Abstract:  This presentation is devoted to the development and analysis of mathematical models to enhance understanding of extreme events in financial markets. This will be undertaken through two specific problems in the mathematics of risk management:

• The analysis of asset-price dynamics in models that capture the possibility of sudden, large changes in prices -- i.e., ``jumps";

• The development and application of tools from mathematical physics to analyze market dynamics leading to a ``crash", and the corresponding matching with tools from Mathematical Finance.