## Dr. Arnaud Lionnet

Oxford Man InstituteIn the follow up to the NAIS supported conference “New Trends in Computational Finance and Related Topics” (see http://icms.org.uk/workshops/CompFin) two research projects were initiated with conference participants, Goncalo dos Reis and Arnaud Lionnet. These projects are on stochastic numerical methods for a certain classes of reaction-diffusion type PDE using Backwards Stochastic Differential Equations (FBSDE). Dr. Lionnet’s will visit the University of Edinburgh to pursue these topics.

## Olavi Nevanlinna

Aalto UniversityMulticentric Calculus – What and Why?

Since 2007, I have been working in my leisure time on spectral computations and, related to that, holomorphic functional calculus which I have been calling “multicentric”. The key observation in the beginning was, informally stated: you cannot generally compute the spectrum but you can compute its complement. Making this somewhat nonsense statement exact, put me onto this path [1]. In short, multicentric calculus [2] aims to transport analysis in a complicated geometry (on the complex plain) into discs. Rather than using local variables (or conformal change of those) I introduce a new global variable which gathers information around several centers instead of just around the origin. This is a many-to-one change of variable and in this way we lose information but to compensate it we simultaneously work with several functions of the new variable. At the end of the computations the results can be transported back to the original setting. This not only opens up new computation!

al! approaches but also leads to new qualitative results, such as the extension of well known a result of von Neumann (1951) on holomorphic calculus for contractions in Hilbert spaces [3]. In this talk I shall recall this extension of von Neumann’s theorem and then, if time permits, I shall discuss preliminary ideas for algebraic structures one meets in this vector valued calculus. For example, we land in a structure where vector valued functions with meromorphic components form a field. How does multiplication look like? Or derivation? How about involutions, etc.

[1] O. Nevanlinna, Computing the spectrum and representing the resolvent, Numer. Funct. Anal. Optim. 30 (9 – 10) (2009) 1025 – 1047

[2] O. Nevanlinna, Multicentric holomorphic calculus, Comput. Methods Funct. Theory 12 (1) (2012) 45 – 65.

[3] Olavi Nevanlinna: Lemniscates and K-spectral sets Journal of Functional Analysis 262 (2012) 1728 – 1741

## Gil Ariel

Bar Ilan UniversityGil Ariel has a M.Sc. in physics from Tel-Aviv University where he was working on charged polymers and Ph.D. from the Courant Institute on multiscale applications in statistical physics. After three years as a Bing instructor at the University of Texas at Austin Gil joined the faculty at Bar-Ilan University as a senior-lecturer (equivalent to an assistant professor). His research concentrates on numerical methods for multiscale stochastic and highly-oscillatory systems and stochastic multiscale methods for complex many-body systems in Biology. From the numerical point of view, he develops efficient computational methods for stochastic and highly oscillatory dynamical systems with fast-slow components. In mathematical biology, he collaborates with experimentalists on bridging the scale between the behavior of individual organisms, such as bacteria or insects, and the dynamics of swarms or large crowds. Gil recently started working on parallel computational methods for multiscale fast-slow dynamics.