An applied lecture with integrated exercises which introduces in some of the most recent topics in Financial and Actuarial mathematics. It is based on recent and original research carried out at FAM.

• Economic introduction: flash crashes, their origin and relevance; high frequency trading.
• Mathematical introduction: Euler's product formula, Riemann's zeta function. Ideas from Riemann's habilitation thesis: analytical continuation via Jacobi's theta transformation formula; explicit formula for prime numbers via zeroes of the zeta function in the critical strip.
• Compound Poisson process: definition, applications in actuarial mathematics and geometry.
• Levy processes: an introduction via Poisson point processes.
• Brownian excursions: a PPP on a function space.
• Limit order book:  PPP meet number theory. Laplace-Mellin transform of a flash crash distribution via Riemann's Xi function.
• Stochastic partial differential equations: solution via time reverted local time; discussion of Hölder continuity via this example.