Kahler-Einstein metrics on symmetric Fano T-varieties

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being Kahler-Einstein but admitting deformations without Kahler-Einstein metric.Comment: 12 pages, main theorem slightly improved + minor correction

Normal singularities with torus actions

We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (QQ-)factorial, or (QQ-)Gorenstein singularities. We also give partial criteria for X to be Cohen-Macaulay or log-terminal. Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varieties in the case where the action is of complexity one.Comment: 23 page

Shanks-transform accelerated PML-based series expansions for the 1-D periodic 3-D Green's functions of multilayered media

A PerfectlyMatched Layer (PML) based formalismis proposed to derive fast converging series expansions for the 1D periodic 3D Green’s functions of layered media. The Shanks transform is applied to accelerate the PML-based series

Hedging Strategies Under Temporary Market Impact

This thesis is concerned with the problem of hedging derivatives under temporary market impact. We are going to examine the problem of hedging derivatives in an optimal control setting similar to [14] and [10]. \ud \ud By considering specific combinations of volatility and liquidity specifications, we are able to obtain analytic solutions. Under the optimal control, the holdings in the stock exhibit a mean reverting behavior. While the reversion-speed is similar to the results in the literature, we are able to find a new characterization of the target portfolio as weighted average of future Δ-hedge portfolios. If the stock-price process is given by a Brownian motion, this characterization collapses to the Δ-hedge. In that case, the final hedging error is proportional to the variance of the Δ-hedge. \ud \ud Further, we extend the analysis by including drift using asymptotic expansions. We find that under the optimal control, the wealth process is mean-reverting towards the expectation of the derivative. The hedging portfolio adjusts more to the inclusion of drift if future hedging is less aggressive. The comparison with numerical solutions suggests that the approximations perform reasonably well