Probabilistic Operator Algebra
Seminar
Winter-Spring-Summer 2007
Fall seminar was moved to Fields,
Schedule for Winter 2008
Mini-Workshop on Non-commutative Probability Theory
Tuesday July 24, 2007. The morning talks will be in
Jeffery 422, the afternoon talks in Jeffery 234.
Tuesday, July 24, 9:30 - 10:15, Jeff 422
Jonathan Novak (Queen's)
Random walks near the boundary of a Weyl chamber and
truncated random matrices
ABSTRACT: We will review the random turns vicious walkers
model of statistical mechanics, which involves multiple
non-intersecting walkers on a one dimensional lattice. We
will show how this can equivalently be phrased as a single
particle lattice walk in a certain Weyl chamber. When we
require the particle to remain "near" the boundary of the
chamber in a certain precise sense, we give an exact formula
for the number of walks as an average over an ensemble of
truncated random unitary matrices. Then, using the fact
that such ensembles approximate the Ginibre ensemble of
non-selfadjoint Gaussian random matrices, we find an
asymptotic formula for the number of walks.
Tuesday, July 24, 10:30 - 11:15, Jeff 422
Michael Skeide (Universita degli Studi del Molise)
Invariant commutative subalgebras of CP-maps
Tuesday, July 24, 11:30 - 12:15, Jeff 422
Rolf Gohm (University of Reading)
Characteristic Functions as Intertwiners between Dilations
ABSTRACT We describe how a conjugacy result in quantum
probability (G), motivated by the scattering theory of
Markov chains (KM), led to a generalization of the operator
theoretic concept of characteristic function as an
intertwiner between dilations (DG1,DG2).
[DG1] S.Dey, R.Gohm: Characteristic Functions for Ergodic Tuples
Integr. equ. oper. theory 58 (2007), 43-63
[DG2] S.Dey, R.Gohm: Characteristic Functions of Liftings
arXiv: 0707.1417
[G] R.Gohm: Noncommutative Stationary Processes
LNM 1839, Springer (2004)
[KM] B.Kuemmerer, H.Maassen: A Scattering Theory for Markov
Chains IDAQP vol.3 (2000), 161-176
Tuesday, July 24, 3:00 - 3:45, Jeff 234
Claus Koestler (UIUC)
Is there a braided extension of free probability?
ABSTRACT: The braid group has a presentation in terms of
`square roots of free generators'. We show that this
presentation leads to a tower of commuting squares in the
braid group von Neumann algebra and extends semicircular
systems coming from free group generators. Our results give
strong evidence for the conjecture that there exists a
braided extension of free probability. This is joint work
with Rolf Gohm.
Tuesday, July 24, 4:00 - 4:45, Jeff 234
Jamie Mingo (Queen's)
Abstract: In a recent paper on random matrices it was found
convenient to introduce the notion of a partitioned
permutation. This is a pair (V, pi) where pi is a usual
permutation of [n], a finite set with n elements, and V is a
partition of [n] such that each cycle of pi is contained in
a block of V.
I will show how to put a partial order on this set and if
time permits explain my conjecture as to what the Moebius
function might be.
Wednesday, May 23, 4:00 - 5:30, Jeff 422
Emily Redelmeier (Queen's)
A CLT for a band matrix model
ABSTRACT: In this talk we will discuss the paper "A CLT for
a band matrix model", by Greg W. Anderson and Ofer Zeitouni.
This paper examines the eigenvalue distribution and
cumulants of a random matrix model whose entries are not
necessarily identically distributed. We look in particular
at the limit of the empirical eigenvalue distribution.
Friday, May 18, 4:00 - 5:30, Jeff 422
Joseph Lin, Queen's University
Limiting Eigenvalue Distributions of a Class of Large
Dimensional Random Matrices, Part II
Abstract: This talk is based on the paper "On the Empirical
Distribution of Eigenvalues of a Class of Large Dimensional
random Matrices" by Silverstein and Bai in 1995. The
limiting eigenvalue distributions of random Hermitian
matrices of the form A + XTX* will be studied with Stieltjes
Transforms.
Wednesday, May 9, 4:00 - 5:30, Jeff 422
Joseph Lin, Queen's University
Limiting Eigenvalue Distributions of a Class of Large
Dimensional Random Matrices
Abstract: This talk is based on the paper "On the Empirical
Distribution of Eigenvalues of a Class of Large Dimensional
random Matrices" by Silverstein and Bai in 1995. The
limiting eigenvalue distributions of random Hermitian
matrices of the form A + XTX* will be studied with Stieltjes
Transforms.
Wednesday, May 2, 4:00 - 5:30, Jeff 422
Marek Bozejko, Wroclaw, Poland
The q-CCR relations for q > 1, with applications to
combinatorial problems on partitions and to
Bessis-Moussa-Villani conjecture
Abstract: We present a construction of q-CCR relations:
A(f)A*(g) - A*(g)A(f) = q^{N} I , for f, g in real
Hilbert space, where q > 1 and N is the number operator.
This is an extension of our model which was done with Roland
Speicher and myself for q in [-1,1].
As an applications we give a new combinatorial formula for
2-partitions V:
cr(V) + pbr(V) = 1/2 ip(V) ,
where cr(V) is the number of crossing in partition V, pbr(V)
is the number of embracing introduced by de Medicis,
Viennot and Nica and ip(V) is the sum of inner points of
2-partitions V considered by Effros,Popa and Yoshida and
myself.
Also we prove that Bessis-Moussa-Villani conjecture (B-M-V),
(which is STILL OPEN for 3 x 3 matrices! and t=trace) for a
large class of self-adjoint operators A and B on a Hilbert
space: (BMV) t(exp(A+ixB)) = F(x) is a positive definite
function on real line.
Namely, (B-M-V) is true for self-adjoint operators A=G(f),
B=G(g), where G(f) = a(f) + a*(f) is generalized Brownian
motion and t is the vacuum state. This extends recent
results of M. Fannes and D. Petz, which proved (B-M-V)
conjecture for the free Brownian motion.
Wednesday, April 25, 4:00 - 5:30, Jeff 422
Michael Neagu, Queen's
Asymptotic freeness of random permutation matrices with
restricted cycle lengths, part III of III
ABSTRACT: In my January talks, I presented the asymptotic
freeness of an independent family of uniformly distributed
random permutation matrices with cycle lengths restricted to
certain sets of positive integers. The methods involved in
proving this result involve two main ideas.
The first (which I discussed in detail my January talks)
addresses the computation of the asymptotic behaviour of the
expected number of cycles of a fixed given length in a
random permutation with restricted cycle lengths.
In my third talk, I will focus on the second tool of the
proof, which involves the analysis of directed, edge-colored
graphs and their congruences. In particular, a quantity
which appears "out of the blue" and has considerable
importance in the proof is what I call the "loop
characteristic" of a such a graph, very similar in nature to
the classical Euler characteristic.
Wednesday, April 18, 3:00 - 4:00, Jeff 422
Benoit Collins, U. of Ottawa
A linearization theorem for non-commutative probability
spaces and applications to Connes embeddability conjecture
Abstract: We prove a von Neumann non-commutative probability
space analogue of a C*-algebra linearization trick by
Haagerup-Thorbjornsen. We apply it to derive a new
equivalent statement of Connes embeddability conjecture. It
is in terms of possible spectral measures of sum of some
selfadjoint elements with matrix coefficients. This is
joint work with Ken Dykema.
Wednesday, April 18, 4:30 - 5:30, Jeff 422
Jonathan Novak, Queen's
Truncations of Random Unitary Matrices and Shifted Schur
Functions
ABSTRACT: We investigate the distribution of the trace of
square submatrices of a random unitary matrix. Using the
so-called "Colour-Flavour Transformation" from Lattice Gauge
Theory we prove that the moments of this distribution are
proportional to the number of pairs of skew Young tableaux
of certain shapes. This allows us to prove that the moments
of the trace enumerate nonintersecting walks on the integer
lattice. We also show a connection with the shifted Schur
functions introduced by Okounkov and Olshanski.
Wednesday, April 4, 4:30 - 6:00, Jeff 115
Jamie Mingo (Queen's)
Second Order Cumulants of Products
Second order cumulants do for fluctuations what first order
cumulants do for moments. An important tool in the analysis
of classical cumulants is the formula of Shiryaev and Leonov
(1969) which expresses the cumulants of products in terms of
cumulants of the factors. In 2000 Krawczyk and Speicher gave
the free version of this formula. In this talk we will show
how the formula can be extended to second order cumulants.
This is joint work with Roland Speicher and Edward Tan.
Wednesday March 21 4:30 - 6:00, Jeff 115
Amna Grgar (Queen's)
The Mobius function of Non-crossing partitions.
Abstract: The Mobius function is one of the important
functions in mathematics, specifically in free probability
theory. I will give an introduction to the Mobius function
and I will show how to compute the Mobius function of
non-crossing partitions.
Wednesday March 7, 4;30 - 6:00, Jeff 115
Roland Speicher (Queen's)
Second order freeness and fluctuations of random matrices
Abstract: I will motivate the notion of second and higher
order freeness and give a survey on the main results of that
theory.
Wednesday February 28 4:30 - 6:00, Jeff 115
Maria Grazia Viola, Queen's
The Free Central Limit Theorem
Abstract: The concept of freeness in free probability
replaces the concept of independence in classical
probability. Many of the results in classical probability
can be generalized to the free independence case. An
example of such a generalization is the central limit
theorem. We will show how prove this result, whose original
proof is due to Voiculescu. We will emphasize as in free
probability non-crossing pairing partitions replace the
general set of pairings used in classical probability.
Wednesday February 14 4:30 - 6:00, Jeff 115
Benoit Collins, University of Ottawa
Convergence of unitary matrix integrals
ABSTRACT: In this joint work with A. Guionnet and
E. Maurel-Segala, we introduce the Schwinger-Dyson equation
associated to a potential V on the set of tracial states of
the free *-algebra generated by X_1=X_1^*, ...,
X_n=X_n^*. If the potential is zero, the solutions of this
equation are free product states. We prove that, for a
prescribed spectral measure of the X_i's and for V small
enough, the solution of this equation exists, is still
unique and even analytic in V. Our techniques come from
random matrix theory. As a by-product, we provide new
examples of Connes-embeddable von Neumann
algebras. Moreover, we solve the problem of convergence of
unitary matrix integrals, and obtain new results on free
entropy.
Wednesday February 7, 4:30 - 6:00, Jeff 115
Jonathan Novak, Queen's
Truncations of random unitary matrices, Young
tableuax, and the Colour Flavour Transformation.
ABSTRACT: We will extend some known power series expansions
for unitary matrix integrals to the case where the matrices
in the integrand have been "truncated." The coefficients in
these power series expansions turn out to be quantities
related to Young Tableaux and Schur functions, and our
method relies heavily on tools from the representation
theory of the symmetric groups. We also make use of the
Weingarten expansion formula, and the "Colour-Flavour
Transformation" of lattice gauge theory. If time permits,
we will discuss how the Colour-Flavour Transformation is a
manifestation of the algebraic concept of Howe Duality.
Wednesday, January 31, 4:30 - 6:00, Jeff 115
Michael Neagu, Queen's
Asymptotic freeness of random permutation matrices with
restricted cycle lengths, Part II of II
Wednesday, January 24, 4:30 - 6:00, Jeff 115
Michael Neagu, Queen's
Asymptotic freeness of random permutation matrices with
restricted cycle lengths, Part I of II
ABSTRACT: Almost from the initial development of free
probability theory, it was shown by Voiculescu that, in many
important cases, independent families of NxN random matrices
are asymptotically *-free in the large N limit.
In my talks, I will present another instance of this
asymptotic freeness phenomenon, namely that of an
independent family of uniformly distributed random NxN
permutation matrices with cycle lengths restricted to fixed
sets of positive integers (sets which are either finite or
``large'' in a sense which I will make precise). I will also
show that this family is asymptotically *-free from various
types of independent families of Gaussian matrices and that
in all cases, under an additional assumption, the
convergence in *-distribution actually holds almost surely.
Wednesday, January 17, 4:30 - 6:00, Jeff 115
Jamie Mingo, Queen's
Free Cumulants and the R-transform, Part II
Abstract: Voiculescu's R-transform does for free
independence what the the logarithm of the Fourier transform
does for ordinary independence of random variables. The
R-transform is a compositional inverse of the Cauchy or
Stieltjes transform (up to a small adjustment). I will
present both the analytical and combinatorial approach to
the R-transform.
Schedule for Fall 2006
Schedule for Winter 2006
Schedule for Fall 2005
Schedule for Winter 2005
Schedule for Fall 2004
Schedule for Winter 2004
Schedule for Fall 2003