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.

In the fourth and fifth year, we obtained the following results: 

• We introduced a new type of “Hodge theory” – Theory  of atoms. This is a new type of quasitanakian category by which we defined new birational and symplectic invariants.
• Based on the Theory of atoms, we obtained obstruction to rationality of orbifolds and stacks.
• Combined with classical invariants we obtained even stronger invariants.
• We have given a new proof of a theorem of Kapranov. Using that we described new stratification on moduli space of objects.
• We extend classical Hodge theory to a new definable structures. Many of the classical results of Hodge theory were extended to this new definable structure.
• We developed a new Fukaya-Seidel category using perverse sheaf of categories. We proved Homological Mirror Symmetry (HMS) for Fukaya-Seidel category in these new terms.
• Curvature properties in the geometries with skew-symmetric torsion and special holonomy SU(3), G2 and Spin(7) are established and the reflection on the exterior derivative and covariant derivative of the torsion are obtained. If the torsion is closed, it is shown that these geometries are generalized Ricci solitons.
• The quaternionic contact heat equation is investigated and an entropy-type formula is established towards finding geometric inequalities, in particular, a quaternionic contact version of the famous Li-Yau inequality, which is one of the main ingredients in the famous Hamilton Ricci flow and Perelman entropy.
• Paraquaternionic conformal deformations of a paraquaternionic contact manifold as well as a pqc curvature tensor are defined. It is proved that the pqc curvature is invariant under paraquaternionic contact deformations. It is shown that the pqc curvature vanishes if and only if the pqc manifold is locally paraquaternionic contact equivalent to the paraquaternionicHeisenberg group as well as to the paraquaternionic psedosphere.