Dr. Dmytro Shklyarov
January 2014 - October 2014
Dmytro Shklyarov is currently a Postdoctoral Fellow at the Mathematics Institute of the University of Freiburg. He holds a Candidate of Science in Mathematics degree from the Institute for Low Temperature Physics of the NAS of Ukraine and a PhD in Mathematics from Kansas State University (US). His initial area of expertise is quantum group theory; his latest research is in the field of derived non-commutative geometry. His research interests include derived and higher categories, Hodge theory and its generalizations, topological field theories and their generalizations/refinements, Donaldson-Thomas invariants and wall-crossing.
- Matrix factorizations and higher residue pairings. Preprint available at http://arxiv.org/pdf/1306.4878.pdf
- Non-commutative Hodge structures: Towards matching categorical and geometric examples. To appear in Trans. Amer. Math. Soc. Preprint available at http://arxiv.org/abs/1107.3156)
- Hirzebruch-Riemann-Roch-type formula for DG algebras. Proc. Lond. Math. Soc. (3) 106 (2013), no. 1, 1–32.
- Covariant q-differential operators and unitary highest weight representations for Uqsun,n. [joint with G. Zhang] J. Math. Phys. 46 (2005), no. 6, 24 pp.
- Fock representations and quantum matrices. [joint with S. Sinel’shchikov and L. Vaksman] Internat. J. Math. 15 (2004), no. 9, 855–894.
Mckay correspondence for vanishing cohomology
A popular question in complex and algebraic geometry is whether a given topological, cohomological, categorical etc. invariant of a resolution of a singular space is independent of the specific shape of the resolution and, thus, reflects properties of the singular space itself. Questions of this sort are the focus of the area of geometry known as Mckay correspondence. The subject emerged in the early 1980’s and its further development has been greatly influenced by the needs of Mirror Symmetry which entered mathematics soon afterwards. Mirror symmetry, in its original form, compares holomorphic and symplectic invariants of mirror pairs of Calabi-Yau manifolds which quite often arise as resolutions of singular spaces. The primary goal of the project is to establish a Mckay correspondence for basic Hodge theoretic invariants of the so-called Landau-Ginzburg models. The latter form the basis of modern versions of Mirror Symmetry