Observations on G2 Moduli Spaces
Monday, 5.2.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In his two seminal articles, Dominic Joyce not only constructed the first examples of closed manifolds with G2-holonomy metrics, but also proved that the moduli space of all G2-metrics on a closed manifold is itself a finite-dimensional manifold. The statement is, however, only a local one, and the global topological properties of these moduli spaces have remained quite mysterious ever since. Indeed, up to now, we only know that they may be disconnected by the work of Crowley, Goette, and Nordström; the question whether all path components are contractible or not has not been answered yet.\n\nIn this talk, I will give a short introduction to G2 metrics and their moduli spaces and outline a construction of a non-trivial element in the second homotopy of the easier-acceesible observer moduli space of G2 metrics on one of Joyce's examples.\nIf time permits, I will indicate why and how this non-trivial example might also descend to the (full) moduli space.\n\nThis talk is based on ongoing joint work with Sebastian Goette.\n
Convergence of computational homogenization methods based on the fast Fourier transform
Tuesday, 6.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Since their inception in the mid 1990s, computational methods based on the fast Fourier transform (FFT) have been established as efficient and powerful tools for computing the effective mechanical properties of composite materials. These methods operate on a regular grid, employ periodic boundary conditions for the displacement fluctuation and utilize the FFT to design matrix-free iterative schemes whose iteration count is (most often) bounded independently of the grid spacing.\nIn the talk at hand, we will take a look both at the convergence of the used iterative schemes and the convergence of the underlying spectral discretization. Remarkably, despite the presence of discontinuous coefficients, the spectral discretization enjoys the same convergence rate as a finite-element discretization on a regular grid. Moreover, the convergence behavior of the effective stresses profit from a superconvergence phenomenon apparently inherent to computational homogenization problems.\n---------------------------------------------------------
Hyperbolicity and rigidity in moduli spaces of polarized manifolds
Tuesday, 6.2.24, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Neues zur Additivität der Kodaira-Dimension
Friday, 9.2.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Approximation of the local time of a sticky diffusion and applications
Friday, 9.2.24, 12:00-13:00, Raum 125, Ernst-Zermelo-Str. 1
We show that the local time of a sticky diffusion can be approximated by certain kind of high-frequency path statistics. This generalizes results of Jacod for smooth diffusions. We prove various form of the result that depend on the type of normalizing sequence we use. We then use the result to: \n1. devise a consistent stickiness parameter estimator, \n2. assess the behavior of number of crossing statistics, \n3. (if time allows) assess the convergence rate of discrete-time hedging strategies in a "sticky Black-Scholes model".\n
On strong approximation of SDEs with a discontinuous drift coefficient
Tuesday, 13.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coefficients of the equation.\nHowever, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients.\nIn the last decade an intensive study of numerical approximation of SDEs with nonglobally Lipschitz continuous coefficients has begun. In particular, strong approximation\nof SDEs with a drift coefficient that is discontinuous in space has recently gained a lot of\ninterest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control\nproblems. Classical techniques of error analysis are not applicable to such SDEs and well\nknown convergence results for standard methods do not carry over in general.\nIn this talk I will present recent results on strong approximation of such SDEs.\nThe talk is based on joint work with Thomas M¨uller-Gronbach (University of Passau)
Counterexamples to Hedetniemi's Conjecture
Monday, 26.2.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
The tensor product \(G\btimes H\) of two finite graphs \(G\) and \(H\) is defined by \(V(G\btimes H) = V(G)\btimes V(H)\) and two vertices \((g_1,h_1)\) and \((g_2,h_2)\) being connected if \(g_1 E_G g_2\) and \(h_1 E_H h_2\). In 1966 Hedetniemi formulated the conjecture that \(\bchi(G\btimes H) = min(\bchi(G),\bchi(H))\). Only in 2019, more then 50 years later, Shitov discovered the existence of counterexamples. We will follow his proof and introduce some interesting notions along the way.