Perverse sheaves on Grassmannians

We give a complete quiver description of the category of perverse sheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety.Comment: AMS-LaTeX, 35 pages, 11 figure

Patient-specific data fusion for cancer stratification and personalised treatment

According to Cancer Research UK, cancer is a leading cause of death accounting for more than one in four of all deaths in 2011. The recent advances in experimental technologies in cancer research have resulted in the accumulation of large amounts of patient-specific datasets, which provide complementary information on the same cancer type. We introduce a versatile data fusion (integration) framework that can effectively integrate somatic mutation data, molecular interactions and drug chemical data to address three key challenges in cancer research: stratification of patients into groups having different clinical outcomes, prediction of driver genes whose mutations trigger the onset and development of cancers, and repurposing of drugs treating particular cancer patient groups. Our new framework is based on graph-regularised non-negative matrix tri-factorization, a machine learning technique for co-clustering heterogeneous datasets. We apply our framework on ovarian cancer data to simultaneously cluster patients, genes and drugs by utilising all datasets.We demonstrate superior performance of our method over the state-of-the-art method, Network-based Stratification, in identifying three patient subgroups that have significant differences in survival outcomes and that are in good agreement with other clinical data. Also, we identify potential new driver genes that we obtain by analysing the gene clusters enriched in known drivers of ovarian cancer progression. We validated the top scoring genes identified as new drivers through database search and biomedical literature curation. Finally, we identify potential candidate drugs for repurposing that could be used in treatment of the identified patient subgroups by targeting their mutated gene products. We validated a large percentage of our drug-target predictions by using other databases and through literature curation

Reducible quasi-periodic solutions for the Non Linear Schr\"odinger equation

The present paper is devoted to the construction of small reducible quasi--periodic solutions for the completely resonant NLS equations on a $d$--dimensional torus \T^d. The main point is to prove that prove that the normal form is reducible, block diagonal and satisfies the second Melnikov condition block wise. From this we deduce the result by a KAM algorithm.Comment: 48 page

Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)

We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by SLOCC invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete {\em et al.} are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).Comment: 49 pages, 16 figure