ABSTRACT. This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If a1,..., ak are ∗-free R-diagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −nt ai1 · · · ain defines a dilation semigroup on the non-self-adjoint operator algebra generated by a1,..., ak. We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1,..., ak. Moreover, we show that Dt satisfies an optimal ultracontractive property: �Dt: L 2 → L ∞ � ∼ t −1 for small t> 0. 1.

Abstract. A notion of unique ergodicity relative to the fixed–point subalgebra is defined for automorphisms of unital C ∗ –algebras. It is proved that the free shift on any reduced amalgamated free product C ∗ –algebra is uniquely ergodic relative to its fixed–point subalgebra, as are autormorphisms of reduced group C ∗ –algebras arising from certain automorphisms of groups. A generalization of Haagerup’s inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C ∗ –algebras is proved. 1.

ABSTRACT. We consider the resolvent (λ − a) −1 of any R-diagonal operator a in a II1-factor. Our main theorem (Theorem 1.1) gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the R-transform of the operator |λ − c | 2 where c is Voiculescu’s circular operator, and give an asymptotic formula for the negative moments of |λ − a | 2 for any R-diagonal a. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce partition structure diagrams in Section 4, a new combinatorial structure arising in free probability. 1.

ABSTRACT. A non-crossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings 1 n1 0 m1

free probability theory; functional inequalities such as logarithmic Sobolev inequalities and Haagerup inequalities in spaces of regular functions; stochastic analysis; and enumerative combinatorics of set partitions. I have three published papers, three submitted for publication, and many other projects in progress. I was awarded an NSF grant DMS-0701162 (2007-2010). I have co-authors and collaborators at institutions in the U.S., Canada, the U.K., France, and Poland. I have had a measure of experience supervising research: in 2007, I ran a very successful Research Experience for Undergraduates project, which has produced two papers whose results are described below; in addition, I am currently supervising a PhD student at MIT, whose work is in random matrix theory and combinatorics. 1. OVERVIEW A unifying theme to most of my work is Gaussian structures. Recall that the standard normal law (Gaussian distribution) is given by γ(dx) = (2π) −1/2 exp(−x 2 /2)dx. By Gaussian structures I mean mathematical objects that owe their properties to some (apparent or hidden) connection with the measure γ. Most of my work has to do with either L p-spaces of Gaussian random variables or Gaussian-type random matrices. For example, the following kinds of objects interest me:

Abstract. In this paper, we are interested in sequences of q-tuple of N ×N random matrices having a strong limiting distribution (i.e. given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the q-tuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in non-commutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjørnsen. We also show that ap-tuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.

We derive a formula for the moments and the free cumulants of the multiplication of k free random variables in terms of k-equal and k-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution µ ⊠k, given by Kargin in [5], which show that the support grows at most linearly with k. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida [11]. Keywords: Free probability; Free multiplicative convolution; Non-crossing partitions.

Abstract. We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, Hd denotes the subspace of the von Neumann algebra of a free group FI spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on Mn(Hd), which improves and generalizes previous results by Kemp-Speicher (in the scalar case) and Buchholz and Parcet-Pisier (in the non-holomorphic setting). Namely the norm of an element of the form P i=(i1,...,i d) ai ⊗ λ(gi1... gi d) is less than 4 5 √ e(‖M0 ‖ 2 + · · · + ‖Md ‖ 2) 1/2, where M0,..., Md are d+1 different block-matrices naturally constructed from the family (ai) i∈I d for each decomposition of I d ≃ I l ×I d−l with l = 0,..., d. It is also proved that the same inequality holds for the norms in the associated non-commutative Lp spaces when p is an even integer, p ≥ d and when the generators of the free group are more generally replaced by ∗-free R-diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.

ABSTRACT. This paper contains sharp estimates for the small-time behaviour of a natural class of one-parameter semigroups in free probability theory. We prove that the free Ornstein-Uhlenbeck semigroup Ut, when restricted to the free Segal-Bargmann (holomorphic) space H0 introduced in [K] and [Bi1], is ultracontractive with optimal bound ‖Ut: H 2 0 → H ∞ 0 ‖ ∼ t −1. This was shown, as an upper bound, in [KS]; the lower bound is our main theorem here. These results are extended to a large class of non-commutative holomorphic spaces generated by R-diagonal operators in a W ∗-probability space. A surprising corollary is the fact that these holomorphic spaces (including H0) are not complex interpolation scale (even in the finite-rank setting), contra to their commutative analogues. 1.

Abstract. Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL(2,C) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory free stochastic measures this provides a combinatorial proof of the Molien-Weyl formula in this setting. Invariant theory has played a major role in 19th century mathematics. It has seen a revival in the last decades and one of the recent generalizations is noncommutative invariant theory. The study of noncommutative invariants of SL(n,C) has been initiated by Almkvist, Dicks, Formanek and Kharchenko [6, 5, 2], see [1] for a survey. An approach using Young tableaux was realized by Teranishi [16] and the symbolic method was adapted from the classical to the noncommutative setting by Tambour [15]. The latter provides the ground on which we establish a natural basis of the noncommutative invariants which is in bijection with certain noncrossing partitions. It arose after computer experiments and subsequent consulting of Sloane’s database [13]. This bijection is applied to provide a