research

I work in the general area of noncommutative algebra. In particular, one area of focus is that of Artin Schelter-regular quadratic algebras. Such algebras can be thought of as noncommutative analogues of the polynomial ring. A good way to think about some of these things is to think of activities in your daily life that requires a certain order. E.g., when doing laundry, you need to wash clothes first and then dry them, not the other way around! Our daily life is full of such instances! I've used noncommutative algebraic geometry to describe graded skew Clifford algebras. I've also worked on a notion of rank for noncommutative quadratic forms and my two advisees (Leah Frauendienst & Jessica Cain) and I have continued that work. I have devoted some time to associating noncommutative algebraic geometry to Lie algebras as well. Moreover, I have explored the area of invariant group theory in the context of quadratic AS-regular algebras (see arXiv paper with J. Gaddis & X. Wang below). I am currently continuing my work in noncommutative algebraic geometry and examining twisted homogeneous coordinate rings with R. Chandler, H. Tran, and X. Wang. With my AIM - SQuaRE group, we have been focused on twists of an algebra's multiplicative structure and have associated those to 2-cocycle twists or Drinfeld twists of bialgebras/Hopf algebras.

In my work in noncommutative algebraic geometry, I often come across quadrics - here's a neat quadric!  (courtesy of Herwig Hauser, Universities of Innsbruck & Vienna, Austria)

Published/Accepted Papers

1. Twisting Manin's universal quantum groups and comodule algebras (with H. Huang, V. C. Nguyen, C. Ure, K. Vashaw, and X. Wang), Adv. in Math. Vol. 445 (2024), https://doi.org/10.1016/j.aim.2024.109651. (arXiv)

2. A cogroupoid associated to preregular forms (with H. Huang, V. C. Nguyen, C. Ure, K. Vashaw, and X. Wang), to appear in the J. Noncommutative Geom. (arXiv) 

3. A Primer on Twists in the Noncommutative Realm with a Focus on Algebra, Representation Theory, and Geometry  (with P. Ocal and K. Ueyama), Contemporary Math. 801 (2024), 179-194. (arXiv)

4. Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation (with H. Huang, V. C. Nguyen, C. Ure, K. Vashaw, and X. Wang), Transf. Groups (2022), doi: 10.1007/s00031-022-09779-9. (arXiv, journal article) 

5. Reflection Groups and Rigidity of Poisson Algebras (with Jason Gaddis & Xingting Wang), Algebr Represent Theor 26, 329-358 (2023). (arXiv, journal article)

6. A notion of rank for noncommutative quadratic forms on four generators (with Leah Frauendienst & Jessica Cain), J. Alg and its Apps. 21(10), (2022). (arXiv)

7. Associating geometry to the Lie Superalgebra sl(1|1) & to the color Lie algebra sl_2^c (with S. Sierra, S. Spenko, M. Vancliff, & E. Wiesner), Proc. Amer. Math. Soc. 147(10), (2019), 4135-4146. (arXiv, journal article)

8. Graded Clifford algebras and Graded Skew Clifford Algebras & their role in the classification of Artin-Schelter Regular Algebras, Adv. in App. Cliff. Algs 27 # 3 (2017), 2856-2871. (arXiv, journal article)

9. Point modules over regular graded skew Clifford algebras (with M. Vancliff), J. Algebra 420 (2014), 54-64. (arXiv, journal article)

10. Generalizing the notion of rank to noncommutative quadratic forms (with M. Vancliff) in "Noncommutative Birational Geometry, Representations and Combinatorics,'' Eds. A. Berenstein and V. Retakh, Contemporary Math. 592 (2013), 241-250. (arXiv, journal article)

11. Point Modules over regular graded skew Clifford algebras. (Dissertation)

Submitted Papers

1. Quantum-symmetric equivalence is a graded Morita invariant (with H. Huang, V. C. Nguyen, K. Vashaw, and X. Wang) arXiv 

2. Regular algebras of dimension four associated to coordinate rings of rank-two quadrics. (with R. Chandler, H. Tran, and X. Wang) arXiv

In preparation

1. Quantum Grassmanians and their Properties. (with J. Gaddis and D. Yee)

Upcoming Invited Talks/Workshops

• Talk at QuaSy-Con II, May 24-26 2024 (conference website) 

• AIM-SQuaRE #3, June 2024

• SRiM @ MSRI/SLMath, July 2024 (website)

Talks/Workshops (since 2020)

1. MSRI Feb 5-9 2024, Introductory Workshop in Noncommutative Algebraic Geometry

2. JMM 2024 - San Francisco 

3. Graduate Seminar @ TTU on challenges of grad school in math (slides) (10/24/2023)

4. June 2-5 2023 @ CMS, Ottawa, Canada

5. Apr 3-7 2023 @ AIM (SQuaRES), San Jose, U.S.A. 

6. March 2023 @ Seattle Noncommutative Algebra Day, Seattle, U.S.A.

7. Nov 2022 @ TTU Graduate Seminar, Cookeville, U.S.A.

8. Sept. 2022 @ Banff International Research Station, Banff, Canada (video of talk) 

9. Twists of algebras and 2-cocycle twists of certain Hopf algebras, June 2022 @ S. Paul Smith's 65th Birthday Conference in Seattle

10. Zhang twists of algebras and cocycle twists of Hopf algebras, Mar 2022 @ SNAD in Seattle (virtual)

11. Deformation quantization, Apr 2021 @ TTU Graduate Student Seminar 

12. Reflection groups of the homogenized nth Weyl Poisson algebra (slides) Jan 2021 @ Virtual JMM

13. An Introduction to Poisson algebras, Oct 2020 @ TTU Graduate Student Seminar (video)

Funding/Grant Applications

• AMS Research Enhancement Grants for PUI Faculty (PI, submitted 03/2024)

• SLMath - Summer Research in Mathematics (Co-PI, submitted 10/2023 - Funded)

• AMS Research Enhancement Grants for PUI Faculty (PI, submitted 03/2023 - Did not receive funding)

• AWM Travel Grant (PI, Submitted 03/2023 - Funded)

• IAS Summer Collaborators Program (Co-PI, submitted 12/2022 - Did not receive funding)

• MSRI - Summer Research in Mathematics (Co-PI, submitted 12/2022 - Did not receive funding)

• AIM-SQuaRE Grant (Co-PI, submitted 12/2021 - Funded 2022-2024)