The students who participate in the seminar give talks on functional analytic aspects of noncommutative geometry. Talks devoted to noncommutative algebraic geometry and to "pure" functional analysis (preferably with an algebraic or geometric flavour) are also welcome. The topics of talks are usually taken from the literature, but sometimes the participants present their own results. Occasionally, talks are given by the seminar advisor or by an invited speaker.
The seminar takes place on Thursdays, 17:00-18:20, Faculty of Mathematics, HSE (Usacheva, 6), room 210.
If you wish to give a talk, please send you suggestions to the seminar advisor (pirkosha at gmail dot com).
Abstract. The talk is based on a recent preprint by Jared T. White. A Banach algebra A is said to be topologically left Noetherian if each closed left ideal of A is topologically generated by finitely many elements. We give a necessary and sufficient condition for the group L1-algebra of a compact group to be topologicaly left Noetherian in terms of the topology on the group. We also characterize topologically finitely-generated ideals in the algebra of approximable operators on a Banach space, and we present some further results on this algebra..
Abstract. The Schwartz algebra of rapidly decreasing functions on Rn has homological dimension n (a result by Ogneva and Helemskii, 1980s). I will try to explain what the previous sentence means and why people calculate such things, and how.
Abstract. Starting with some quantum group one can define a non-commutative complex geometry formalism. This setting allows one to define Hermitian and Kähler structures, Lefschetz, Hodge and Laplace operators and to prove Hodge and Lefschetz decompositions and the hard Lefschetz theorem. I am going to give an overview of the topic after a recent preprint of Réamonn Ó Buachalla.
Abstract. The goal of the talk is to apply methods of derived categories to multivariable spectral theory of linear operators. More specifically, I will extend J. L. Taylor's holomorphic functional calculus theorem (1970) to the setting of derived categories. I believe that this is exactly the environment in which Taylor's theorem is most naturally formulated and proved.
Abstract. The Krylov and Bogolyubov theorem on the existence of an f-invariant measure could give an application to the ergodic theory of dynamical systems, whose manifold is often given by physical equations (and possibly, assumptions), and which is often smooth in physical contexts. In this talk I will try to prove a theorem due to D. Ruelle which states the existence of an f-stable manifold. Also, I will try to introduce the idea of perturbation which gives a justification to numerical methods in relevant areas of physics (many-body problems).
Abstract. The algebra ℱ(Bn) of holomorphic functions on the free ball first appeared in G. Popescu's paper "Free holomorphic functions on the unit ball of B(H)n". In the same paper, Popescu introduced a "universal" (or, rather, a pseudouniversal) property of this algebra, which is in a sense a generalization of the universal property of the "usual" algebra of holomorphic functions on the ball in Cn. Unfortunately, the above-mentioned "universal" property of ℱ(Bn) cannot be formulated without using adjoint operators on Hilbert spaces, and this property is not universal from the category-theoretic point of view.
I will talk about Popescu's algebra ℱ(Bn), about its pseudouniversal property, and will explain why this algebra is worth studying. Then I will try to discuss another algebra which has a "genuine" universal property and which is also a reasonable candidate for an algebra of holomorphic functions on the free ball. Unfortunately, this algebra (in contrast to Popescu's algebra) has no explicit description so far.
Abstract. Following a paper by Davidson and Popescu, we consider noncommutative disc algebras constructed out of free products of discrete subsemigroups of R+, and we discuss the corresponding generalized Cuntz algebras. We show how to reconstruct the discrete semigroup from the generalized Cuntz algbera. We also prove a dilation theorem for representations of discrete semigroups, which yields an analog of the von Neumann inequality.
Abstract. Let G be a compact abelian group. For a given function f on G one can easily measure how close f is to a constant function via standard deviation. Furthermore, the Lp-norms of the Fourier transform of f provide us with information about correlations between f and characters of G. The sequence of Gowers norms can be used to generalize these observations to certain functions of higher order (for example, complex exponents with polynomial phase). We are going to discuss some properties of Gowers norms and their applications, such as the famous Green-Tao theorem on arithmetic progressions of primes.
Abstract. A well-known theorem states that an action of an operator T on a Banach space can be extended to an action of holomorphic functions in a neighborhood of the spectrum of T. In 1970, J.L. Taylor introduced the joint spectrum of an n-tuple of commuting operators and proved the existence of a holomorphic calculus in a neighborhood of this spectrum. Let us observe that an n-tuple of commuting operators is the same thing as a representation of an n-dimensional abelian Lie algebra. We generalize Taylor's spectrum to an arbitrary Lie algebra g in terms of homology, and we show that, if g is either semisimple or nilpotent, then the Taylor spectrum of any finite-dimensional g-module coincides with the set of isomorphism classes of its simple submodules. On the other hand, we give an example showing that this is not always true for a solvable Lie algebra. Thus the Taylor spectrum is a new invariant of Lie algebra representations, and it is worth studying.
Abstract. The classical Poincaré theorem (1907) asserts that the polydisk and the ball in Cn are not biholomorphically equivalent for n>1. Our goal is to prove a noncommutative (or, more exactly, "quantum") version of this result. The proof makes heavy use of the joint spectral radius, which was advertised in B.Nazarov's talk a week ago.
Abstract. The joint spectral radius of a set of linear operators first appeared in the joint work of G.-C. Rota and G. Strang (1960) in the study of properties of normed algebras. In the same year, independently of the first two authors, Furstenberg and Kirsten rediscovered this concept, having eventually transformed it into the Lyapunov exponent.
The joint spectral radius has found many applications in mathematical physics, wavelet theory, combinatorics, formal language theory, and so on.
