Journal Article: Colored five-vertex models and Lascoux polynomials and atoms
Buciumas, Valentin, Scrimshaw, Travis and Weber, Katherine (2020). Colored five-vertex models and Lascoux polynomials and atoms. Journal of the London Mathematical Society, 102 (3), 1047-1066. doi: 10.1112/jlms.12347
Journal Article: Uniform description of the rigged configuration bijection
Scrimshaw, Travis (2020). Uniform description of the rigged configuration bijection. Selecta Mathematica, 26 (3) 42. doi: 10.1007/s00029-020-00564-8
Journal Article: Multiline queues with spectral parameters
Aas, Erik, Grinberg, Darij and Scrimshaw, Travis (2020). Multiline queues with spectral parameters. Communications in Mathematical Physics, 374 (3), 1743-1786. doi: 10.1007/s00220-020-03694-4
Journal Article: Categorical relations between Langlands dual quantum affine algebras: exceptional cases
Oh, Se-jin and Scrimshaw, Travis (2019). Categorical relations between Langlands dual quantum affine algebras: exceptional cases. Communications in Mathematical Physics, 368 (1), 295-367. doi: 10.1007/s00220-019-03287-w
Kashiwara's Crystals And Generalizations
Quantum groups are important objects in various aspects of mathematical physics, such integrable systems. Representations of quantum groups have nice bases called crystal bases, which allows us to translate problems in representation theory into a combinatorial framework called (Kashiwara) crystals. Crystals have appeared in a diverse set of mathematical topics, including geometry, probability theory, and statistical mechanics. A recent trend has been to generalize these to other applications, such as Lie superalgebras.
There are many questions available for honours, Masters, and PhD students in crystals and releated fields, including algebraic combinatorics, representation theory, and algebraic geometry. I am happy to talk about any of these subjects to find a project that interests you.
Colored five-vertex models and Lascoux polynomials and atoms
Buciumas, Valentin, Scrimshaw, Travis and Weber, Katherine (2020). Colored five-vertex models and Lascoux polynomials and atoms. Journal of the London Mathematical Society, 102 (3), 1047-1066. doi: 10.1112/jlms.12347
Uniform description of the rigged configuration bijection
Scrimshaw, Travis (2020). Uniform description of the rigged configuration bijection. Selecta Mathematica, 26 (3) 42. doi: 10.1007/s00029-020-00564-8
Multiline queues with spectral parameters
Aas, Erik, Grinberg, Darij and Scrimshaw, Travis (2020). Multiline queues with spectral parameters. Communications in Mathematical Physics, 374 (3), 1743-1786. doi: 10.1007/s00220-020-03694-4
Categorical relations between Langlands dual quantum affine algebras: exceptional cases
Oh, Se-jin and Scrimshaw, Travis (2019). Categorical relations between Langlands dual quantum affine algebras: exceptional cases. Communications in Mathematical Physics, 368 (1), 295-367. doi: 10.1007/s00220-019-03287-w
Existence of Kirillov–Reshetikhin crystals for near adjoint nodes in exceptional types
Naoi, Katsuyuki and Scrimshaw, Travis (2021). Existence of Kirillov–Reshetikhin crystals for near adjoint nodes in exceptional types. Journal of Pure and Applied Algebra, 225 (5) 106593, 106593. doi: 10.1016/j.jpaa.2020.106593
Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras
Salisbury, B. and Scrimshaw, T. (2021). Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras. Journal of Algebra, 573, 148-168. doi: 10.1016/j.jalgebra.2020.12.035
Seong Im, Mee and Scrimshaw, Travis (2021). The Regularity of Almost-Commuting Partial Grothendieck-Springer Resolutions and Parabolic Analogs of Calogero--Moser Varieties . Journal of Lie Theory, 31 (1), 127-148.
Double Grothendieck polynomials and colored lattice models
Buciumas, Valentin and Scrimshaw, Travis (2020). Double Grothendieck polynomials and colored lattice models. International Mathematics Research Notices. doi: 10.1093/imrn/rnaa327
Colored five-vertex models and Lascoux polynomials and atoms
Buciumas, Valentin, Scrimshaw, Travis and Weber, Katherine (2020). Colored five-vertex models and Lascoux polynomials and atoms. Journal of the London Mathematical Society, 102 (3), 1047-1066. doi: 10.1112/jlms.12347
Crystal structures for symmetic Grothendieck polynomials
Monical, Cara, Pechenik, Oliver and Scrimshaw, Travis (2020). Crystal structures for symmetic Grothendieck polynomials. Transformation Groups. doi: 10.1007/s00031-020-09623-y
Kirillov–Reshetikhin crystals B1, s for slˆn using Nakajima monomials
Gunawan, Emily and Scrimshaw, Travis (2020). Kirillov–Reshetikhin crystals B1, s for slˆn using Nakajima monomials. Algebras and Representation Theory, 23 (4), 1609-1635. doi: 10.1007/s10468-019-09904-5
Uniform description of the rigged configuration bijection
Scrimshaw, Travis (2020). Uniform description of the rigged configuration bijection. Selecta Mathematica, 26 (3) 42. doi: 10.1007/s00029-020-00564-8
Multiline queues with spectral parameters
Aas, Erik, Grinberg, Darij and Scrimshaw, Travis (2020). Multiline queues with spectral parameters. Communications in Mathematical Physics, 374 (3), 1743-1786. doi: 10.1007/s00220-020-03694-4
Crystal structures for canonical Grothendieck functions
Hawkes, Graham and Scrimshaw, Travis (2020). Crystal structures for canonical Grothendieck functions. Algebraic Combinatorics, 3 (3), 727-755. doi: 10.5802/alco.111
Existence of Kirillov–Reshetikhin crystals for multiplicity-free nodes
Biswal, Rekha and Scrimshaw, Travis (2020). Existence of Kirillov–Reshetikhin crystals for multiplicity-free nodes. Publications of the Research Institute for Mathematical Sciences, 56 (4), 761-778. doi: 10.4171/PRIMS/56-4-4
On higher level Kirillov–Reshetikhin crystals, Demazure crystals, and related uniform models
Lenart, Cristian and Scrimshaw, Travis (2019). On higher level Kirillov–Reshetikhin crystals, Demazure crystals, and related uniform models. Journal of Algebra, 539, 285-304. doi: 10.1016/j.jalgebra.2019.07.036
Lubovsky, Arthur and Scrimshaw, Travis (2019). Alcove path model for B(∞). Journal of Pure and Applied Algebra, 223 (11), 4778-4800. doi: 10.1016/j.jpaa.2019.02.015
Oh, Se-jin and Scrimshaw, Travis (2019). Correction to: Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases. Communications in Mathematical Physics, 371 (2), 833-837. doi: 10.1007/s00220-019-03570-w
Identities from representation theory
Oh, Se-jin and Scrimshaw, Travis (2019). Identities from representation theory. Discrete Mathematics, 342 (9), 2493-2541. doi: 10.1016/j.disc.2019.05.020
A uniform approach to soliton cellular automata using rigged configurations
Liu, Xuan and Scrimshaw, Travis (2019). A uniform approach to soliton cellular automata using rigged configurations. Annales Henri Poincare, 20 (4), 1175-1215. doi: 10.1007/s00023-019-00773-8
Categorical relations between Langlands dual quantum affine algebras: exceptional cases
Oh, Se-jin and Scrimshaw, Travis (2019). Categorical relations between Langlands dual quantum affine algebras: exceptional cases. Communications in Mathematical Physics, 368 (1), 295-367. doi: 10.1007/s00220-019-03287-w
Hall-Littlewood polynomials and Hecke action on ordered set partitions
Huang, Jia, Rhoades, Brendon and Scrimshaw, Travis (2019). Hall-Littlewood polynomials and Hecke action on ordered set partitions. Proceedings of the American Mathematical Society, 147 (5), 1839-1850. doi: 10.1090/proc/14157
Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types
Okado, Masato, Schilling, Anne and Scrimshaw, Travis (2018). Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types. Journal of Algebra, 516, 1-37. doi: 10.1016/j.jalgebra.2018.08.031
Virtual crystals and Nakajima monomials
Salisbury, Ben and Scrimshaw, Travis (2018). Virtual crystals and Nakajima monomials. Symmetry, Integrability and Geometry: Methods and Applications, 14 103,. doi: 10.3842/sigma.2018.103
Rigged configurations and the *-involution
Salisbury, Ben and Scrimshaw, Travis (2018). Rigged configurations and the *-involution. Letters in Mathematical Physics, 108 (9), 1985-2007. doi: 10.1007/s11005-018-1063-2
Virtualization map for the Littelmann path model
Pan, Jianping and Scrimshaw, Travis (2017). Virtualization map for the Littelmann path model. Transformation Groups, 23 (4), 1045-1061. doi: 10.1007/s00031-017-9456-3
Type Dn(1) rigged configuration bijection
Okado, Masato, Sakamoto, Reiho, Schilling, Anne and Scrimshaw, Travis (2017). Type Dn(1) rigged configuration bijection. Journal of Algebraic Combinatorics, 46 (2), 341-401. doi: 10.1007/s10801-017-0756-4
Rigged configurations for all symmetrizable types
Salisbury, Ben and Scrimshaw, Travis (2017). Rigged configurations for all symmetrizable types. Electronic Journal of Combinatorics, 24 (1) #P1.30. doi: 10.37236/6028
Rigged configurations as tropicalizations of loop schur functions
Scrimshaw, Travis (2017). Rigged configurations as tropicalizations of loop schur functions. Journal of Integrable Systems, 2 (1). doi: 10.1093/integr/xyw015
Connecting marginally large tableaux and rigged configurations via crystals
Salisbury, Ben and Scrimshaw, Travis (2016). Connecting marginally large tableaux and rigged configurations via crystals. Algebras and Representation Theory, 19 (3), 523-546. doi: 10.1007/s10468-015-9587-y
A crystal to rigged configuration bijection and the filling map for type D-4(3)
Scrimshaw, Travis (2016). A crystal to rigged configuration bijection and the filling map for type D-4(3). Journal of Algebra, 448, 294-349. doi: 10.1016/j.jalgebra.2015.09.047
A rigged configuration model for B(infinity)
Salisbury, Ben and Scrimshaw, Travis (2015). A rigged configuration model for B(infinity). Journal of Combinatorial Theory, Series A, 133, 29-57. doi: 10.1016/j.jcta.2015.01.008
Crystal structure on rigged configurations and the filling map
Schilling, Anne and Scrimshaw, Travis (2015). Crystal structure on rigged configurations and the filling map. Electronic Journal of Combinatorics, 22 (1). doi: 10.37236/4674
Abrams's stable equivalence for graph braid groups
Prue, Paul and Scrimshaw, Travis (2014). Abrams's stable equivalence for graph braid groups. Topology and its Applications, 178, 136-145. doi: 10.1016/j.topol.2014.09.009
Candidate for the crystal B(∞) for the queer Lie superalgebra
Salisbury, Ben and Scrimshaw, Travis (2019). Candidate for the crystal B(∞) for the queer Lie superalgebra. Formal Power Series and Algebraic Combinatorics, Ljubljana, Slovenia, 1 July - 5 July 2019. Formal Power Series and Algebraic Combinatorics.
Crystal structures for symmetric Grothendieck Polynomials
Monical, Cara, Pechenik, Oliver and Scrimshaw, Travis (2019). Crystal structures for symmetric Grothendieck Polynomials. International Conference on Formal Power Series and Algebraic Combinatorics, Ljubljana, Slovenia, 1-5 July 2019. Formal Power Series and Algebraic Combinatorics.
Description of crystals for generalized Kac-Moody algebras using rigged configurations
Salisbury, Ben and Scrimshaw, Travis (2018). Description of crystals for generalized Kac-Moody algebras using rigged configurations. Conference on Formal Power Series and Algebraic Combinatorics , Hanover, NH, United States, 16-20 July 2018. Formal Power Series and Algebraic Combinatorics.
Multiline queues with spectral parameters
Aas, Erik, Grinberg, Darij and Scrimshaw, Travis (2018). Multiline queues with spectral parameters. Conference on Formal Power Series and Algebraic Combinatorics, Hanover, NH, United States, 16-20 July 2018. Formal Power Series and Algebraic Combinatorics.
Note for students: The possible research projects listed on this page may not be comprehensive or up to date. Always feel free to contact the staff for more information, and also with your own research ideas.
Kashiwara's Crystals And Generalizations
Quantum groups are important objects in various aspects of mathematical physics, such integrable systems. Representations of quantum groups have nice bases called crystal bases, which allows us to translate problems in representation theory into a combinatorial framework called (Kashiwara) crystals. Crystals have appeared in a diverse set of mathematical topics, including geometry, probability theory, and statistical mechanics. A recent trend has been to generalize these to other applications, such as Lie superalgebras.
There are many questions available for honours, Masters, and PhD students in crystals and releated fields, including algebraic combinatorics, representation theory, and algebraic geometry. I am happy to talk about any of these subjects to find a project that interests you.