Oleksandr Tsymbaliuk
Gibbs Assistant ProfessorDepartment of Mathematics, Yale University
12 Hillhouse Ave, Office 219C, New Haven, CT 06511, USA
Email: oleksandr.tsymbaliuk@yale.edu
Research Interests
My research interests are in Representation theory (particularly, in Cherednik algebras, quantum affine and
toroidal algebras, shuffle algebras, shifted quantum affine algebras) and its connection to Algebraic geometry
(via Laumon spaces, Nakajima quiver varieties, and Coulomb branches)
My research is supported by NSF Grant No. DMS1502497 (changed to DMS1821185 in 2017)
CV: here
Employment and Education
Gibbs Assistant Professor, Yale University, 2017‒present
Research Assistant Professor, Simons Center for Geometry and Physics, 2014‒2017
PhD, Mathematics, Massachusetts Institute of Technology, 2014
MS, Mathematics, Moscow State University, 2009
MS, Mathematics, Independent University of Moscow, 2009
Publications

Multiplicative slices, relativistic Toda and shifted quantum affine algebras
(with M. Finkelberg)
Submitted; arXiv:1708.01795 (125 pp, minor update on 01/25/2018) (arXiv) 
Homomorphisms between different quantum toroidal and affine Yangian algebras
(with M. Bershtein)
Submitted; arXiv:1512.09109 (30 pp, significant update on 01/23/2018) (arXiv) 
Several realizations of Fock modules for toroidal $\ddot{U}_{q,d}(\mathfrak{sl}_n)$
Algebras and Representation Theory, DOI: 10.1007/s1046801797615 (journal) (arXiv) 
Classical limits of quantum toroidal and affine Yangian algebras
Journal of Pure and Applied Algebra 221 (2017), 2633‒2646 (journal) (arXiv) 
The affine Yangian of $\mathfrak{gl}_1$
revisited
Advances in Mathematics 304 (2017), 583‒645 (journal) (arXiv) 
Bethe subalgebras of $U_q(\widehat{\mathfrak{gl}}_n)$
via shuffle algebras
(with B. Feigin)
Selecta Mathematica 22 (2016), no. 2, 979‒1011 (journal) (arXiv) 
Infinitesimal Hecke algebras of $\mathfrak{so}_N$
Journal of Pure and Applied Algebra 219 (2015), 2046‒2061 (journal) (arXiv) 
Infinitesimal Cherednik algebras as Walgebras
(with I. Losev)
Transformation Groups 19 (2014), no. 2, 495‒526 (journal) (arXiv) 
Representations of infinitesimal Cherednik algebras
(with F. Ding)
Representation Theory (electronic) 17 (2013), 557‒583 (journal) (arXiv) 
Equivariant Ktheory of Hilbert schemes via shuffle algebra
(with B. Feigin)
Kyoto Journal of Mathematics 51 (2011), no. 4, 831‒854 (journal) (arXiv) (updated) 
Quantum affine GelfandTsetlin bases and quantum toroidal algebra via Ktheory of affine Laumon spaces
Selecta Mathematica 16 (2010), no. 2, 173‒200 (journal) (errata) (arXiv) (updated)
Teaching Experience
 Spring 2018: Lecturer for MATH 667 (Topics in Quantum Groups and Representation Theory) at Yale webpage
 Fall 2017: Instructor for MATH 120 (Calculus of Functions of Several Variables) at Yale webpage
 Fall 2016: Head instructor for MAT 118 (Mathematical Thinking) at SBU webpage
 Fall 2015: Lecturer for MAT 126 (Calculus B) at SBU
 Fall 2014: Recitation leader for MAT 303 (Calculus IV with Applications) at SBU
 Spring 2014: Teaching assistant for 18.100B (Real Analysis) at MIT webpage
 Winter 2014: Mentor in the MIT Directed Reading Program (Representation Theory) webpage
 Fall 2012: Recitation leader for 18.02 (Multivariable Calculus) at MIT webpage
 2011‒2013: Grader for MIT courses 18.100B (Real Analysis), 18.125 (Real and Functional Analysis), 18.01 (Calculus), 18.782 (Introduction to Arithmetic Geometry), 18.705 (Commutative Algebra), and 18.737 (Algebraic Groups)
Mentoring at MIT PRIMES program
I had mentored Fengning Ding during 2011‒2013 in the MIT PRIMES program. With our project, Fengning won the 4th Prize at 2012 Intel STS US national competition (\$40,000 award) and became 2012 Davidson Fellow Laureate (\$50,000 award).
MIT PRIMES is a free, yearlong afterschool research program for high school students from the Boston area. Program participants work with MIT researchers on exciting unsolved problems in mathematics, computer science, and computational biology.
Selected Talks that include Notes

Temple University, Algebra Seminar, November 2015
Relation between quantum toroidal algebras of $\mathfrak{sl}_n$ and affine Yangians of $\mathfrak{sl}_{nm}$ (handwritten notes) 
Yale University, Geometry, Symmetry and Physics Seminar, April 2015
Shuffle realization of $\ddot{U}_{q,d}(\mathfrak{sl}_n)$ and Bethe subalgebras of $U_q(\widehat{\mathfrak{gl}}_n)$ (handwritten notes) 
Northeastern University, Graduate student seminar, April 2014
The affine Yangian and the quantum toroidal of $\mathfrak{gl}_1$ (handwritten notes) 
MITNEU, Graduate seminar on Quantum cohomology and Representation theory, February 2014
Geometric representation theory of the Hilbert schemes (pdf notes I) (pdf notes II) (pdf notes III) 
Northeastern University, Graduate student seminar, April 2013
Infinitesimal Cherednik algebras (handwritten notes) 
HarvardMIT, Graduate student seminar in Geometric Representation theory, September 2011
Category $\mathcal{O}$ at the negative level (pdf notes) 
MIT, Infinite Dimensional Algebra Seminar, March 2010
DingIohara algebras and their action on the Ktheory of the Hilbert scheme (handwritten notes) 
Clay Mathematics Institute, Workshop "Macdonald Polynomials and Geometry", March 2010
GelfandTsetlin bases via Laumon spaces (handwritten notes)