Veranstaltungen

Gravitational and electromagnetic perturbations for the full subextremal range, |Q|<M, of Reissner-Nordström spacetimes, as solutions to the Einstein-Maxwell equations, have been shown to be linearly stable. We address the aforementioned problem for the extremal, |Q|=M, Reissner-Nordström spacetime, and contrary to the subextremal case we see that instability results hold, manifesting along the future event horizon of the black hole, H^+. In particular, depending on the number of translation invariant derivatives of derived gauge-invariant quantities, we show decay, non-decay, and polynomial blow-up estimates asymptotically along H^+. As a consequence, we show that solutions to the generalized Teukolsky system of positive and negative spin satisfy analogous estimates. Stronger and unprecedented instabilities are realized for the negative spin solutions, with the extreme curvature component, $\underline{a}$, not decaying asymptotically along the event horizon.

What kinds of scientific computing problems are suited to be solved on a quantum device with quantum advantage? It turns out that by transforming a partial differential equation (PDE) into a higher-dimensional space, certain important issues can be resolved while at the same time not incurring a curse of dimensionality, when performed with a quantum algorithm. In this talk, I’ll explore ways in which quantum algorithms can be used to efficiently solve not just linear PDEs but also certain classes of nonlinear PDEs, like nonlinear Hamilton-Jacobi equations and scalar hyperbolic equations, based on the level-set formalism. Using another transformation, PDEs with uncertainty can be tackled. I’ll also introduce a simple new way–called Schrodingerisation– to simulate general linear partial differential equations via quantum simulation. Using a simple new transform and introducing one extra dimension, any linear partial differential equation can be recast into a system of Schrodinger’s equations – in real time — in a straightforward way. This approach is not only applicable to PDEs for classical problems but also those for quantum problems – like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit. In this talk, I’ll explore ways in which quantum algorithms can be used to efficiently solve not just linear PDEs but also certain classes of nonlinear PDEs, like nonlinear Hamilton-Jacobi equations and scalar hyperbolic equations, based on the level-set formalism. Using another transformation, PDEs with uncertainty can be tackled. I’ll also introduce a simple new way–called Schrodingerisation– to simulate general linear partial differential equations via quantum simulation. Using a simple new transform and introducing one extra dimension, any linear partial differential equation can be recast into a system of Schrodinger’s equations – in real time — in a straightforward way. This approach is not only applicable to PDEs for classical problems but also those for quantum problems – like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit.

I will present natural symbolic representations of intrinsically ergodic, but not necessarily expansive, principal algebraic actions of countably infinite amenable groups and use these representations to find explicit generating partitions (up to null-sets) for such actions. This is joint work with Hanfeng Li.

Most complex manifolds have a trivial group of holomorphic symmetries. To the contrary $\Cn$ for $n\ge 2$ has a huge automorphism group which was studied a lot by Rudin and Rosay in the late 1980’s. Answering a question by Rudin, Andersen and Lempert proved that a certain infinite dimensional subgroup of automorphisms is dense, however meagre, in the holomorphic automorphism group. Their result was improved by Forstneric and Rosay to show that any local phase flow of a time dependend holomorphic vector field can be approximated on compacta by a family of holomorphic automorphisms. This remarkable result let to an enormous number of applications for complex geometry. In 2002 Varolin generalized this to complex manifolds calling it density property and gave first examples of such highly symmetric objects. Kaliman and the speaker developed effective tools for proving the density property. The list of such manifolds is rather long nowadays and is growing steadily. The geometric applications of the density property are as well becoming more and more. We try to give an overview of this area of research also called Andersen-Lempert theory. There is a version of volume density property. Natural generalizations to symplectic holomorphic manifolds are awaiting us in the future.

The elephant random walk (ERW) is a self-reinforced discrete-time process on multidimensional lattice which was introduced in the early 2000s by Schütz and Trimper in order to study how long-range memory affects the behavior of the random walk. In this talk, I will present the reinforcement mechanism and how martingale theory can be used to obtain results on the asymptotic behavior of the ERW, as well as some of its related processes (barycenter, moderate Cramér deviation...).

For a group or semigroup or ring G, solving equations where the coefficients are elements in G and the solutions take values in G can be seen as akin to solving systems of linear equations in linear algebra, Diophantine equations in number theory, or more generally polynomial systems in algebraic geometry. Inspired by number theory, the collection of questions surrounding the solvability of equations and the description of their solutions is often referred to as `Diophantine Problems in groups’. In this talk I will give a survey about solving equations in infinite non-abelian groups, with emphasis on the hyperbolic ones, and extensions thereof.

We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions. This method is also extended to molecular dynamics with Coulomb interactions, in the framework of Ewald summation. We will show its superior performance compared to the current state-of-the-art methods (for example PPPM) for the corresponding problems, in the computational efficiency and parallelizability.

We show in this talk, how an oracle for approximate integer programming can be used to solve an integer program exactly. The method is based on convex optimzation techniques. It yields the best known complexity bounds for general integer programming and novel complexity results for integer programs in equality form in which the domain is bounded by a polynomial in the dimension. Joint work with Daniel Dadush and Thomas Rothvoss

In 2018 Marius Graber and Petra Schwer introduced the notion of <i>Coxeter Shadows</i>, being a subset of the Coxeter group elements combinatorially defined via foldings of galleries in the Coxeter complex. These shadows have shown to have versatile applications in algebra and geometry, making them an object of interest for current research. In this talk we will retrace the roots of their invention to the early 20th century, and by examining vivid examples discover the beauty of folded galleries.

A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir functional associated with a state-space representation of the Volterra series expansion available for any analytic fading memory filter. It is hence called the Volterra reservoir kernel. Even though the state-space representation and the corresponding reservoir feature map are defined on an infinite-dimensional tensor algebra space, the kernel map is characterized by explicit recursions that are readily computable for specific data sets when employed in estimation problems using the representer theorem. We showcase the performance of the Volterra reservoir kernel in a popular data science application in relation to bitcoin price prediction.

I’ll explain why and how to twist K3 surfaces, how to control these twists in terms of Hodge theory and how to view families of twisted K3 surfaces geometrically. The Brauer group and the associated Brauer family give rise to families that behave very much like twistor spaces.