The unifying theme of our research is categorical Kaehler Geometry initiated by M. Kontsevich and L. Katzarkov. During the first year we have obtained results in the following directions:

• Developing foundations of new Birational invariants – spectra: Noncommutative, quantum, Givental Steenbrink. These invaraints are part of a new categorical singularity theory.
• Study relations between these spectra and their relations with other parts of Mathematics and theoretical Physics.
• Developing the foundations of Quantum toric geometry. This is a part of a bigger program – creating Chimeric Geometry.
• Developing theory of spectral networks and initiating theory of invariants they produce. This will lead to a new approach to classical uniformation questions.
• Developing the theory of moduli spaces of objects. We also investigate the stratification inflicted on this moduli spaces by Noncommutative spectra.
• New results on special metrics – classical and categorical.

In the second and third year we obtained the following results:

• The non-rationality of many 3- and 4-dimensional Fanos unknown before is proved.
• We extended the methods and results to the case of Fano complete intersections in weighted projective spaces and more general Fano stacks.
• We extended these results to the case of algebraically non-closed fields making a connection between Physics and Logics from one side and Arithmetics on the other.
• Proof of Blow up Formulae.
• Proof of Semi-continuity of Spectra.
• The existence of Moduli spaces of objects for 3- and 4-dimensional CY manifolds is proved.
• The proof is based on a recent work by T. Pantev, B. Toen, G. Vessozi, M. Vaquette and makes a connection between shifted symplectic structures and derived categories.
• We expect that the stratification defined on these moduli spaces and the spectra for the corresponding categories are closely related. In that respect, we outline the connection between non-commutative spectrum and Vafa-Witten invariants.
• We defined the Quantum Toric Geometry in the new cite of o-minimal structures.
• We defined the theory of deformations for quantum stacks.
• The existence of Fukaya-Seidel category with coefficients is established.
• The proof of the Homological Mirror Symmetry (HMS) for Fukaya-Seidel category with coefficients is obtained in several important cases.
• We developed the study of the discriminanats of toric varieties.
• New examples of complete Einstein Riemannian manifolds of negative scalar curvature are constructed as a hyperbolic extension of a complete paraholomorphic paracomplex Riemannian Einstein space of negative scalar curvature. For example, we show that the hyperbolic extension of the product of two n-dimensional discs with the Poincaré metric is a complete odd dimensional Riemannian Einstein space with negative scalar curvature.
• An Obata type theorem on the 3-Sasakian sphere is proved. All solutions to the quaternionic contact Yamabe equation on the qc sphere of dimension 4n + 3 as well as on the quaternionic Heisenberg group are determined.
• A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown. Consequently, all extremals of the Folland-Stein inequality on the quaternionic Heisenberg groups are explicitly determined.
• A new version of the CR almost Schur Lemma is established. It gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold, which in the torsion-free case is a better estimate compared with the known ones.
• Quaternionic contact (qc) versions of the so called Almost Schur Lemma are also established.
• Curvature properties in the geometries with skew-symmetric torsion and special holonomy SU(3) are established and the reflection on the exterior derivative and covariant derivative of the torsion are obtained.