Veranstaltungen

Geometric (pre)quantization can be performed only for integral symplectic manifolds. In 1989 Alan Weinstein conjectured that using the notion of Diffeology, as well as Noncommutative Geometry methods, one might obtain the representations required to “quantise the unquantisable” from a torus bundle rather than a line bundle. In joint work with P. Antonini, we showed that the obstruction to integrality can be lifted by adding extra dimensions and passing to the diffeological category. In fact, the added dimensions force the C*-algebra associated with this construction to be nothing else than the crossed product algebra associated with a torus action.

In the 1990s, two physisists, Gopakumar and Vafa, proposed an ideal way to count curves in a Calabi-Yau threefold, that is conjecturally equivalent to other curve counting theories such as Gromov-Witten theory. It is very recent that Maulik and Toda gave a mathematically rigorous definition of GV invariants. In this talk, I will review a recent progress on the GV theory, including chi-independence for GV invariants on local curves and degree two GW/GV equivalence for some smooth curves with generic normal bundles. This is based on a joint work with T. Kinjo and another work with B. Davison.

In this talk, I will discuss vortex dynamics in the planar Euler equations, focusing on two key aspects. First, I will present a rigorous derivation of leapfrogging quartets of concentrated vortex patches near singular time-periodic relative equilibria of the point vortex system, using KAM theory. In the second part, I will show how to extend these techniques to desingularize time-periodic vortex orbits when the Euler equation is set in a generic bounded simply-connected domain. Specifically, we can prove that for a single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches.

We will discuss the smooth closing lemma for Hamiltonian diffeomorphisms with invariant Lagrangians.  Based on joint work with Erman Cineli & Shira Tanny.

The toroidal compactifications of A_g

Bounded cohomology is a functional analytic analogue of group cohomology, with many applications in rigidity theory, geometric group theory, and geometric topology. A major drawback is the lack of excision, and because of this some basic computations are currently out of reach; in particular the bounded cohomology of some “small” groups, such as the free group, is still mysterious. On the other hand, in the past few years full computations have been carried out for some “big” groups, most notably transformation groups of R^n, where the ordinary cohomology is not yet completely understood. I will report on this recent progress, which will include joint work with Caterina Campagnolo, Yash Lodha and Marco Moraschini, and joint work with Nicolas Monod, Sam Nariman and Sander Kupers.

We present our recent developments on the asymptotic expansions of high-frequency multiple scattering iterations in the exterior of sound-hard scatterers. As in the sound-soft case, these expansions lead into wavenumber dependent estimates on the derivatives (of all orders) of the multiple scattering iterations which, in turn, allow for the design and analysis of Galerkin boundary element methods (BEM) for their frequency independent approximation. We also present preliminary theoretical developments related to the accurate approximation of the remaining infinite tail in the Neumann series formulation of multiple scattering problems. Time permitting, in the second part of the talk, we present our preliminary results on the frequency independent approximation of the sound-soft scattering amplitude based on Bayliss-Turkel type local approximations to the Dirichlet-to-Neumann operator. Joint with: Y. Boubendir (NJIT) and S. Lazergui (NJIT)

In 1961, Jovan Karamata proved a remarkable identity involving the Dilogarithm function and intersection of diagonals of regular polygons. We reframe the problem and give different proofs for the result. We also investigate what happens when we consider different approaches to it.

Wachstumsmodelle, ovales Billard, gekoppelte Pendel, Newton-Verfahren im Komplexen: Solche dynamische Systeme lassen sich mit einem (open-source) Programm simulieren. Die zugrundeliegende Mathematik ist in genügendem Masse elementar zugänglich, damit sie Anregungen für den erweiterten Mathematikunterricht bieten kann. Sei es in Freifächern, Intensivwochen oder auch für selbständige Arbeiten interessierter Schüler.

In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance and financial services. For example, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks, to provide efficient volatility smoothing or to detect financial asset price bubbles. Theoretical foundations of deep learning for finance, however, are far less developed. In this talk, we start by revisiting some recently developed deep learning methods. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties. Our results allow to obtain more explicit convergence guarantees, thereby making employed neural network methods more trustable. Based on joint works with F. Biagini, A. Jacquier, A. Mazzon, T. Meyer-Brandis, C. Schwab, R. Wiedemann

I will report on joint work with Andrea Brini in which we establish expected properties of the refined Gromov-Witten generating series of local P2 coming from B-model heuristics: modularity, the extended holomorphic anomaly equation, orbifold regularity and the leading order conifold asymptotics. The first three statements generalise results of Lho-Pandharipande and Coates-Iritani while the proof of the last property is new even in the unrefined limit. I will start off explaining why these four features are expected in the first place and will mention some of their consequences. Afterwards I will discuss aspects of their proof mostly focusing on the arguments proving the leading order conifold asymptotics.