Understanding the size of the partial sums of the Möbius function is one of the most fundamental problems in analytic number theory. This motivated the 1944 paper of Wintner, where he introduced the concept of a random multiplicative function: a probabilistic model for the Möbius function. In recent years, it has been uncovered that there is an intimate connection between random multiplicative functions and the theory of Gaussian Multiplicative Chaos, an area of probability theory introduced by Kahane in the 1980's. We will survey selected results and discuss recent research on the distribution of partial sums of random multiplicative functions when restricted to integers with a large prime factor.

In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.

Ever wondered how ideas from physics can used in real-world scenarios? Come to this talk to understand what is an option and how they are traded in markets. I will recall some basic notions of stochastic calculus and derive the Black-Scholes (BS) equation for plain vanilla options. The BS equation can be solved using standard path integral techniques, that also allow to price more exotic derivatives. Finally, I will discuss whether the assumptions behind Black-Scholes dynamics are reasonable in real-world markets (spoiler: they're not), volatility smiles and term structures of the implied volatility.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

A consequence of the works of Costello and Lurie is that the Hochschild chain complex of a Calabi-Yau category admits the structure of a framed E_2 algebra (the genus zero operations). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative
Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain a generalization of string topology to manifolds with boundary, as well as interesting operations on Hochschild homology of Fano varieties. This is joint work with Chris Brav.