Oleksandr (Sasha) Tsymbaliuk
Assistant ProfessorDepartment of Mathematics, Purdue University
150 N. University Street, Office 620, West Lafayette, IN 47907, USA
Email: otsymbal@purdue.edu
Russian War and Invasion of Ukraine
The truth shall win over the evil of russia's genocide of the peaceful Ukraine, but the cost of human life that civilian Ukrainians will pay for this is going to be incredibly high. You may support UKRAINIAN ARMY at savelife.in.ua
Ukrainian news in English: Ukrainian Pravda, Censor.net, Ukrainian Witness
The horrible actions of the russian troops: webpage
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 Yangians) and its connection to Algebraic Geometry
(via Laumon spaces, Nakajima quiver varieties, Coulomb branches) and Integrable Systems
(the Todalike integrable systems and the quantum inverse scattering method)
My research has been supported by NSF Grants No. DMS1502497 (changed to DMS1821185), DMS2001247 (changed to DMS2037602), and DMS2302661
CV: here
Employment
Assistant Professor, Purdue University, 2020‒present
Gibbs Assistant Professor, Yale University, 2017‒2020
Research Assistant Professor, Simons Center for Geometry and Physics, 2014‒2017
Education
PhD, Mathematics, Massachusetts Institute of Technology, 2014
MS, Mathematics, Independent University of Moscow, 2009
MS, Mathematics, Moscow State University, 2009
Kharkiv Physics and Mathematics Lyceum #27, Ukraine, 2004
Publications (arXiv, Purdue format)

Affine standard Lyndon words: Atype
(with Y. Avdieiev)
Submitted; Preprint arXiv:2305.16299 (32pp, last update on 05/25/2023) (arXiv) 
Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$
(with Y. Hu)
Submitted; Preprint arXiv:2305.00810 (46pp, last update on 05/01/2023) (arXiv) 
Difference operators via GKLOtype homomorphisms: shuffle approach and application to quantum Qsystems
Letters in Mathematical Physics 113 (2023), Paper No. 22, 43pp (journal) (arXiv) 
Shuffle approach towards quantum affine and toroidal algebras
This book is based on the lecture notes from my crashcourse on shuffle algebras delivered in March 2019 in Tokyo
Book accepted by SpringerBriefs in Mathematical Physics (2023)
Preprint arXiv:2209.04294 (124pp, last update on 04/13/2023) (arXiv) 
Transfer matrices of rational spin chains via novel BGGtype resolutions
(with R. Frassek and I. Karpov)
Communications in Mathematical Physics 400 (2023), 1‒82 (journal) (arXiv) 
Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
(with R. Frassek)
Communications in Mathematical Physics 392 (2022), 545‒619 (journal) (arXiv) 
Surface defects in gauge theory and KZ equation
(with N. Nekrasov)
Letters in Mathematical Physics 112 (2022), Paper No. 28, 53pp (journal) (arXiv) 
Quantum loop groups and shuffle algebras via Lyndon words
(with A. Negut)
Submitted; Preprint arXiv:2102.11269 (75pp, last update on 10/18/2021) (arXiv) 
Lax matrices from antidominantly shifted Yangians and quantum affine algebras: Atype
(with R. Frassek and V. Pestun)
Advances in Mathematics 401 (2022), Paper No. 108283, 73pp (journal) (arXiv) 
Shuffle algebra realizations of type A super Yangians and quantum affine superalgebras for all Cartan data
Letters in Mathematical Physics 110 (2020), 2083‒2111 (journal) (arXiv) 
Duality of Lusztig and RTT integral forms of $U_v(L\mathfrak{sl}_n)$
Journal of Pure and Applied Algebra 225 (2021), no. 1, Paper No. 106469, 14pp (journal) (arXiv) 
Shifted quantum affine algebras: integral forms in type A
(with M. Finkelberg; appendices joint with A. Weekes)
Arnold Mathematical Journal 5 (2019), 197‒283 (journal) (arXiv) 
PBWD bases and shuffle algebra realizations for
$U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(mn))$
and their integral forms
Selecta Mathematica (New Series) 27 (2021), Paper No. 35, 48pp (journal) (arXiv) 
On Sevostyanov's construction of quantum difference Toda lattices
(with R. Gonin)
International Mathematics Research Notices (2019), no. 12, 8885‒8945 (journal) (arXiv) 
Multiplicative slices, relativistic Toda and shifted quantum affine algebras
(with M. Finkelberg)
Representations and Nilpotent Orbits of Lie Algebraic Systems (special volume in honour of the 75th birthday of Anthony Joseph), Progress in Mathematics 330 (2019), 133‒304 (journal) (arXiv) 
Classical limits of quantum toroidal and affine Yangian algebras
Journal of Pure and Applied Algebra 221 (2017), no. 10, 2633‒2646 (journal) (arXiv) 
Several realizations of Fock modules for toroidal $\ddot{U}_{q,d}(\mathfrak{sl}_n)$
Algebras and Representation Theory 22 (2019), 177‒209 (journal) (arXiv) 
Homomorphisms between different quantum toroidal and affine Yangian algebras
(with M. Bershtein)
Journal of Pure and Applied Algebra 223 (2019), no. 2, 867‒899 (journal) (arXiv) 
Bethe subalgebras of $U_q(\widehat{\mathfrak{gl}}_n)$
via shuffle algebras
(with B. Feigin)
Selecta Mathematica (New Series) 22 (2016), 979‒1011 (journal) (arXiv) 
The affine Yangian of $\mathfrak{gl}_1$
revisited
Advances in Mathematics 304 (2017), 583‒645 (journal) (arXiv) 
Infinitesimal Hecke algebras of $\mathfrak{so}_N$
Journal of Pure and Applied Algebra 219 (2015), no. 6, 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 (New Series) 16 (2010), 173‒200 (journal) (errata) (arXiv) (updated)
Teaching Experience
 Fall 2023: Lecturer for MA 59500L (Lie Algebras) at Purdue
 Fall 2022: Lecturer for MA 26500 (Linear Algebra) at Purdue webpage
 Spring 2022: Lecturer for MA 26500 (Linear Algebra) at Purdue webpage
 Spring 2021: Lecturer for MA 59800CIDLA (Infinite dimensional Lie algebras and applications) at Purdue webpage
 Fall 2020: Lecturer for MA 26500 (Linear Algebra) at Purdue webpage
 Fall 2019: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale webpage
 Spring 2019: Lecturer for MATH 754 (Infinite dimensional Lie algebras and applications) at Yale webpage
 Fall 2018: Lecturer for MATH 353 (Introduction to Representation Theory) at Yale webpage
 Fall 2018: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale webpage
 Spring 2018: Lecturer for MATH 667 (Topics in Quantum Groups) at Yale webpage
 Fall 2017: Lecturer 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: Section leader for MAT 303 (Calculus IV with Applications) at SBU
 Spring 2014: Teaching assistant for Math 18.100B (Real Analysis) at MIT webpage
 Winter 2014: Mentor in the MIT Directed Reading Program (Representation Theory) webpage
 Fall 2012: Section leader for Math 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 first two years of MIT PRIMES program.
This resulted in the joint paper:
Representations of infinitesimal Cherednik algebras,
Representation Theory (electronic) 17 (2013), 557‒583
(journal) (arXiv)
With our project "Representations of infinitesimal Cherednik algebras", 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 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.
Mentoring at Yulia's Dream program
Since May 2022, I have been mentoring a team of three high school students from Ukraine in the Yulia’s Dream program, an initiative under MIT PRIMES.
This consisted of two reading courses and the research project which resulted in the joint paper:
Affine standard Lyndon words: Atype,
Preprint arXiv:2305.16299 (32pp, last update on 05/25/2023)
(arXiv)
Yulia's Dream is a research program for exceptional high school students from Ukraine. It is dedicated to the memory of
Yulia Zdanovska, a 21yearold graduate of the National University of Kyiv, a silver medalist at the 2017 European Girls'
Mathematical Olympiad, and a teacher for the “Teach for Ukraine” program who was killed by a Russianfired missile in her home city of Kharkiv.
Notes for Old Selected Talks

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)