This seminar is an informal forum where members of the DIANA group meet to discuss topics of interest. We meet on a weekly basis. The programme for these meetings will be advertised below, and by email.
If you wish to be added to (or removed from) our email list, please contact tobias.beran@univie.ac.at: subscribe or unsubscribe.
The the seminar takes place every Friday at 09:45 am in SE 07 and streamed via moodle and will be announced by email weekly.
Anyone interested is welcome to attend.
| Date | Speaker | Title |
|---|---|---|
| October 16 | Dr. James Grant | Introduction to integrable systems. |
| October 23 | No seminar. | |
| October 30 | Prof. Günther Hörmann | One or two exotic aspects from our recent work on Schrödinger-type equations. [Notes1], [Notes2] |
| November 6 | Simon Rössler | An algebraic way of introducing PDOs onmanifolds Part 1 |
| November 13 | Simon Rössler | An algebraic way of introducing PDOs onmanifolds Part 2 |
| November 20 | Simon Rössler | An algebraic way of introducing PDOs onmanifolds Part 3 |
| November 27 | Dr. Shantanu Dave | Characterising pseudo-differential operators |
| December 4 | No seminar. Defensio dissertationis of Alice Mikikits-Leitner takes place in Room C2.09 at 11am. | |
| December 11 | Alexander Bihlo | Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures |
| December 18 | Prof. Roland Steinbauer | Normally hyperbolic operators. AbstractBuilding on the global approach to PDOs recently discussed in a series of talks by Simon Rössler we present some basic facts on normally hyperbolic operators. These lurk in the background of work some of us have recently been doing on wave equations.For those who are interested I include a detailed summary: Normally hyperbolic operators are the Lorentzian anlogue of generalised Laplacians (10.1.23 in Simon’s talk): Let $E$ be a vector bundle over a Lorentzian manifold $(M, g)$. A second order PDO $P$ on $E$ is called normally hyperbolic if its principal symbol is given by (minus) the metric, more precisely we have $\sigma_2(P)(\xi) =−g(\xi, \xi)Id_E$. The most basic example of a normally hyperbolic operator is the metric d’Alembertian acting on $C^\infty(M)$ defined by $\Box g:=−div\circ grad$, which in local coordinates takes the form $\Box g=g^{ij}\partial_i \partial_j$ and in case of $M$ being Minkowski space is the usual wave operator. Another example is the connection d’Alembert or Bochner-Laplace operator: Let $\nabla$ be a connection on $E$. Together with the Levi-Civita conection on $M$ it induces a connection on $T^*M\otimes E$ again denoted by $\nabla$. We then define $\Box^\nabla:=−trace_g\otimes Id_E(\nabla\circ\nabla)$. We will prove the Weitzeböck formula which says that up to an order zero term every normally hyperbolic operator is the connection d’Alembertian for a suitable connectionon $E$. |
| January 8 | David Rottensteiner | Foundations of Harmonic Analysis on the Heisenberg Group. AbstractThe Heisenberg group $\H^n$ is the "simplest" non-commutative Lie group and plays an important role in several branches of mathematics. After constructing the group and its Lie algebra we focus on the representations of $\H^n$. We will prove a famous theorem by Marshall Harvey Stone and John von Neumann, classifying the irreducible unitary representations of the Heisenberg group. Along the way we present concepts like the twisted convolution and integrated representations. We finally introduce the group Fourier transform for $\H^n$ and prove the Plancherel theorem, which establishes an isometric isomorphism between $L2(\H^n)$ and the space $L2(\R^*,HS(L2(\R^n)); \mu)$ of square-integrable functions from $\R^*$ into the space of Hilbert-Schmidt operators on $L2(\R^n)$. Here, $\mu$ is the so-called Plancherel measure. |
| January 15 | David Rottensteiner | Foundations of Harmonic Analysis on the Heisenberg Group. AbstractThe Heisenberg group $\H^n$ is the "simplest" non-commutative Lie group and plays an important role in several branches of mathematics. After constructing the group and its Lie algebra we focus on the representations of $\H^n$. We will prove a famous theorem by Marshall Harvey Stone and John von Neumann, classifying the irreducible unitary representations of the Heisenberg group. Along the way we present concepts like the twisted convolution and integrated representations. We finally introduce the group Fourier transform for $\H^n$ and prove the Plancherel theorem, which establishes an isometric isomorphism between $L2(\H^n)$ and the space $L2(\R^*,HS(L2(\R^n)); \mu)$ of square-integrable functions from $\R^*$ into the space of Hilbert-Schmidt operators on $L2(\R^n)$. Here, $\mu$ is the so-called Plancherel measure. |
| January 22 | Stefan Fürdös | Hörmander's necessary condition for local solvability of PDO. AbstractUntil the middle of the 20th century it was a general belief that linear PDE always have solutions---at least locally and for suitably smooth coefficients and right hand sides. However, Hans Lewy in 1957 discovered that the first order PDE $ L u = [-i D_1 + D_2 - 2(x_1+ix_2) D_3] u = f $ does not allow for a classical (more precisely: $C^1$ with Hölder continuous derivatives) solution in the neighbourhood of any point in $R^3$ for certain right hand sides $f$. It was Lars Hörmander in 1960 who provided an easily checkable and geometric, necessary condition for local solvability (with test functions as right hand sides and distributional solutions): The principal symbol $p$ of a solvable operator satisfies $\{p,\bar p\}=0$ on the zeroes of $p$, where $\{ , \}$ denotes the Poisson bracket. In this talk we present the main aspects of the long and technical proof. |
| January 29 | Stefan Fürdös | Hörmander's necessary condition for local solvability of PDO. AbstractUntil the middle of the 20th century it was a general belief that linear PDE always have solutions---at least locally and for suitably smooth coefficients and right hand sides. However, Hans Lewy in 1957 discovered that the first order PDE $ L u = [-i D_1 + D_2 - 2(x_1+ix_2) D_3] u = f $ does not allow for a classical (more precisely: $C^1$ with Hölder continuous derivatives) solution in the neighbourhood of any point in $R^3$ for certain right hand sides $f$. It was Lars Hörmander in 1960 who provided an easily checkable and geometric, necessary condition for local solvability (with test functions as right hand sides and distributional solutions): The principal symbol $p$ of a solvable operator satisfies $\{p,\bar p\}=0$ on the zeroes of $p$, where $\{ , \}$ denotes the Poisson bracket. In this talk we present the main aspects of the long and technical proof. |