Autumn 2008 schedule

# MathematicsDepartmentColloquium : Spring 2009 schedule

 Monday, January 26, 1530-1630.

SPEAKER : Tony Dorlas, DIAS, Ireland.

TITLE : Quantum channels with long-term memory.

ABSTRACT : In this talk, I introduce the concept of a quantum channel for transmitting classical information. First I explain how information can in principle be transmitted securely over a quantum channel. Then I discuss Schor's example of an error correcting code and the analogue of Shannon's theorem for the optimal rate of faithful transmission of information. For a memoryless channel, this is known as the HSW theorem, but I shall also consider channels with memory.
 Monday, February 16, 1530-1630.

SPEAKER : Peter Hansbo, Chalmers (OBS! This is Peter's inaugural lecture for his promotion to professor in Computational Engineering)

TITLE : Discontinuous finite element methods : useless or what ?

ABSTRACT : Discontinuous Galerkin (DG) methods have seen a huge boost in popularity over the last few years. But of what use are they ? Discontinuous approximations do not increase the approximation power compared to continuous approximations of the same polynomial degree, but they do introduce a large number of extra unknowns. In this talk I will give my view of the possible benefits of DG methods with applications in solid mechanics, fluid mechanics, and electromagnetics.

 Monday, February 23, 1530-1630.

SPEAKER : Milagros Izquierdo Barrios, Linköping.

TITLE : Art and mathematics : The Möbius band.

ABSTRACT : A Möbius band is the mathematical object obtained when pasting the two extremes of a piece of paper after twisting one extreme up side down. The Möbius band has inspired many artists during the 20th century. It evokes eternity and interplay. In the talk we use the Möbius band in two ways: as a mathematical tool to see art and as an artistic tool to illustrate mathematical concepts.

 Monday, March 9, 1530-1630

SPEAKER : Sophie Grivaux, Universite de Lille.

ABSTRACT : If X is a separable infinite-dimensional Banach space, and T is a bounded operator on X, the Invariant Subspace (Subset) Problem is to know whether there exists a closed subspace (subset) M of X distinct from 0 and X which is invariant by T. The Invariant Subspace Problem has been answered in the negative by Enflo and Read. Read has been able to produce counterexamples on classical Banach spaces such as l_1, but the question is still open when X is a reflexive Banach space, in particular when X is a Hilbert space. We will describe in an elementary way the main features of a "Read's type operator", and introduce and discuss some Hilbertian versions of these operators which have few invariant closed sets.

 Monday, May 11, 1530-1630

CANCELLED !! (This talk has unfortunately had to be cancelled)

 Monday, May 18, 1530-1630

SPEAKER : Michael Klibanov, UNC Charlotte.

TITLE : Carleman estimates for theory and numerical estimates for ill-posed and inverse problems.

ABSTRACT : In 1939 Swedish mathematician Torsio Carleman in an 8 page paper has proposed a monumental and very elegant technique, which carries his name since then. His motivation was to avoid the analiticity condition of the Holmgren uniqueness theorem. This technique was forgotten for 19 years until A. Calderon (1958) has published another paper. Since then the method of Carleman estimates plays a fundamental role in the PDE theory. It works for the so-called unique continuation theorems for ill-posed Cauchy problems. Some examples are Cauchy problem for an elliptic equation, and Cauchy problems for parabolic and hyperbolic equations with the Cauchy data on a part of the lateral boundary. In 1981 Bukhgeim and Klibanov simultaneously and independently have proposed to modify Carleman estimates for proofs of unqiueness theorems for multidimensional coefficient inverse problems (MCIPs), since such theorems were proven only under some kind of analiticity or smallness conditions before that. Later the Klibanov and Malinsky have introduced Carleman estimates in control theory (1991) for proofs of the Lipschitz stability for hyperbolic equations and inequalities with the Cauchy data at the lateral boundary.

In this talk we will present some ideas of Carleman estimates for: (1) uniqueness for MCIPs, (2) Lipschitz stability for the hyperbolic case, (3) convergence analysis for the quasi-reversibility numerical method for above ill-posed Cauchy problems and (4) a globally convergent numerical method for some MCIPs (#4 is a joint work with Dr. L. Beilina, Chalmers University of Technology and Gothenburg University). Note that the quasi-reversibility method was first proposed by R. Lattes and J.-L. Lions in 1969, and they have proved convergence. However, the rate of convergence was not established and this is what will be presented.

 Monday, May 25, 1530-1630

SPEAKER : Alan Sokal, NYU and University College London.

TITLE : Complete monotonicity for inverse powers of some combinatorially defined polynomials.

ABSTRACT : If $P$ is a univariate or multivariate polynomial with real coefficients and strictly positive constant term, and $\beta$ is a positive real number, it is sometimes of interest to know whether $P^{-\beta}$ has all nonnegative Taylor coefficients. Problems of this type go back at least to a celebrated paper of Szeg\H{o} (1933). In this talk I give a combinatorial interpretation of Szeg\H{o}'s result and then generalize it to a statement about complete monotonicity. I go on to give two sufficient conditions for complete monotonicity of inverse powers of polynomials: one applying to determinantal polynomials (including the spanning-tree polynomials of graphs and, more generally, the basis generating polynomials of regular or complex unimodular matroids), and the other applying to quadratic forms (including the basis generating polynomials of rank-2 matroids). Finally, I discuss the relation with the half-plane property, and mention some open questions. This is joint work with Alex Scott.