For a given operator $A$ it is often of interest to determine which operators satisfy the equation $A X = \lambda X A$ for some  complex number $\lambda$. In this situation $\lambda$ is said to be an extended eigenvalue of $A$. Although the case $\lambda = 1$ is much better understood, even when  $\lambda \neq  1$, $X$ shares many properties with $A$. In particular, when $A$ is compact, $X$ has an invariant subspace. In this work we focus on the integral operators of the Volterra type. We will exhibit some of the possible solutions of theoperator equation above and discuss their significance.

Recent results show that the Moreau envelope functions associated with some classes of non-smooth functions (eg. Prox-Regular, Primal-Lower-Nice) have C1 smoothness. In this talk we discuss the local existence and uniqueness of absolutely continuous solutions of some differential inclusions determined by subdifferentials  of  related  envelope functions.

 
. An option is a right without obligation. Options are widely used in the financial industry to mitigate risks or to leverage investment opportunities. If you believe that there is no free lunch then an option always come with a price. How to price an option is a highly active area of research in mathematical finance with an extensive literature. We will try to give an elementary introduction to the widely used classical idea of pricing options by replicating portfolios stemming from Black, Scholes and Merton. We then indicate the difficulties of this method in investment practice and propose an alternative based on information theoretical techniques. This talk is accessible to undergraduate students.