My research focuses on representation theory, quantum groups, tensor categories, categorification, and exploring applications of these areas to low-dimensional topology and mathematical physics.
My research can best be classified with in the areas of Representation Theory (for example, of braided bialgebras, monoidal categories or 2-categories), Quantum Algebra (Hopf algebras, Nichols algebras, etc), Diagrammatic Tensor Categories, and further work in Noncommutative Algebra (Koszul algebras, noncommutative symmetric functions).
Part of my recent work relates to categorification (joint work with Vanessa Miemietz and You Qi). My work with Vladimir Retakh falls into the area of noncommutative geometry.
My preprints are also available on the arXiv.
List of Publications and Preprints
Research articles in mathematics
[1] Juliet Cooke and Robert Laugwitz. Presentations for small reflection equation algebras of type A. arXiv preprint arXiv:2412.16004, 2024. doi: 10.48550/arXiv.2412.16004.
[2] Johannes Flake, Robert Laugwitz, and Sebastian Posur. Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras. arXiv preprint arXiv:2412.15056, 2024. doi: 10.48550/arXiv.2412.15056.
[3] Cameron Kemp, Robert Laugwitz, and Alexander Schenkel. Shifted poisson structures on higher Chevalley-Eilenberg algebras. arXiv preprint arXiv:2412.12804, 2024. doi: 10.48550/arXiv.2412.12804.
[4] Johannes Flake, Robert Laugwitz, and Sebastian Posur. Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers. arXiv preprint arXiv:2410.08702, 2024. doi: 10.48550/arXiv.2410.08702.
[5] Cameron Kemp, Robert Laugwitz, and Alexander Schenkel. Infinitesimal 2-braidings from 2-shifted Poisson structures. Journal of Geometry and Physics, 212:105456, 2025. doi: 10.1016/j.geomphys.2025.105456.
[6] Johannes Flake, Robert Laugwitz, and Sebastian Posur. Projection formulas and induced functors on centers of monoidal categories. arXiv preprint arXiv:2402.10094, 2024. doi: 10.48550/arXiv.2402.10094.
[7] Robert Laugwitz, Chelsea Walton, and Milen Yakimov. Reflective centers of module categories and quantum K-matrices. arXiv preprint arXiv:2307.14764, 2023. doi: 10.48550/arXiv.2307.14764.
[8] Samuel Hannah, Robert Laugwitz, and Ana Ros Camacho. Frobenius monoidal functors of Dijkgraaf-Witten categories and rigid Frobenius algebras. SIGMA Symmetry Integrability Geom. Methods Appl., 19:Paper No. 075, 42, 2023. doi: 10.3842/SIGMA.2023.075.
[9] Robert Laugwitz and Guillermo Sanmarco. Finite-dimensional quantum groups of type Super a and non-semisimple modular categories. arXiv preprint arXiv:2301.10685, 2023. doi: 10.48550/arXiv.2301.10685.
[10] Robert Laugwitz and Vanessa Miemietz. Differential graded cell 2-representations. Arkiv för Matematik (to appear), 2022. doi: 10.48550/arXiv.2212.07753.
[11] Robert Laugwitz and Vanessa Miemietz. Pretriangulated 2-representations via dg algebra 1-morphisms. arXiv preprint arXiv:2205.09999, 2022. doi: 10.48550/arXiv.2205.09999.
[12] Robert Laugwitz and Chelsea Walton. Constructing non-semisimple modular categories with local modules. Comm. Math. Phys., 403(3): 1363–1409, 2023. ISSN 0010-3616. doi: 10.1007/s00220-023-04824-4.
[13] Johannes Flake, Robert Laugwitz, and Sebastian Posur. Indecomposable objects in Khovanov-Sazdanovic’s generalizations of Deligne’s interpolation categories. Adv. Math., 415:Paper No. 108892, 70, 2023. ISSN 0001-8708. doi: 10.1016/j.aim.2023.108892.
[14] Johannes Flake, Nate Harman, and Robert Laugwitz. The indecomposable objects in the center of Deligne’s category . Proc. Lond. Math. Soc. (3), 126(4):1134–1181, 2023. ISSN 0024-6115. doi: 10.1112/plms.12509.
[15] Mikhail Khovanov and Robert Laugwitz. Planar diagrammatics of self-adjoint functors and recognizable tree series. Pure Appl. Math. Q., 19(5):2409–2499, 2023. ISSN 1558-8599.
[16] Robert Laugwitz and Chelsea Walton. Constructing non-semisimple modular categories with relative monoidal centers. Int. Math. Res. Not. IMRN, 2022(20):15826–15868, 2022. ISSN 1073-7928. doi: 10.1093/imrn/rnab097.
[17] Robert Laugwitz and Chelsea Walton. Braided commutative algebras over quantized enveloping algebras. Transform. Groups, 26(3): 957–993, 2021. ISSN 1083-4362. doi: 10.1007/s00031-020-09599-9.
[18] Johannes Flake and Robert Laugwitz. On the monoidal center of Deligne’s category . J. Lond. Math. Soc. (2), 103(3):1153–1185, 2021. ISSN 0024-6107. doi: 10.1112/jlms.12403.
[19] Robert Laugwitz and Vladimir Retakh. Noncommutative shifted symmetric functions. Mosc. Math. J., 20(1):93–126, 2020. ISSN 1609-3321. doi: 10.17323/1609-4514-2020-20-1-93-126.
[20] Robert Laugwitz and You Qi. A categorification of cyclotomic rings. Quantum Topol., 13(3):539–578, 2022. ISSN 1663-487X. doi: 10.4171/qt/172.
[21] Robert Laugwitz. The relative monoidal center and tensor products of monoidal categories. Commun. Contemp. Math., 22(8):1950068, 53, 2020. ISSN 0219-1997. doi: 10.1142/S0219199719500688.
[22] Robert Laugwitz and Vanessa Miemietz. Cell 2-representations and categorification at prime roots of unity. Adv. Math., 361:106937, 66, 2020. ISSN 0001-8708. doi: 10.1016/j.aim.2019.106937.
[23] Robert Laugwitz. Comodule algebras and 2-cocycles over the (braided) Drinfeld double. Commun. Contemp. Math., 21(4):1850045, 46, 2019. ISSN 0219-1997. doi: 10.1142/S0219199718500451.
[24] Robert Laugwitz and Vladimir Retakh. Algebras of quasi-Plücker coordinates are Koszul. J. Pure Appl. Algebra, 222(9):2810–2822, 2018. ISSN 0022-4049. doi: 10.1016/j.jpaa.2017.11.001.
[25] Robert Laugwitz. On Fomin-Kirillov algebras for complex reflection groups. Comm. Algebra, 45(8):3653–3666, 2017. ISSN 0092-7872. doi: 10.1080/00927872.2016.1243698.
[26] Robert Laugwitz. Pointed Hopf algebras with triangular decomposition. Algebr. Represent. Theory, 19(3):547–578, 2016. ISSN 1386-923X. doi: 10.1007/s10468-015-9588-x.
[27] Robert Laugwitz. Koszulness of enveloping algebras associated to generalized Yang–Baxter equations. arXiv preprint arXiv:1410.4528, 2014. doi: 10.48550/arXiv.1410.4528.
[28] Robert Laugwitz. Braided Drinfeld and Heisenberg doubles. J. Pure Appl. Algebra, 219(10):4541–4596, 2015. ISSN 0022-4049. doi: 10.1016/j.jpaa.2015.02.031.
Other articles, mathematical outreach and education
[2] Michelle Cheng and Robert Laugwitz. The braids on your blanket. Journal of Humanistic Mathematics, 14(2):286–337, 2024. ISSN 2159-8118. doi: 10.5642/jhummath.YMZO2460.
Mentorship
All at University of Nottingham, unless otherwise stated.
Postdocs:
- Juliet Cooke (January 2022 to December 2022)
PhD students:
- Martin Ray (October 2024 to present), Topic: Diagrammatic tensor categories
- Jonathan Davies (October 2022 to present), Topic: Monoidal 2-categories
- Cameron Kemp (June 2022 to present, joint supervision with Alexander Schenkel), Topic: Derived algebraic geometry, deformation quantization, and higher categories
MMath students:
- Samuel Knutsen (2023-2024), "Categorical Formalisms for Supervised Learning"
- Martin Ray (2023-2024), "The classification of finite reflection groups and semisimple Lie algebras using Coxeter graphs", afterwards: PhD, University of Nottingham
MSci students:
- Joseph Aziz (2021-2022), "The 3−j and 6−j symbols in loop quantum gravity", afterwards: PhD student at Royal Holloway University, Physics
- Shikhar Agrawal (2021-2022), "Invariants of knots obtained from quantum algebra", afterwards: Postgraduate study at Indian Institute of Technology, Delhi
- Nicodemos Andreou (2020-2021), "Invariants of knots obtained from quantum algebra", afterwards: PhD student at University of Derby, Physics
- Timontheus Keanu (2019-2020), "Understanding the Actions of Hopf Algebras on Algebras via the Taft Algebra and Other Examples", afterwards: PhD student at Victoria University of Wellington, Mathematics
Undergraduate research internship students:
- Cameron Howat (Summer 2023), "Generalized Temperley-Lieb algebras", afterwards: Masters student, University of Warwick, Mathematics
- Niamh Brereton (Summer 2022), "Undergraduate perceptions of mathematical proofs", afterwards: PhD student, University of Birmingham, Mathematical Education
- Martin Ray (Summer 2022), "Generalized Temperley-Lieb algebras", afterwards: PhD, University of Nottingham
- Xuanzuo Chen (Summer 2021), "Construction of quantum groups", afterwards: PhD student, University of Oxford, Mathematics
- Riccardo Mattarei (Summer 2021), "Construction of quantum groups", afterwards: PhD student, Cornell University, Mathematics