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| Date |
Speaker |
Title |
| October 15 | Dr. James D.E. Grant | Null injectivity radius estimates |
| October 22 | Dr. Shantanu Dave | Regularity via Hilbert transforms |
| October 15 | cancelled | |
| October 29 | Vera Ganglberger | The kernel theorem and microlocal analysis for distributions on manifolds |
| November 5 | Prof. Peter Wagner | Distributions supported by hypersurfaces |
| November 12 | Dr. Paolo Giordano | Infinitesimals without Logic AbstractMathematicians quarrelled for generations about the
existence of infinitesimals before the appearing of several
rigorous theories: NSA, synthetic differential geometry,
surreal numbers, Weil functors, Levi-Civita fields... All of
them are full either of mathematical logic and foundational
problems or follow pure algebraic techniques, not always having
a clear intuitive interpretation. Is it possible to define a
ring of infinitesimals using only first-year elementary
calculus? We will see that a very simple and intuitive approach
can answer positively to this question and that its properties
are surprisingly powerful, opening the possibility to develop
an intrinsic differential geometry both for ordinary manifold
and infinite dimensional spaces. For more information look for
"Fermat reals" (with quotation marks) in some search engine. |
| November 19 | Mag. Martina Glogowatz | Factorization of second-order strictly hyperbolic operators with non-smooth coefficients and microlocal diagonalization AbstractWe will start with a motivation for this topic give
a brief introduction in one-way wave propagation in
inhomogeneous acoustic media with smooth background data.
In the following we will remove the smoothness conditions of the operator.
Working in spaces of Colombeau generalized functions
we will then state a factorization procedure and a microlocal diagonalization. |
| November 26 | | Four special short lectures |
| December 3 | Dipl.-Ing. Eduard Nigsch | Thesis defence: A nonlinear theory of generalized tensor fields on Riemannian manifolds |
| December 10 | Prof. Roland Steinbauer | Approaches to Differential Operators AbstractSome talks of last year’s Diana seminar have focused on the algebraic approach to partial differential operators. There one defines a PDO as a linear map on smooth functions or sections that interacts appropriately with the $C^\infty$-module structure, a condition which can be expressed by the vanishing of certain commutator brackets. This approach is very general as it allows one to define PDOs on modules over a commutative algebra (and even allows one to see PDOs as a concept of commutative algebra). In particular, it is possible to define PDOs on Colombeau spaces in an algebraic way and this is where my interest lies.
In this talk, however, I will rather stick to some basics and discuss in detail the equivalence of several approaches to PDOs on open subsets of Euclidean space. If time permits I will also sketch the basics of the algebraic theory of PDOs and make some preliminary remarks on algebraic PDOs on $\mathcal D′$ and $\mathcal G$, which I find surprising.
For the first part of the talk I have prepared some LaTeX notes which I attach. These are essentially a modification and extension of some parts of my lecture course at this year’s Novi Sad summer school. The corresponding notes may be found at: |
| December 17 | Ivana Vojnovic and Milena Stojkovic | New proofs of the uniform boundedness principle and the Schauder theorem |
| January 14 | Jan-Hendrik Treude | Riemannian volume comparison AbstractIt is a fundamental question in modern differential geometry to understand the curvature tensor of a Riemannian manifold. One approach to this question is to compare the behaviour of certain geometric quantities on an arbitrary Riemannian manifold to the behaviour of these quantities on another Riemannian manifold which one understands very well, e.g. Euclidean space, the sphere or hyperbolic space.
In my talk, I want to explain a statement from this area, which is known as "Bishop's Volume Comparison Theorem". Here one assumes that a Riemannian manifold obeys a lower bound on the Ricci curvature, and compares the volumes |
| of geodesic balls to the volumes of geodesic balls of the same radius in a Riemannian manifold of constant curvature, where the lower bound on the Ricci curvature is sharp. | | |
| January 21 | Matthias Winter | Strongly continuous evolution systems |