Past Functional Analysis Seminar

4 March 2014
17:00
to
18:15
Andrew Morris
Abstract

We consider the layer potentials associated with operators $L=-\mathrm{div} A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent.  A ``Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi--Nash--Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.

  • Functional Analysis Seminar
18 February 2014
17:00
to
18:15
Tony Dooley
Abstract
Contractions of Lie groups have been used by physicists to understand how classical physics is the limit ``as the speed of light tends to infinity" of relativistic physics. It turns out that a contraction can be understood as an approximate homomorphism between two Lie algebras or Lie groups, and we can use this to transfer harmonic analysis from a group to its ``limit", finding relationships which generalise the traditional results that the Fourier transform on $\R$ is the limit of Fourier series on $\TT$. We can transfer $L^p$ estimates, solutions of differential operators, etc. The interesting limiting relationship between the representation theory of the groups involved can be understood geometrically via the Kirillov orbit method.
  • Functional Analysis Seminar
11 February 2014
17:00
to
18:15
Andreas Rosen
Abstract
We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A\in L_\infty, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.
  • Functional Analysis Seminar
4 February 2014
17:00
to
18:15
Chin Pin Wong
Abstract
An important aspect in the study of Kato's perturbation theorem for substochastic semi- groups is the study of the honesty of the perturbed semigroup, i.e. the consistency between the semigroup and the modelled system. In the study of Laplacians on graphs, there is a corresponding notion of stochastic completeness. This talk will demonstrate how the two notions coincide.
  • Functional Analysis Seminar
19 November 2013
17:00
to
18:30
Tomasz Kania
Abstract
We address the following two questions regarding the maximal left ideals of the Banach algebras B(E) of bounded operators acting on an infinite-dimensional Banach space E: i) Does B(E) always contain a maximal left ideal which is not finitely generated? ii) Is every finitely-generated maximal left ideal of B(E) necessarily of the form {T\in B(E): Tx = 0}? for some non-zero vector x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals mentioned above, a positive answer to the second question would imply a positive answer to the first. Our main results are: Question i) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; Question ii) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E); the answer to Question ii) is positive for many, but not all, Banach spaces. We also make some remarks on a more general conjecture that a unital Banach algebra is finite-dimensional whenever all its maximal left ideals are finitely generated; this is true for C*-algebras. This is based on a recent paper with H.G. Dales, T. Kochanek, P. Koszmider and N.J. Laustsen (Studia Mathematica, 2013) and work in progress with N.J. Laustsen.
  • Functional Analysis Seminar
12 November 2013
17:00
to
18:12
Martin Kolb
Abstract
We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.
  • Functional Analysis Seminar

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