Georgia Institute of Technology College of Sciences

This talk is concerned with new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities in a range of parameters for which no explicit results of symmetry have previously been known. The method proceeds via spectral estimates. This is joint work with Jean Dolbeault and Maria Esteban.

A discussion of the papers "RNA folding at elementary step resolution" by Flamm et al (2000) and "Modeling RNA folding paths with pseudoknots: Application to hepatitis delta virus ribozyme" by Isambert and Siggia (2000).

L-moments are expectations of certain linear combinations of order statistics. They form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. L-moments are in analogous to the conventional moments, but are more robust to outliers in the data and enable more secure inferences to be made from small samples about an underlying probability distribution. They can be used for estimation of parametric distributions, and can sometimes yield more efficient parameter estimates than the maximum-likelihood estimates. This talk gives a general summary of L-moment theory and methods, describes some applications ranging from environmental data analysis to financial risk management, and indicates some recent developments on nonparametric quantile estimation, "trimmed" L-moments for very heavy-tailed distributions, and L-moments for multivariate distributions.

The prevalence of rare events accompanied by disastrous economic and social consequences, the so-called Black-Swan events, makes today's world far different from just decades ago. In this talk, I shall address the issue of modeling the wealth process of an insurer in a stochastic economic environment with dependent insurance and financial risks. The asymptotic behavior of the finite-time ruin probability will be studied. As an application, I shall discuss a portfolio optimization problem. This talk is based on recent joint works with Raluca Vernic and Zhongyi Yuan.