# Extended Probabilistic Operator Algebras Seminar

Abstracts

1 | Steve Avsec | Symmetries of Noncommutative Brownian Motions | Classically, Freedman's theorem states that a sequence of random variables $(X_1, X_2, \ldots)$ is rotatable if and only if $X_j = g_j \sigma$ where $(g_1, g_2, \dots)$ are iid Gaussian variables and $\sigma$ is independent. A sequence is rotatable if the joint distribution of every subsequence is invariant under orthogonal transformations. Freedman's theorem along with the classical Gaussian functor implies that the only random process with rotatable increments is (conditionally) a Brownian motion. In this talk, we will discuss noncommutative random processes with rotatable increments as well as those with quantum rotatable increments. |

2 | Michael Brannan | The Connes embedding property for quantum group von Neumann algebras | In this talk I will discuss some joint work with Benoit Collins and Roland Vergnioux, where we consider the problem of constructing a trace-preserving embedding of the von Neumann algebra of a compact quantum group of Kac type into an ultrapower of the hyperfinite II$_1$-factor. As an application, we prove that the II$_1$-factors associated to free orthogonal and free unitary quantum groups have the Connes embedding property. We also compute the free entropy dimension of the standard generators of these von Neumann algebras. |

3 | Ian Charlesworth | Combinatorics of Bi-free Probability | In 2013, Voiculescu introduced the notion of bi-free independence to consider simultaneously left and right actions of non-commutative random variables. Shortly thereafter, Mastnak and Nica took a combinatorial approach to the same idea and proposed bi-free cumulant functionals, giving the concept of combinatorial bi-freeness. Using bi-non-crossing partitions, an analogue of the non-crossing partitions occurring in free probability, we were able to demonstrate that these two proposed notions agree. I will give a definition of bi-non-crossing partitions and the argument unifying the definitions of bi-freeness, and also show how a suitably defined Kreweras complement of bi-non-crossing diagrams yields a multiplicative convolution of bi-free random variables, in much the same way as in the free case. (Based on joint work with Brent Nelson and Paul Skoufranis.) |

4 | Jennifer Good | Nevanlinna-Pick Interpolation for Weighted Hardy Algebras | Inspired by Popescu's work with noncommutative varieties, Muhly and Solel recently extended their theory for Hardy algebras of $W^*$-correspondences to accommodate operator-valued weights, thereby producing weighted generalizations of Popescu's noncommutative disc algebra and related spaces. We will explore how several classical notions, for instance the interpolation results of Nevanlinna and Pick, have appropriate generalizations in this setting. |

5 | Michael Hartglass | Free product graph algebras and atomless loopss | I will present a canonical free-product (C* or von Neumann) algebra associated to a weighted graph. I will then show, along the lines of work by Shlyakhtenko, Skoufranis, Mai, Speicher, and Weber, that certain self-adjoint polynomials coming from loops in this algebra have no atoms in their spectral measure. |

6 | Ben Hayes | Embedding Dimension and Regularity Problems in von Neumann Algebras | Particularly strong forms of having microstates free entropy dimension one with respect to every set of generators have been established in the past, including the notion of strongly 1-bounded by Jung, as well as embedding dimension zero as defined by Shen. We discuss an a version of embedding dimension in the presence in the spirit of Voiculescu's free entropy in the presence. In particular, we prove various strong indecomposability properties of L(F_{n}). The proof of the key new property we establish is essentially linear, based on the study of bimodules over a von Neumann algebra, and may be regarded as an abstraction of Voiculescu's original proof that L(F_{n}) does not have a Cartan subalgebra. |

7 | William Helton | Free Probability and System Realizations | The ``linearization trick" has been used by Haagerup, Anderson, Belinski, Mai and Speicher to compute the distribution function for a noncommutative polynomial p applied to suitable freely independent self adjoint random variables in a type $II_1$ factor. The ``linearization trick" actually goes back to the 1960s and is highly developed in the engineering computer science and free analysis literature for noncommutative rational functions. The talk will describe this and what one must do to use the existing theory for noncommuting rational functions in Free Probability. The work is joint with Anderson, Mai and Speicher. |

8 | Adrian Ioana | Local spectral gap for simple Lie groups | I will present a "local'' spectral gap theorem for translation actions of dense subgroups generated by algebraic elements on arbitrary simple Lie groups. This extends to the non-compact setting works of Bourgain-Gamburd and Benoist-de Saxce. This is joint work with Remi Boutonnet and Alireza Salehi Golsefidy. |

9 | Marius Junge | q-gaussian actions | We consider von Neumann algebras which are generated by q-gaussian random variables and an algebra which is left invariant by the automorphisms granted by second quantization. These objects can be easily constructed using Speicher's central limit theorem, and generalize Shlyakhtenko's A-valued system. The first goal of this talk is to describe the underlying Fock space, then we move on to partial classification results and first results on relative strong solidity under very special additional assumption. Despite the similarities to the case q=0, more combinatorics has to compensate for weaker interaction principles. This is joint work with Bogdan Udrea and Stephen Longfiled, and based on discussions with Steve Avsec. |

10 | Greg Kuperberg | Quantum metrics and quantum graphs in the von Neumann algebra style | I will explain my definition, joint with Nik Weaver, of a W^* quantum metric space, and some results about the closely related notion of a quantum graph that are relevant to quantum information theory. |

11 | Weihua Liu | Extended de Finetti theorems for boolean independence and monotone independence | We construct several new spaces of quantum sequences and their families of maps in sense of So{\l}tan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. we will show extended de Finetti theorems for monotone independence and boolean independence: Roughly speaking, an infinite sequence of random variables is monotonically(boolean) spreadable if and only if the variables are identically distributed and monotone(boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability. |

12 | Adam Marcus | Polynomial Convolutions and Free Probability | This talk will focus primarily on properties of the symmetric additive convolution of polynomials introduced in Nikhil's talk. In particular, we will show a number of interesting parallels to well-known concepts in free probability. This will include a weakened version of ``freeness'' that is strong enough to mimic the usual free additive convolution but weak enough to exist in finite settings. This is joint work with Dan Speilman and Nikhil Srivastava. |

13 | Jamie Mingo | Freeness and Broken Symmetries | Transposing and partial transposing can break the symmetry of a matrix;this may produce freeness. I will give some examples and some general results. |

14 | Brent Nelson | An example of factoriality under non-tracial finite free Fisher information assumptions | Suppose $M$ is a von Neumann algebra equipped with a faithful normal state $\varphi$ and is generated by a finite set $G=G^*$, $|G|\geq 3$. We show that if $G$ consists of eigenvectors of the modular operator $\Delta_\varphi$ and have finite free Fisher information, then the centralizer $M^\varphi$ is a $\II_1$ factor and $M$ is a factor of type depending on the eigenvalues of $G$. We use methods of Connes and Shlyakhtenko to establish the existence of diffuse elements in $M^\varphi$, followed by a contraction resolvent argument of Dabrowski to obtain the factoriality. |

15 | David Penneys | Progress in the classification of small index subfactors | Subfactor theory has many examples of unexpected discrete classification, beginning with Jones' index rigidity theorem. A similar phenomenon is the nonoccurence of the D_{odd} and E_7 principal graphs. This trend continues for non A_\infty subfactors with small index, i.e., with index just above 4. I will discuss recent progress in the small index subfactor classification program. Along the way, I'll talk about joint work with Afzaly and Morrison which pushes the classification of non A_\infty subfactors to index 5+1/4. |

16 | David Renfrew | Eigenvalue densities of Structured Random Matrices | We will discuss several ensembles of non-Hermitian random matrices with block structures, motivated from math and physics. We give a characterization of the limiting spectral measure and compute their spectral radius. |

17 | Dima Shlyakhtenko | Regularity questions for polynomials of non-commutative random variables | In a joint work with Ian Charlesworth, we show that the spectral measure of an arbitrary polynomial of an n-tuple of non-commutative random variables cannot be singular and must be non-atomic, provided that there exists a dual system in the sense of Voiculescu (an assumption slightly stronger than finiteness of free Fisher information). |

18 | Paul Skoufranis | Independences from and Partial Transformations for Bi-Free Pairs of Faces | Since the notion of bi-free pairs of faces was introduced by Voiculescu a couple of years ago, substantial work has been performed surrounding this concept. In this talk, we will discuss several results that can be deduced from the combinatorial side of bi-freeness. In particular, we will discuss how one can deduce bi-freeness from freeness in certain settings, how bi-freeness of pairs of algebras yields bi-freeness for matrices of algebras, how forms of independences (e.g. classical, free, Boolean, monotone) arise from bi-free pairs of faces, and analogues of the $R$- and $S$-transforms in the bi-free setting. |

19 | Nikhil Srivastava | Finite Free Convolutions of Polynomials | We study a convolution operation on polynomials which may be seen as a finite-dimensional analogue of the free convolution of two measures in free probability theory. We show that this operation preserves real-rootedness, and establish bounds on the extreme roots of the convolution of two polynomials via the inverse Cauchy transforms. We use these properties to study the expected characteristic polynomials of random regular graphs, and in particular to establish the existence of bipartite Ramanujan graphs of every degree and every size. Joint work with A. Marcus and D. Spielman. |

20 | Dan Voiculescu | Partial transforms for some simple bi-free convolutions. | Partial transforms for some simple bi-free convolutions. |

21 | David Zimmermann | Logarithmic Sobolev Inequalities, Convolutions, and Applications | We give a brief survey of the background and various applications of the logarithmic Sobolev inequality (LSI) for a probability measure. We then examine the LSI for Gaussian convolutions in $\mathbb{R}^n$. We conclude with an application of this result to random matrices. |

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