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Publications (arXiv, Purdue format)

• Orthosymplectic R-matrices (with K. Hong)
  Preprint arXiv:2408.16720 (31pp, last update on 08/29/2024) (arXiv)
• Fusion and specialization for type ADE shuffle algebras (with A. Negut)
  Submitted; Preprint arXiv:2408.02411 (28pp, last update on 08/05/2024) (arXiv)
• Standard Lyndon loop words: weighted orders (with S. Khomych, N. Korniichuk, K. Molokanov)
  Submitted; Preprint arXiv:2407.19939 (32pp, last update on 08/12/2024) (arXiv) (updated)
• Two-parameter quantum groups and R-matrices: classical types (with I. Martin)
  Submitted; Preprint arXiv:2407.01450 (45pp, last update on 07/05/2024) (arXiv) (updated)
• Orthosymplectic Yangians (with R. Frassek)
  Submitted; Preprint arXiv:2311.18818 (64pp, last update on 12/30/2023) (arXiv) (updated)
• Orthosymplectic superoscillator Lax matrices (with R. Frassek)
  Letters in Mathematical Physics 114 (2024), Paper No. 49, 39pp (journal) (arXiv)
• Affine standard Lyndon words: A-type (with Y. Avdieiev)
  International Mathematics Research Notices (2024), no. 21, 13488‒13524
  (journal) (arXiv)
• Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$ (with Y. Hu)
  International Mathematics Research Notices (2024), no. 7, 6259‒6302 (journal) (arXiv)
• Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems
  Letters in Mathematical Physics 113 (2023), Paper No. 22, 43pp (journal) (arXiv)
• Shuffle approach towards quantum affine and toroidal algebras
  SpringerBriefs in Mathematical Physics (2023), xi+130pp, DOI:10.1007/978-981-99-3150-7, ISBN:978-981-99-3150-7
  (book) (arXiv)
• Transfer matrices of rational spin chains via novel BGG-type 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)
  Advances in Mathematics 439 (2024), Paper No. 109482, 69pp (journal) (arXiv)
• Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type (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 $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ 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 W-algebras (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 K-theory of Hilbert schemes via shuffle algebra (with B. Feigin)
  Kyoto Journal of Mathematics 51 (2011), no. 4, 831‒854 (journal) (arXiv) (updated)
• Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces
  Selecta Mathematica (New Series) 16 (2010), 173‒200 (journal) (errata) (arXiv) (updated)

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Teaching Experience

• Fall 2024: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Spring 2024: Lecturer for MA 59500 (Quantum Groups and Applications) at Purdue web-page
• Spring 2024: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Fall 2023: Lecturer for MA 59500L (Lie Algebras) at Purdue web-page
• Fall 2022: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Spring 2022: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Spring 2021: Lecturer for MA 59800CIDLA (Infinite dimensional Lie algebras and applications) at Purdue web-page
• Fall 2020: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Fall 2019: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale web-page
• Spring 2019: Lecturer for MATH 754 (Infinite dimensional Lie algebras and applications) at Yale web-page
• Fall 2018: Lecturer for MATH 353 (Introduction to Representation Theory) at Yale web-page
• Fall 2018: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale web-page
• Spring 2018: Lecturer for MATH 667 (Topics in Quantum Groups) at Yale web-page
• Fall 2017: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale web-page
• Fall 2016: Head instructor for MAT 118 (Mathematical Thinking) at SBU web-page
• 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 web-page
• Winter 2014: Mentor in the MIT Directed Reading Program (Representation Theory) web-page
• Fall 2012: Section leader for Math 18.02 (Multivariable Calculus) at MIT web-page
• 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)

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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 (joint with F. Ding)
  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 year-long after-school 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 2022, I have been mentoring high school students from Ukraine in the Yulia’s Dream program, an initiative of MIT PRIMES.

During 2022-2023, this consisted of two reading courses with an expository paper

• 2022 PRIMES YULIA’S DREAM expository paper by Y. Avdieiev, K. Drozdova
  Expository report (2022), 12pp (preprint)

and the research project which resulted in the joint paper:

• Affine standard Lyndon words: A-type (joint with Y. Avdieiev)
  International Mathematics Research Notices (2024), no. 21, 13488‒13524, DOI:10.1093/imrn/rnae197
  (journal) (arXiv)

During 2023-2024, this consisted of two reading courses and the research project which resulted in the joint paper:

• Standard Lyndon loop words: weighted orders (joint with S. Khomych, N. Korniichuk, K. Molokanov)
  Preprint arXiv:2407.19939 (32pp, last update on 08/12/2024) (arXiv) (updated)

Yulia's Dream is a research program for exceptional high school students from Ukraine. It is dedicated to the memory of Yulia Zdanovska, a 21-year-old 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 Russian-fired missile in her home city of Kharkiv.

Other mentored high-school projects

I have also mentored several summer projects by high-school students, which resulted in:

• Correlation functions of quantum toroidal $\mathfrak{gl}_1$ algebra by H. Cui
  Journal of Mathematics Research 13 (2021), no. 2, 7‒20 (paper)
• Correlation functions of quantum toroidal $\mathfrak{gl}_n$ algebra by H. Cui
  Journal of Mathematics Research 14 (2022), no. 4, 94‒105 (paper)
• Specialization homomorphisms among shuffle algebras by C. Wang
  Interim report (2022), 47pp (preprint)

Mentoring at Purdue REU program

Since September 2023, I have been mentoring at Purdue REU program. This resulted in the joint paper:

• Two-parameter quantum groups and R-matrices: classical types (joint with I. Martin)
  Submitted; Preprint arXiv:2407.01450 (45pp, last update on 07/05/2024) (arXiv) (updated)

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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)
• MIT-NEU, 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)
• Harvard-MIT, 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
  Ding-Iohara algebras and their action on the K-theory of the Hilbert scheme (handwritten notes)
• Clay Mathematics Institute, Workshop "Macdonald Polynomials and Geometry", March 2010
  Gelfand-Tsetlin bases via Laumon spaces (handwritten notes)