and homotopy and
> proceed to the aforementioned results. The talk should be generally
> accessible, and ask questions if anything is unclear.

9/29/20 Raymond Cheng

Title
Function theory and $l^p_A$.
Abstract
$l^p_A$ is the space of analytic functions on the open unit disk whose Taylor coefficients belong to the familiar sequence space $l^p$.  Despite its simple definition and classical origins, surprisingly little is known about this space, except when p=1 or 2.  We’ll do a non-technical survey of what is known or not known, mostly in comparison with the Hardy spaces, touching on zero sets, boundary behavior, factorization, shift-invariant subspaces, and multipliers.

The Conference

Date: June 16-17

The conference will feature 6 one hour (50 minutes) lectures and 12 half hour (20 minutes) lectures over two days. We will try to plan some amount of socialization time, which in the worst case will be a big zoom call. Please email me if you want to give a talk, although slots are limited, and will be selected for a balance of topics.

Tentative Schedule

Tuesday, June 16

10:00AM-10:50AM Igor Klep: Noncommutative convexity and partial convexity
11:00AM-11:20AM Mario Kummer: Spectral sets and derivatives of the psd cone
11:30AM-11:50AM Eric Evert: Existence of best low rank approximations for tensors of order three
12:00PM-12:20PM Jurij Volcic: Singularities, irreducibility, and Bertini’s theorem for noncommutative polynomials
12:30PM-2:00PM Social Sesquihour
2:00PM-2:20PM Ryan Tully-Doyle: Boundary realizations
2:30PM-2:50PM Sayan Das: On Fundamental groups of certain property (T) factors
3:00PM-3:20PM Roy Araiza: An Abstract Characterization for Projections in Operator Systems
3:30PM-4:20PM Bill Helton: Optimization over Convex Sets in Matrix Variables
4:30PM-5:20PM Martino Lupini: Definable algebraic invariants in (noncommutative) topology

Wednesday, June 17

10:00AM-10:20AM Tirthankar Bhattacharyya: Distinguished varieties with respect to the bidisc
10:30AM-10:50AM Tapesh Yadav: Convergence of certain lower triangular random matrices to the Volterra operator
11:00AM-11:50AM Michael Hartz: The column-row property for complete Pick spaces
12:00PM-12:20PM Eli Shamovich: Noncommutative Choquet simplifies
12:30PM-2:00PM Social Sesquihour
2:00PM-2:50PM Constanze Liaw: Point masses of matrix-valued Aleksandrov-Clark measures
3:00PM-3:20PM Robert Martin: Non-commutative rational functions in the full Fock space
3:30PM-3:50PM Evgenios Kakariadis: Rigidity of nonselfadjoint operator algebras
4:00PM-4:20PM Alberto Dayan: Interpolating Matrices
4:30PM-5:20PM Kelly Bickel: Some Remarks on Crouzeix’s Conjecture

Hour Lectures

  • Bill Helton
    Optimization over Convex Sets in Matrix Variables
    The talk concerns optimization over free sets, in particular over a set defined by inequalities on functions of matrix variables. The functions are typically (noncommutative) polynomials or rational functions and the sets include matrices of all sizes, hence are dimension free sets. In the talk the sets will mostly be convex.Extreme points are getting to be understood. But fixing a level n of a free spectrahedron and optimizing a linear functional over it leads to surprises.Missing in Zoom is my favorite activity: conferring  by the coffee pot.As  a lame substitute, this talk will mention a number of open questions, some not very hard  and some half baked as well as more conventional challenges.The work here is  with Evert, Fu, Klep, McCullough,  Volcic, Watts, Yin
  • Kelly Bickel
    Title: Some Remarks on Crouzeix’s Conjecture
    Abstract: In the early 2000s, Michel Crouzeix proved that for each matrix A, its numerical range W(A) is a 11.08-spectral set for A and that for 2×2 matrices, 11.08 can be replaced with 2. His conjecture that 11.08 can always be replaced with 2 has become known as Crouzeix’s conjecture. More recently, Crouzeix and Palencia have shown that W(A) is always a (1+sqrt{2})-spectral set and their proof techniques suggest that researchers interested in the conjecture should examine the so-called “extremal functions,” namely the functions analytic on the interior of W(A) and continuous on its closure that maximize the ratio of || f(A)|| to the supremum of |f(z)| on W(A).In this talk, we will present an overview of Crouzeix’s conjecture and highlight both what is known and what questions remain open. The talk will include several new results about, for example, special cases of Crouzeix’s conjecture when the eigenvalues of A are well separated and the structure of extremal functions both for Crouzeix’s conjecture and for more general spectral set problems. Throughout the talk, we will illustrate the results using compressions of the shift on finite dimensional model spaces. This is joint work with Pamela Gorkin, Anne Greenbaum, Thomas Ransford, Felix Schwenninger, and Elias Wegert.
  • Igor Klep
    Title: Noncommutative convexity and partial convexity
    Abstract: Motivated by classical notions of partial convexity, and bilinear matrix inequalities, we present the theory of free noncommutative (nc) sets that are defined by (low degree) polynomials with constrained terms. Given a tuple of polynomials G, a free set is called G-convex if it
    closed under isometric conjugation by isometries intertwining G. We establish an Effros-Winkler Hahn-Banach separation theorem for G-convex sets; they are delineated by linear pencils in the coordinates of G and the variables x.
    We shall also consider partial convexity for nc functions. For instance, we will explain that nc rational functions that are partially convex admit butterfly-type realizations that necessitate square roots.
  • Michael Hartz
    Title: The column-row property for complete Pick spaces
    Abstract: Complete Pick spaces form a class of Hilbert function spaces including the Hardy space on the disc, the classical Dirichlet space and also the Drury-Arveson space on the ball. They were first studied because of their connection to interpolation problems. More recently, it turned out that some apparently function theoretic properties of the Hardy space in fact extend to more general complete Pick spaces. In a number of cases, these results required an additional hypothesis, called the column-row property.
    I will talk about a result showing that the column-row property is automatic for complete Pick spaces. Moreover, I will sketch applications to weak products, interpolating sequences and de Branges–Rovnyak spaces on the ball.
  • Constanze Liaw
    Title: Point masses of matrix-valued Aleksandrov-Clark measures
    Abstract:Families of matrix-valued Aleksandrov-Clark measures arise from purely contractive matrix functions on the unit disc by means of the Herglotz-Riesz representation formula. While a description of their absolutely continuous parts closely follows the scalar Aleksandrov-Clark theory, results and proofs regarding the singular parts of the measures are more complicated. Our newest results characterize point masses of matrix-valued Aleksandrov-Clark measures in terms of directional Caratheodory angular derivatives.
  • Martino Lupini
    Title: Definable algebraic invariants in (noncommutative) topology
    Abstract: I will discuss how classical homological algebraic invariants from algebraic topology of (noncommutative) spaces, such as Cech cohomology and analytic K-homology, can be enriched with additional descriptive set-theoretic information. The resulting definable versions provide finer algebraic invariants for topological spaces and C*-algebras than the purely algebraic ones.

Half Hour Lectures

  • Evgenios Kakariadis
    Title: Rigidity of nonselfadjoint operator algebras
    Abstract: In the past 20 years nonselfadjoint algebras have been proven to provide complete invariants for geometric structures. This follows from a combination of techniques from Complex Analysis, Functional Analysis and Algebra. In this talk I will survey on rigidity results of nonselfadjoint operator algebras related to subproduct systems and semigroups. In some cases this is in stark contrast to what happens with C*-algebras.
  • Alberto Dayan
    Title: Interpolating Matrices
    Abstract: In the first part of this talk we will extend some well known characterizations of interpolating sequences on the disk to sequences of matrices with spectra in the unit disk. The second half of the talk will deal with interpolating sequences of pair of matrices with joint spectra in the bidisk.
  • Roy Araiza
    Title: An Abstract Characterization for Projections in Operator Systems
    Abstract: Given an abstract operator system V it is not clear how one would go about defining the notion of a projection. During this talk I will present an answer and some recent results on this question. This is done by first considering abstract compression operator systems associated with a positive contraction in V and then determining when we have a realization of V in such an abstract compression operator system. It then follows that there is a one-to-one correspondence between abstract and concrete projections, and in particular, that every abstract projection is a concrete projection in the C*-envelope of V. I will then conclude with some applications to quantum information theory. In particular, the study of certain correlation sets. This is joint work with Travis Russell (West Point)
  • Mario Kummer
    Title: Spectral sets and derivatives of the psd cone
    Abstract: A spectrahedron is the solution set to a linear matrix inequality. Consider a spectrahedral cone K in n-space which is symmetric with respect to permuting the coordinates. According to an observation by Bauschke, Güler, Lewis and Sendov the set S(K) of all symmetrich nxn matrices, whose spectrum is in K, is a hyperbolicity cone. We give a representation theoretic sufficient condition on K for S(K) being a spectrahedral cone. Applying this to Brändén’s spectrahedral representation of elementary symmetric polynomials yields a spectrahedral representation of all derivative relaxations of the cone of positive semidefinite matrices.
  • Tirthankar Bhattacharyya
    Title: Distinguished varieties with respect to the bidisc
    Abstract: A variety in $\mathbb C^2$ is the zero set of a polynomial. A variety is said to be distinguished with respect to the bidisc $\mathbb D^2$ if its intersection with the bidisc is non-empty and if the intersection of the variety with the topological boundary of the bidisc is the same as the intersection of the variety with the distinguished boundary of the bidisc ($\mathbb T^2$). Such varieties hold an important place in Hilbert space operator theory. An example is the Neil parabola $\{ (z,w) : z^3 = w^2\}$. There is a very well-known characterization of distinguished varieties with respect to the bidisc by Agler and McCarthy. This talk will present a newly obtained characterization based on https://arxiv.org/abs/2001.01410.
  • Jurij Volcic
    Title: Singularities, irreducibility, and Bertini’s theorem for noncommutative polynomials
    The free locus of a noncommutative polynomial f consists of all matrix tuples X such that f(X) is singular. Free loci naturally arise from problems about solution sets of noncommutative inequalities, domains of noncommutative rational functions and factorization in free algebra. From the geometric perspective, a free locus is a collection of algebraic hypersurfaces inside spaces of matrix tuples (one for each matrix size). This talk connects the irreducibility of a noncommutative polynomial with the irreducibility of said hypersurfaces in its free locus. Among the given consequences are a real Nullstellensatz, and a Bertini-type irreducibility theorem on the “eigenlevel sets” of a noncommutative polynomial.
  • Tapesh Yadav
    Title: Convergence of certain lower triangular random matrices to the Volterra operator
    Abstract: We observe some sufficient conditions for convergence of random lower diagonal matrices to Volterra operator. Mode of convergence is in almost sure sense in SOT and WOT like fashion.
  • Ryan Tully-Doyle
    Title: Boundary realizations
    Abstract: In a sequence of papers beginning in 2012, Bickel and later Bickel and Knese developed a refinement of Agler’s famous realization theory from 1990 on Schur functions in several variables that both gave control over the size of the Hilbert spaces involved and explicitly formulated pieces of the realization in terms of the function. In a separate line of work, Bickel, Pascoe, and Sola have recently produced a detailed analysis of the boundary behavior and regularity of two variable rational functions. This talk will give an overview of some of these ideas and discuss preliminary work on blending them – investigating how realizations behave at boundary singularities when the function is well-behaved there.
  • Eric Evert
    Title: Existence of best low rank approximations for tensors of order three
    Abstract: In this talk we give deterministic bounds for existence of best low rank approximations for tensors of order three. In pursuit of these bounds, we view a third order tensor as a matrix tuple and study a “joint generalized eigenvalue problem” for the matrix tuple. We establish a connection between between tensors which which fail to have a a best low rank approximation, and matrix tuples which are defective in the sense of algebraic and geometric multiplicities for joint generalized eigenvalues. Using a Bauer-Fike type bound for joint generalized eigenvalues together with this characterization yields the desired existence bounds.
    This talk is based on joint work with Lieven De Lathauwer. ​
  • Eli Shamovich
    Title: Noncommutative Choquet simplifies
    Abstract: In this talk, I will present joint work with Matt Kennedy. Choquet simplifies are infinite-dimensional versions of classical simplices. They arise naturally as the collections of all probability Borel measures on a compact Hausdorff space and as the collections of invariant measures of a dynamical system. Namioka and Phelps characterized Choquet simplices via nuclearity of the associated functions system. In the non commutative generalisation one replaces function systems with operator systems and non commutative convex sets with matrix/nc convex sets. In this talk, I will define nc Choquet simplices, as well as nc analogs of Bauer and Poulsen simplices. It turns out, that as in the classical case, every nc Bauer simplex is the nc state space of a C^*-algebra. Lastly, I will discuss dynamical applications, in particular a non commutative version of a theorem of Glasner and Weiss.
  • Sayan Das
    Title: On Fundamental groups of certain property (T) factors
    Abstract: Calculation of Fundamental groups of II$_1$ factors is, in general, an extremely improtant and hard problem. In this direction, a conjecture due to A. Connes states that the Fundamental group of the group von Neumann algebra of $L(G)$, where $G$ is an i.c.c. property (T) group, is trivial.In this talk, I shall provide the first examples of property (T) group factors with trivial fundamental group. This talk is based on a recent joint work with Ionut Chifan,  Cyril Houdayer and Krishnendu Khan.
  • Rob Martin
    Title: Non-commutative rational functions in the full Fock space
    Abstract: The full Fock space can be viewed as the Non-commutative (NC) Hardy space of square-summable power series in several NC variables. Any power series in Fock space defines a free non-commutative function in the NC unit row-ball of all strict row contractions. In this talk we characterize when an NC rational function belongs to the Fock space, we obtain analogues of classical Hardy space results for their inner-outer factorizations and we discuss spectral properties of NC rational multipliers of the Fock space. This is joint work with Michael T. Jury (University of Florida) and Eli Shamovich (Ben-Gurion University of the Negev).

The After Party

Date 6/18, 11AM

We will discuss various aspects of the seminar series and the conference, what could be done better, should we change the time, and so on. Moreover, we will decide if we want to continue the seminar series, or repeat the conference, especially if certain conferences in August are cancelled.

Some notes about online activities

Benefits

  • Cheaper to attend
  • Less time wasted in transit
  • Better for the environment
  • Easier for those with high-in person responsibilities (for example, having children, ducks, chickens, dogs, students etc.)

Drawbacks

  • Divided attention span
  • Requires making clear slides
  • Inhibition of the social aspect to some degree
  • Worse networking opportunities for extremely junior people

After the outbreak ends

It is important to think critically about whether or not we want to continue with some amount of online activity in the future.