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.
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Monday, February
16, 1530-1630.
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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.
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Monday, February
23, 1530-1630.
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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.
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Monday, March
9, 1530-1630
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SPEAKER :
Sophie Grivaux, Universite de Lille.
TITLE : About Read's counterexamples
to the Invariant Subspace problem.
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.
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Monday, May
11, 1530-1630
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CANCELLED !! (This talk has unfortunately had to be cancelled)
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Monday, May
18, 1530-1630
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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.
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Monday, May
25, 1530-1630
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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.