Department of Mathematical Sciences

**Thursday, February 1, 2001 (at 3 pm in CAB 377)**: Robert V. Moody.*Title*: Almost periodic measures and pure point diffraction, I.*Abstract*: Diffraction from point sets in*R*has become a particularly interesting topic since the discovery of quasicrystals. The trouble is that the necessary and sufficient conditions for pure point diffraction are not known, even though we have some good mechanisms (namely the cut and project method) for producing interesting (aperiodic) sets of this type. This pair of talks addresses this problem. The diffraction is described in terms of the Fourier transform of the auto-correlation measure arising from the point set itself. The main idea is to connect the notion of almost-periodicity to the cut and project method. The result is to give a new and more general result on pure point diffraction. The setting necessarily ends up in the domain of harmonic analysis on locally compact Abelian groups, and we will use this as our setting for these talks.^{n}- February 6, 2001: Robert V. Moody.
*Title*: Almost periodic measures and pure point diffraction, II. - No seminar this weak, but the following (job) talk is of interest:
**Thursday, February 15, 2001 (at 3:30 pm in CAB 657)**: Alexander Litvak (Technion; Israel).*Title*: Asymptotic theory of convex bodies.*Abstract*: From the point of view of geometry, geometric functional analysis studies centrally symmetric convex bodies with non-empty interior. However, the requirement that the body should be centrally symmetric is not natural in geometry. In this talk, we discuss the asymptotic behavior of various parameters of high-dimensional convex bodies and their projections as the dimensions tend to infinity. We compare the properties of centrally symmetric convex bodies with those of non-symmetric convex bodies. Surprisingly, it turns out that the behavior of a "random" projection is the same in both the symmetric and the non-symmetric case, despite the fact that in general the projected body may be very far from being symmetric. We also present some recent results on norms of sequences of random variables, which have applications in geometric functional analysis and are of independent interest in probability as well.**Coffee, cookies, and doughnuts will be served at 3:15 pm in CAB 649.** **Thursday, February 22, 2001 (at 3 pm in CAB 657)**: Zhiguo Hu (University of Windsor).*Title*: The support of operators in*VN(G)*and a decomposition of*A(G)*.*Abstract*: Let*G*be a locally compact group, and let*VN(G)*be the von Neumann algebra generated by the left regular representation of*G*. It is known that if an element*T*of*VN(G)*is the left convolution operator by a bounded, complex Borel measure on*G*, then the support of*T*can be covered by countably many compact subsets of*G*. Suppose that*G*is non-discrete and that*H*is an open subgroup of*G*. In this talk, we will discuss, for an arbitrary operator*T*in*VN(G)*, how many cosets*gH*we need at least to cover the support of*T*. As an application of our estimate, we obtain a decomposition of*VN(G)*which leads to a useful decomposition for the Fourier algebra*A(G)*.- No seminar again, but another interesting job talk:
**Tuesday, February 27, 2001 (at 3:30 pm in CAB 657)**: Roman Vershynin (Weizmann Institute of Science; Israel).*Title*: Coordinate restrictions of operators: convexity meets harmonic analysis and signal processing.*Abstract*: Properties of a linear operator*T*in*R*can generally be improved by restricting^{n}*T*onto a suitable coordinate subspace of a controllable dimension. A typical example of this situation arises in harmonic analysis, where the coordinate vectors are given by the characters of a group. We will discuss a recent extension of the so-called ``principle of restricted invertibility'' due to J. Bourgain and L. Tzafriri. We obtain a subsapce of optimal dimension where*T*is a nice isomorphism. This yields an optimal solution to the problem of harmonic density going back to W. Schachermeyer. Another important application is to finite-dimensional convextiy, where the coordinate vectors come from the remarkable John's decomposition. If the coordinate subspace is chosen at randon, the norm of*T*can be reduced to nearly an optimal value. This is proved using noncommutative theory methods by G. Pisier. We will see how this result implies a solution to an open problem in signal processing (recovering random data losses). A common feature shared by all these results is that the dimension*n*plays no role there, and is replaced by the Hilbert-Schmidt norm of*T*.**Coffee, cookies, and doughnuts will be served at 3:15 pm in CAB 649.** - And yet another interesting job talk...
**Thursday, March 1, 2001 (at 3:30 pm in CAB 657)**: Ioannis Gasparis (Oklahoma State University; USA)*Title*: A continuum of totally incomparable, hereditarily indecomposable Banach spaces.*Abstract*: A Banach space*X*is said to be hereditarily indecomposable if no infinite-dimensional subspace of*X*can be decomposed into the direct sum of two further infinite-dimensional closed subspaces. Gowers and Maurey solved the famous "unconditional basic sequence problem" by constructing the first example of a hereditarily indecomposable Banch space. In this talk, we shall present a method of constructing asymptotic*l*, reflexive, hereditarily indecomposable Banach spaces. The definition of the norm in such spaces involves certain sets of finitely supported signed measures on the set of positive integers. In order to describe those sets of measures, we apply techniques from descriptive set theory such as trees, Schreier families, and Ramsey type theorems. This method allows us to exhibit a family of cardinality equal to the continuum, whose memebers are totally incomparable, reflexive, hereditarily indecomposable Banach spaces._{1}**Coffee, cookies, and doughnuts will be served at 3:15 pm in CAB 649.** - March 6, 2001: Vaclav Zizler (Academy of Sciences of the Czech Republic).
*Title*: Optimal injections of dual balls into cubes.*Abstract*: We will discuss various types of optimal injections of weak star dual balls into cubes, starting from nonlinear injections into the sets of countably supported elements and ending with linear injections into Hilbert spaces. These properties are shown to be equivalent to various types of smoothness of spaces. Some applications in the theory of uniform Eberlein compacts will be shown. Several open problems in this area will be discussed. We will use a few recent joint papers by M. Fabian, G. Godefroy, P. Hájek, V. Montesinos, and V. Zizler. The presentation will be such that the talk will be easily accessible for graduate students in functional analysis and related fields. - Not a seminar talk, but an interesting colloquium:
**Thursday, March 8, 2001 (at 3:30 pm in CAB 657)**: Eberhard Kaniuth (Universität Paderborn; Germany)*Title*: Primitive ideal spaces of group algebras of nilpotent discrete groups.*Abstract*: For a discrete group*G*, let*l*denote the convolution algebra of absolutely summable, complex valued functions on^{1}(G)*G*and*C*the enveloping^{*}(G)*C*-algebra of^{*}*l*. The primitive ideal space, Prim(^{1}(G)*C*, of^{*}(G))*C*is the set of all kernels of irreducible representations of^{*}(G)*C*, endowed with the hull-kernel topology. The talk will mainly focus on nilpotent groups and survey topics such as the description of primitive ideals in terms of characters of^{*}(G)*G*and topological properties of Prim(*C*. We shall also discuss the primitive ideal space of^{*}(G))*l*.^{1}(G)**Refreshments will be served at 3:00 pm in CAB 649.** - March 13, 2001: Günter Schlichting (Technische Universität München; Germany).
*Title*: On the structure of homogeneous Riemannian spaces.*Abstract*: We characterize the possible stability subgroups*H*of Lie groups*G*acting transitively on a Riemannian space by various compactness conditions on*H*. This applies, in particular, to orbits arising from unitary actions of*G*on Hilbert spaces. - March 20, 2001: Volker Runde.
*Title*: Connes-amenability and normal, virtual diagonals for measure algebras.*Abstract*: Let*G*be a locally compact group. It has been shown recently - by H. G. Dales, F. Ghahramani, and A. Ya. Helemskii - that the measure algebra*M(G)*is amenable if and only if*G*is discrete and amenable. As for von Neumann algebras, amenability seems to be too strong a notion to develop a reasonably rich theory for measure algebras. It turns out that (again as for von Neumann algebras) a modified notion of amenability - called*Connes-amenability*-, which takes the dual space structure into account, is the "right" notion of amenability to deal with measure algebras:*M(G)*is Connes-amenable if and only if*G*is amenable. - March 27, 2001: Ngai-Ching Wong (National Sun Yat-Sen University; Taiwan).
*Title*: Shifts on locally compact spaces.*Abstract*: Let*X*and*Y*be locally compact Hausdorff spaces, and let*C*and_{0}(X)*C*be the Banach spaces of continuous functions on_{0}(Y)*X*and*Y*vanishing at infinity, respectively. For a linear isometry*T*from*C*into_{0}(X)*C*of finite corank, we show that there is a cofinite subset_{0}(Y)*Y*of_{1}*Y*such that is a weighted composition operator and*Tf*o_{Y1}= h · f*X*is homeomorphic to a quotient space of*Y*modulo a finite subset. When_{1}*X*=*Y*, such a*T*is called an isometric quasi-*n*-shift on*C*. In this case, the action of_{0}(X)*T*can be implemented as a shift on a tree-like structure, called a*T*-tree, in*M(X)*with exactly*n*joints. The*T*-tree is total in*M(X)*when*T*is a shift. With this tools, we can analyze the structure of*T*. In particular, we give some partial answers to an open problem of Gutek*et. al.*asserting whether*X*is separable if*C*admits an isometric_{0}(X)*n*-shift. - April 10, 2001: Matthias Neufang.
*Title*: On Mazur's property and property (X).*Abstract*: We completely characterize those von Neumann algebras whose preduals have Mazur's property and show that, for preduals of von Neumann algebras, Mazur's property is equivalent to property (X) as introduced by Godefroy and Talagrand. We also generalize these two properties to the level of arbitrary cardinals and study these generalizations. As it turns out, property (X) of level for an arbitrary cardinal coincides with the original property (X) unless is measurable. We give several applications of our results to concrete spaces, such as*L*for a locally compact group^{1}(G)*G*and*T(H)*, the trace class operators on a Hilbert space*H*.

Last update: 4/11/2001.