Technische Universität Chemnitz

Fakultät für Mathematik

09107 Chemnitz

E-mail: dmytro.shklyarov at mathematik.tu-chemnitz.de

Lehre/Teaching

WS 2017/18:
Funktionentheorie
Optimization for Non-Mathematicians

SS 2017:
Darstellungstheorie

Forschung/Research

Research interests:
non-commutative geometry,
cyclic homology,
differential graded categories,
matrix factorizations,
(homological) mirror symmetry,
topological quantum field theories,
Hodge theory,
representation theory and quantum groups.

My initial area of expertise was quantum group theory. In this area, my research focused on Leonid Vaksman's theory of Quantum Bounded Symmetric Domains. My current research is devoted to what might be called categorical (or derived) noncommutative geometry and its interplay with algebraic geometry, Hodge theory, singularity theory, mirror symmetry, and topological field theories. A brief overview of my activity in this area can be found here.

Veröffentlichungen/Publications

Almost all of my papers can be found in the arXiv (a few old preprints can also be found here). Here is the full publication list:

**On Hochschild invariants of Landau-Ginzburg orbifolds.**arXiv:1708.06030-
**Calabi-Yau structures on categories of matrix factorizations.**J. Geom. Phys. 119 (2017), 193--207. **Exponentially twisted cyclic homology.**Int. Math. Res. Not. IMRN 2017, no. 2, 566--582.-
**Matrix factorizations and higher residue pairings.**Adv. Math. 292 (2016), 181--209. -
**On a Hodge theoretic property of the Kuenneth map in periodic cyclic homology.**J. of Algebra 446 (2016), 132--153. -
**Non-commutative Hodge structures: Towards matching categorical and geometric examples.**Trans. Amer. Math. Soc. 366 (2014), no. 6, 2923--2974. -
**Hirzebruch-Riemann-Roch-type formula for DG algebras.**Proc. Lond. Math. Soc. (3) 106 (2013), no. 1, 1--32. -
**2-Representations and Equivariant 2D Topological Field Theories.**arXiv:0812.4482 **On Serre duality for compact homologically smooth DG algebras.**arXiv:0702590**Covariant q-differential operators and unitary highest weight representations for U**(with G. Zhang) J. Math. Phys. 46 (2005), no. 6, 24 pp._{q}su_{n,n.}**Fock representations and quantum matrices.**(with S. Sinel'shchikov and L. Vaksman) Internat. J. Math. 15 (2004), no. 9, 855--894.**Berezin transform on the quantum unit ball.**(with G. Zhang) J. Math. Phys. 44 (2003), no. 9, 4344--4373.**On a q-analogue of the Penrose transform.**(with S. Sinel'shchikov, A. Stolin, and L. Vaksman) Ukrain. Fiz. Zh. 47 (2002), no. 3, 288--292.**Non-compact quantum groups and quantum Harish-Chandra modules.**(with S. Sinel'shchikov, A. Stolin, and L. Vaksman) Supersymmetry and quantum field theory (Kharkov, 2000). Nuclear Phys. B Proc. Suppl. 102/103 (2001), 334--337.**On a q-analog of the Fock inner product.**Preprint available as part of arXiv:0109198**On uniqueness of a covariant Wick-type deformation of the quantum disk.**Preprint available as part of arXiv:0109198**Hidden symmetry of some algebras of q-differential operators.**(with S. Sinel'shchikov and L. Vaksman) Noncommutative structures in mathematics and physics (Kiev, 2000), 309--320, NATO Sci. Ser. II Math. Phys. Chem., 22, Kluwer Acad. Publ., Dordrecht, 2001.-
**On q-analogues of certain prehomogeneous vector spaces: comparison of several approaches.**Mat. Fiz. Anal. Geom. 8 (2001), no. 3, 325--345. **Geometric realizations for some series of representations of the quantum group SU**(with S. Sinel'shchikov and L. Vaksman) Mat. Fiz. Anal. Geom. 8 (2001), no. 1, 90--110._{2,2}.**q-Analogues of some bounded symmetric domains.**(with S. Sinel'shchikov and L. Vaksman) Czechoslovak J. Phys. 50 (2000), no. 1, 175--180.**q-Analogues for Green functions for powers of the invariant Laplacian in the unit disc.**Mat. Fiz. Anal. Geom. 7 (2000), no. 3, 345--365.**A q-analogue of the Berezin quantization method.**(with S. Sinel'shchikov and L. Vaksman) Lett. Math. Phys. 49 (1999), no. 3, 253--261.**Quantum matrix ball: the Bergman kernel.**(with S. Sinel'shchikov and L. Vaksman) arXiv:9909036**Quantum matrix ball: differential and integral calculi.**(with S. Sinel'shchikov and L. Vaksman) arXiv:9905035**On function theory in quantum disc: q-differential equations and Fourier transform.**(with S. Sinel'shchikov and L. Vaksman) arXiv:9809002**On function theory in quantum disc: invariant kernels.**(with S. Sinel'shchikov and L. Vaksman) arXiv:9808047**On function theory in quantum disc: covariance.**(with S. Sinel'shchikov and L. Vaksman) arXiv:9808037**Function theory in the quantum matrix ball: an invariant integral.**(with S. Sinel'shchikov and L. Vaksman) arXiv:9803110**Integral representations of functions in the quantum disk.**(with L. Vaksman) (Russian) Mat. Fiz. Anal. Geom. 4 (1997), no. 3, 286--308.

My PhD thesis can be downloaded from this link. (Here are the slides that I used during my PhD defense and here is my advisor's homepage.)