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-mapsTuesday,

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), 161-176Tuesday,

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 214:30 - 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 284:30 - 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 144:30 - 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 IIWednesday,

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