I'll present a review of what is known about effective low-energy theory arising as a result of compactification of string theory down to 4 dimensions. In particular, I'll discuss why Calabi-Yau manifolds naturally arise as internal compactification manifolds, what is known in this case about quantum corrections, and why these manifolds are not enough from the phenomenological point of view. If time permits, I'll also present basic facts about flux compactifications allowing to get more realistic models.

I'll discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I'll show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of (0,4) SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular. I'll show how to construct its modular completion and to make manifest the modular invariance of the twistorial construction of D-instanton corrected hypermultiplet moduli space.

I will discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I will show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of (0,4) SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular. I will show how to construct its modular completion and prove the modular invariance of the twistorial construction of D-instanton corrected hypermultiplet moduli space.

I'll discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I'll show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of (0,4) SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular. I'll show how to construct its modular completion and prove the modular invariance of the twistorial construction of D-instanton corrected hypermultiplet moduli space.

I'll review the phenomenon of wall-crossing in theories with N=2 supersymmetry and its relation to the instanton contributions to the low energy effective actions in gauge and string theories. In the latter case, it is relevant for the description of D-brane instantons. Adding NS5-brane instantons to the story, one arrives to considering certain generalized theta series. I'll derive the properties of these theta series following from the mutual consistency of wall-crossing, D-brane and NS5-brane instantons, and show how these results can be used to get a new type of quantum dilogarithm identities.

Genome processing relies on the intracellular localization and dynamic assembly of higher-order nucleoprotein complexes. In bacteria, the mechanism of assembly for the most widespread partition systems, ParABS, responsible for active DNA segregation remains elusive. We have combined super-resolution, genome-wide, biochemical and modeling approaches to investigate quantitatively the formation of the nucleoprotein complex organized around the centromere-like sequences, parS. We found that the active confinement of nearly all ParB proteins around parS, observed at the single molecule resolution, relies on a network of synergistic interactions involving protein-protein and protein-DNA interactions. Our physico-mathematical modeling of ParB binding pattern revealed that ParB binds stochastically in the vicinity of parS over long distances. Based on our findings, and consistent with previous data, we propose a new model that relies on a nucleation and looping mechanism leading to the formation of a dynamic lattice for the partition complex assembly. We thus provide new bases to model the DNA segregation process. Our original assembly model may also apply to many unrelated proteins that self-assemble in superstructures through nucleation centers.