There have been a lot of interests in understanding the behaviour of random multiplicative functions, which are probabilistic models for deterministic arithmetic functions such as the Möbius function and Dirichlet characters. Despite recent advances, the limiting distributions of partial sums of random multiplicative functions remain mysterious even at the conjectural level. In this talk, I shall discuss the so-called $L^2$ regime of twisted sums and provide a precise answer to the distributional problem. This is based on ongoing work with Ofir Gorodetsky.

We consider quadratic deformations of the q-symmetric algebras A_q given by x_i x_j = q_{ij} x_j x_i, for q_{ij} in C*.   We explicitly describe the Hochschild cohomology and compute the weights of the torus action (dilating the x_i variables). We describe new families of filtered deformations of A_q, which are Koszul and Calabi—Yau algebras. This also applies to abelian category deformations of coh(P^n), and for n=3 we give examples having no homogeneous coordinate ring.  We then focus on the case where n is even and the deformations are obtainable from deformation quantisation of toric log symplectic structures on P^n.  In this case we construct formally universal families of quadratic algebras deforming A_q, obtained by tensoring filtered deformations and Feigin—Odesskii elliptic algebras. The universality is a consequence of a beautiful combinatorial classification of deformations via "smoothing diagrams", a collection of disjoint cycles and segments in the complete graph on n vertices, viewed as the dual complex for the coordinate hyperplanes in P^{n-1}.  Already for n=5 there are 40 of these, mostly entirely new. Our proof also applies to deformations of Poisson structures, recovering the P^n case of our previous results on general log symplectic varieties with normal crossings divisors, which motivated this project.  This is joint work with Mykola Matviichuk and Brent Pym.

In this talk I will discuss asymmetric orbifolds and will focus on their application to toroidal compactifications of heterotic string theory. I will consider theories in 6 and 4 dimensions with 16 supercharges and reduced rank. I will present a novel formalism, based on the Leech lattice, to construct ‘islands’ without vector multiplets.