# Winter term 2020/21

## 25. März 2021

* 16:15 -17:45 *(A&G)

**Prof. Richard Montgomery** (University of California, Santa Cruz)

**Four Open questions in the N-body problem**

*Abstract*: The classical N-body problem is alive and well. I begin with a pictorial survey of some of its solution curves. I then describe four open questions within the problem and recent progress on these questions.

(Contact: V. Matveev / M. Dafinger)

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 18. März 2021

*14:30-16:00 *(A&G)

**Prof. Emilio Musso** (Politecnico di Torino)

**The Cr-strain functional for Legendrian curves in the 3-dimensional Sphere**

*Abstract*: Let S^{3} be the unit 3-sphere with its standard Cauchy–Riemann (CR) structure. We consider the CR geometry of Legendrian curves in S3, thought of as homogeneous space of its CR-transformations group. More specifically, the focus is on the simplest cr-invariant variational problem for Legendrian curves and on its closed critical curves. The Liouville integrability of such a variational problem is considered. We discuss the admissible contact isotopy classes of closed critical curves with constant bending. Subsequently, we label closed critical curves with non-constant bending with three numerical invariants (quantum numbers). We exhibit that each critical curve can be explicitly reconstructed once that one knows these numerical invariants. We analyze the geometrical meaning of the quantum numbers.

(Contact: V. Matveev / M. Dafinger)

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 11. März 2021

*14:30-16:00 (A&G)*

**Prof. Boris Kruglikov** (UiT the Arctic University of Norway)

**Dispersionless integrability: different approaches and examples**

*Abstract*: I will discuss what is the integrability of dispersionless systems via hydrodynamic reductions, Lax pairs, Zakharov pairs, and background geometry. Several examples of equations of mathematical physics will be considered. No special preliminary knowledge will be assumed.

(Contact: V. Matveev / M. Dafinger)

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 4. März 2021

*(nicht stattgefunden)*

* 16:15 -17:45 *(A&G)

**Prof. Richard Montgomery** (University of California, Santa Cruz)

**Four Open questions in the N-body problem**

*Abstract*: The classical N-body problem is alive and well. I begin with a pictorial survey of some of its solution curves. I then describe four open questions within the problem and recent progress on these questions.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 18. Februar 2021

*14:30-16:00 *(A&G)

**Marc Mars, Miguel Manzano** (Fundamental Physics Department, Universidad de Salamanca)

**Null shells: general matching across null boundaries and connection with cut-and-paste formalism**

*Abstract*: Null shells are a useful geometric construction to study the propagation of infinitesimally thin concentrations of massless particles or impulsive waves. In this talk, I will present the necessary and sufficient conditions that allow for the matching of two spacetimes with respective null embedded hypersurfaces as boundaries. Whenever the matching is possible, it is shown to depend on a diffeomorphism between the set of null generators in each boundary and a scalar function, called step function, that determines a shift of points along the null generators. Generically there exists at most one possible matching but in some circumstances this is not so. When the null boundaries are totally geodesic, the point-to-point identification between them introduces a freedom whose nature and consequences are detailed. The expression for the energy-momentum tensor of a general null shell is also shown.

Finally, the most general shell (with non-zero energy, energy flux and pressure) that can be generated by matching two Minkowski regions across a null hyperplane is presented. This connects the original Penrose’s cut-and-paste construction with the standard matching formalism.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 11. Februar 2021

*14:30-16:00 (A&G)*

**Prof. Dr. Daniel Grieser** (Oldenburg)

**The geodesic flow on singular spaces**

* Abstract: *From a dynamical systems point of view the geodesic flow on a complete Riemannian manifold is rather boring when considered for short times only. The situation changes if we allow the underlying space to have singularities. Then the short time behavior of geodesics near the singularities can be quite interesting, even in the case of pretty simple singularities, like cones and (incomplete) cusps. I will explain results obtained in collaboration with Vincent Grandjean and how the study of these questions naturally involves blow-ups, Hamiltonian systems with degenerate symplectic form and normally hyperbolic dynamical systems.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 4. Februar 2021

*16:15 -17:45 (DS&MP)*

**Markus Lange (**University of British Columbia)

**Exactness of Linear Response in the Quantum Hall Effect**

*Abstract*: In general, linear response theory expresses the relation between a driving and a physical system’s response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm’s law. In this talk I will present, that in the case of the quantum Hall effect, the linear responds is the full responds of the system and all higher order corrections vanish.

(Contact: D. Hasler/ M. Dafinger)

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 21. Januar 2021

*14:30-16:00 (A&G)*

**Dr. Omid Makhmali** (Institute of Mathematics of the Polish Academy of Sciences, Poland)

**Causal structures and their space of null geodesics**

*Abstract*: We define generalized causal structures as a field of projective hypersurfaces over a manifold which can be considered as a Finslerian extension of conformal pseudo-Riemannian geometry. We solve the local equivalence problem for such structures using Cartan's method of equivalence. We investigate geometric structures induced on the space of "null geodesics" of certain classes of causal structures which will involve CR structures of hypersurface type and Finsler metrics of constant flag curvature.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 17. Dezember 2020

*14:30-16:00 (A&G)*

**Prof. Paolo Piccione** (São Paulo, Brasilien)

**Minimal spheres in ellipsoids**

*Abstract*: In 1987, Yau posed the question of whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^{4} are planar, i.e., determined by the intersection with a hyperplane. While this is the case if the ellipsoid is nearly round, Haslhofer and Ketover have recently shown the existence of an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, with min-max methods. Using bifurcation theory and the symmetries that arise in the case where at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with R. G. Bettiol.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 26. November 2020

*14:30-16:00 (A&G)*

**Prof. Karin Hanley Melnick** (University of Maryland)

**A D'Ambra Theorem in conformal Lorentzian geometry**

**A D'Ambra Theorem in conformal Lorentzian geometry**

*Abstract*: D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 19. November 2020

*14:30-16:00 (A&G)*

**Dr. Shaosai Huang** (University of Wisconsin)

**Topological rigidity of the first Betti number and Ricci flow smoothing**

*Abstract*: The infranil fiber bundle is a typical structure appeared in the collapsing geometry with bounded sectional curvature. In this talk, I will discuss a topological condition on the first Betti numbers that guarantees a torus fiber bundle structure (a special type of infranil fiber bundle) for collapsing manifolds with only Ricci curvature bounded below. The main technique applied here is smoothing by Ricci flows. This covers my joint with Bing Wang.

Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

## 12. November 2020

*14:30-16:00 (A&G)*

**Silvan Bernklau** (Universität Jena)

**The spectral mapping theorem for C**_{0}-semigroups

_{0}-semigroups

*Abstract*: Two known proofs of the spectral mapping theorem for eventually norm continuous semigroups are presented, one exploiting individual properties of subcomponents of the spectrum and one relying on a representation of the spectrum in abelian Banach algebras.

(Contact: V. Matveev)