Programme, including abstracts

Monday 25th May

• 13:15-14:15: Bases in Banach spaces: an introductory lecture for graduate students by Andras Zsak.
• 14:30-15:30 and 16:00-17:00: NBFAS lectures by Stephen J. Dilworth (South Carolina, USA), Convergence of some greedy algorithms in Banach spaces.
  Abstract: Let D be a fundamental set in a Banach space X. Greedy algorithms provide an intuitively appealing method for approximating a given vector by a linear combination of n vectors belonging to D. The convergence of one such "Pure Greedy Algorithm" in Hilbert space is well understood. We consider some natural generalizations of this Hilbert space algorithm to the Banach space setting and examine their convergence properties with respect to either the norm or the weak topologies. We also consider the important case in which D is a Schauder basis for X, so that each vector in X has a series representation with respect to the basis. Here the most natural approximation method is to select the n largest basis coefficients in absolute value. We shall consider the convergence properties of this "Thresholding Greedy Algorithm" and discuss the existence of bases for which algorithms of this type are effective.

Tuesday 26th May

• 9:00-10:00: Edward W. Odell (Texas, USA), Embedding Theorems in Banach spaces.
  Abstract: From Banach's book we know that every separable Banach space embeds isometrically into C(0,1). Since then many more embedding theorems have been discovered. We will survey a number of these results, generally of one of two types of problems.
  A) Given a separable Banach space X with a certain property, does it embed into a space Y with better structure (e.g., a basis) and the same or a closely related property?
  B) Given a class of separable Banach spaces C, is there a member Y of C (or a related class) which is universal for C?
  The solution to A) often leads to a solution to B).
• 10:05-11:05: Thomas Schlumprecht (Texas A&M, USA), The universality of l₁ as a dual space.
  Abstract: Let X be a Banach space with a separable dual. Using a method of Bourgain and Delbaen, we prove that X embeds isomorphically into an L_infinity space Z whose dual is isomorphic to l₁.
  If X has a shrinking finite dimensional decomposition and X^* does not contain an isomorph of l₁, then we construct such a Z, as above, not containing an isomorph of c₀. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.
• 11:05-11:30: Coffee.
• 11:30-12:30: Rafal Latala (Warsaw, Poland), Khintchine, Kahane and related inequalities.
  Abstract: The Khintchine inequality states that L^p norms of linear combinations of Rademachers are comparable. Kahane extended this inequality to the case of vector coefficients. We will discuss these and related inequalities, putting emphasis on the best constants and various methods of proofs.
• 12:30-13:30: Lunch.
• 13:30-14:30: Timothy Feeman (Villanova, USA), The Secret Lives of Power Functions.
  Abstract: We explore some not so well known geometric features of a familiar family of functions.
• 14:35-15:35: Charles J. Read (Leeds), Weak amenability and irregular commutative semigroups.
  Abstract: It is well known that if a semigroup is strongly regular then the algebra l¹(S) is weakly amenable. But this is not necessary. In a joint paper with H. Ghlaio, we show that there are commutative irregular semigroups S such that l¹(S) is weakly amenable. This generalises a result of H.G. Dales that there are (noncommutative) semigroups which are not strongly regular, but l¹(S) is weakly amenable. The counterexample involves learning a bit about commutative semigroups; the examples are unitizations of nil semigroups, and they are 0-cancellative.
• 15:40-16:40: Richard Haydon (Oxford).