My current research interests include
affine Hecke algebras,
quantum groups,
quantum symmetric pairs,
modular representation theory,
quiver varieties,
Springer fibers,
geometric representation theory,
categorifications, and
algebraic combinatorics.
I'm an organizer of the colloquium at the Institute of Mathematics, Academia Sinica, the Basic Notion Seminar at the National Taiwan University/Academia Sinica.
Assistant Research Fellow
2020 - present
Institute of Mathematics
Academia Sinica, Taiwan
Summer Collaborators Program
2019
Collaborators: Mee Seong Im, Jieru Zhu, Arik Wilbert
School of Mathematics, Institute of Advanced Study
LT Assistant Professor
2017 - 2020
Mentor: Dan Nakano
Department of Mathematics,
University of Georgia, USA
Visiting Postdoc Fellow
2016 - 2017
Max Planck Institute for Mathematics in Bonn, Germany
Doctor of Philosophy
2011 - 2016 Graduate Instructor (3 years) / Teaching Assistant (.5 year)
Adviser: Weiqiang Wang
Department of Mathematics, University of Virginia, USA
Research Assistant
2010 - 2011
Mentor: Shun-Jen Cheng
Mathematics Division
National Center of Theoretical Sciences, Taiwan
Bachelor of Science / Science in Engineering
2005 - 2009
Double major in Electrical Engineering / Mathematics
National Taiwan University, Taiwan
Activities
Upcoming Activities
Nov 19, 2020,
Algebra Seminar,
Jagiellonian University, Poland
Dec 17, 2020,
Algebra Seminar,
National Cheng Kung University, Taiwan
Dec ??, 2020,
Colloquium,
National Central University, Taiwan
Mar 11 - 13, 2022,
AMS Sectional on Representations of algebraic groups and quantum groups:
a tribute to Cline-Parshall-Scott (CPS),
University of Virginia, USA
Organization
Sep 28, 2020 - now, Basic Notion Seminar, Academia Sinica/National Taiwan University
Mar 11-13, 2022, AMS Sectional, University of Virginia
Aug 2018 - May 2020, Algebra Seminar, University of Georgia
Mar 13-15, 2020, AMS Sectional (canceled due to COVID), University of Virginia
Jun 10-12, 2018, Southeast Lie Theory Conference X, University of Georgia
Past Visitings
Nov 02, 2020, Chien Kuo Senior High School, Taiwan
Mar 12-13 2020, Representations of algebraic groups and quantum groups (at the occasion of Brian Parshall’s retirement), University of Virginia.
Feb 21, 2019, National Taiwan University, Taiwan
Dec 24-26, 2019, National University of Singapore
Dec 20, 2019, Academia Sinica, Taiwan
Dec 23, 2019, National Tsing Hua University, Taiwan
Dec 19, 2019, National Sun Yet-sen University, Taiwan
Nov 9-10, 2019, AMS Sectional, UC Riverside
May 13-15, 2019, Southeast Lie Theory Conference XI, LSU
Jun 2-15, 2019, Summer Collaborators Program, Institute for Advanced Study
Mar 15-17, 2019, AMS Sectional Meeting, Auburn University
January 7-11, 2019, Taipei Conference in Representation Theory VI, Academia Sinica, Taiwan
Dec 7, 2018 - Jan 11, 2019, visiting Shun-Jen Cheng, Academia Sinica, Taiwan.
Oct 19-21, 2018, Representation Theory, Combinatorics, and Geometry, University of Virginia
Jun 4 - 9, 2018, Summer School on Lie Theory, University of Georgia
Jun 26 - Jul 7, 2017, visiting Shun-Jen Cheng, Academia Sinica, Taiwan.
Apr 28, 2017, visiting Andrew Hubery, Universität Bielefeld, Germany.
Feb 14 - 17, 2017, visiting Simon Riche, Université Clermont Auvergne, France.
Nov 27 - Dec 01, 2016, visiting Stefan Kolb, Newcastle University, UK.
Oct 20 - 21, 2016, visiting Peter Fiebig, Emmy Noether Zentrum für Algebra, FAU Erlangen-Nürnberg, Germany.
Sep 23 - 27, 2016, visiting Yiqiang Li, University at Buffalo (SUNY), New York, US.
Jun 08 - Aug 08, 2016, visiting Shun-Jen Cheng, Academia Sinica, Taiwan.
Conference Talks
Oct 24 - 25, 2020, AMS Fall Western Sectional Meeting:
Special Session on Geometry and Representation Theory of Quantum Algebras and Related Topics,
University of Utah (meeting virtually)
May 2019, Southeastern Lie Theory Workshop (contributed), Louisiana, US.
Mar 2019, AMS Sectional Meeting, Auburn University, Alabama, US.
Oct 2018, Representation Theory, Combinatorics, and Geometry (contributed), University of Virginia, Virginia, US
Sep 2017, AMS Sectional Meeting, University at Buffalo (SUNY), New York, US.
Jul 2017, Taipei Workshop on Representation Theory of Lie Superalgebras and Related Topics (invited), Academia Sinica, Taiwan.
May 2016, Algebraic Groups, Quantum Groups and Geometry (invited), University of Virginia, Virginia, US
May 2016, Workshop on Hecke Algebras and Lie Theory (poster), University of Ottawa.
May 2015, Southeastern Lie Theory Workshop (contributed), North Carolina State University.
Jun 2015, Workshop on Lie Theory and Representation Theory (invited), East China Normal University, China.
Mar 2015, AMS Sectional Meeting, Georgetown University.
May 2014, Southeastern Lie Theory Workshop (contributed), University of Georgia.
Jul 2010, Cross-strait Conference on Graph Theory and Combinatorics (contributed), National Taiwan University, 2010.
Seminar Talks and Mini Courses
Nov 09, 2020, Basic Notions Seminar, National Taiwan University, Taiwan
Oct 16 - Nov 14, 2020,
Minicourse on representation theory,
National Center of Theoretical Sciences, Taiwan
Aug 31 - Sep 2, 2020, Taipei workshop on representation theory and related topics (invited), Academia Sinica, Taiwan
Oct 28, 2019, Algebra Seminar, Irreducible components of two-row Springer fibers and Nakajima's quiver varieties, University of Georgia
Jan 28, 2019, Algebra Seminar, On q-Schur algebras of type B via coordinate coalgebras, University of Georgia
Jan 03, 2019, Seminar, Constructing q-Schur algebras of type B via coordinate coalgebras, Academia Sinica, Taiwan
Sep 03, 2018, Algebra Seminar, Master Schur Duality, University of Georgia
Apr 02, 2018, Algebra Seminar, Weyl modules over affine Lie algebras in prime characteristic, University of Georgia
Sep 25, 2017, Algebra Seminar, Quantum groups arising from Schur-type dualities II, University of Georgia
Aug 30, 2017, Oberseminar, From Schur duality to quantum symmetric pairs, University of Georgia
Aug 28, 2017, Algebra Seminar, Quantum groups arising from Schur-type dualities I, University of Georgia
Apr 28, 2017, Algebra Seminar, Affine Hecke algebras and quantum symmetric pairs, Universität Bielefeld, Germany
Feb 17, 2017, Algebra Seminar, Affine Hecke algebras and quantum symmetric pairs, Université Clermont Auvergne
Dec 16, 2016, Oberseminar Representation Theory, Affine Hecke algebras and quantum symmetric pairs, Universität Bonn, Germany.
Dec 12, 2016, Algebra Seminar, Affine Hecke algebras and quantum symmetric pairs, IMJ-PRG, France.
Dec 8, 2016, MPI-Oberseminar, From Schur duality to quantum symmetric pairs, MPIM Bonn, Germany.
Nov 29, 2016, Algebra Seminar, Affine Hecke algebras and quantum symmetric pairs, Newcastle University.
Oct 21, 2016, Emmy Noether Seminar, Affine Hecke algebras and quantum symmetric pairs, Emmy Noether Center, FAU Erlangen-Nürnberg.
Sep 25, 2016, Algebra Seminar, Affine Hecke algebras and quantum symmetric pairs, University at Buffalo, SUNY.
Jul 2016, Mini course: Fifty shades of quantum groups, National Taiwan Normal University, Taiwan.
Dec 2015, Lie Group/Representation Theory Seminar, Affine flag varieties, Hecke algebras, and quantum symmetric pairs, University of Maryland.
Nov 2015, Algebra seminar, Quantum symmetric pairs of affine type: multiplication formulas, canonical bases, and Schur dualities, University of Virginia.
Dec 2014, Seminar, An elementary approach to monomial and canonical bases of quantum affine gl_n, Academia Sinica, Taiwan.
Dec 2014, Mini course on representation theory of Lie algebras, National Central University, Taiwan.
Dec 2014, Seminar, Quantum groups and q-Schur dualities, National Central University, Taiwan.
Jul 2014, Mini course on representation theory of Lie algebras, National Taiwan Normal University, Taiwan.
May 2014, Algebra seminar, Quantum Schur-type dualities of finite and affine type B, University of Virginia.
Nov 2013, Graduate seminar, I park my car in a Weyl alcove by a labeled tree, University of Virginia.
Sep 2013, Algebra seminar, On Weyl modules over affine Lie algebras in prime characteristic, University of Virginia.
Nov 2010, Seminar, Algorithms on parking functions and related multigraphs, National University of Kaohsiung, Taiwan.
Publications
S. Eu, T. Fu, C. Lai,
On the enumeration of parking functions by leading terms, Advances in Applied Mathematics,
35 (2005), 392 - 406.
S. Eu, T. Fu, C. Lai,
Cycle Lemma, Parking Functions and Related ultigraphs, Graphs and Combinatorics,
26 (2010), 345 - 360.
C. Lai,
On Weyl modules over affine Lie algebras in prime characteristic, Transform. Groups, 21 (2016), 1123-1153.
Z. Fan, C. Lai, Y. Li, L. Luo and W. Wang,
Affine flag varieties and quantum symmetric pairs, Mem. Amer. Math. Soc., 265 (2020), no. 1285.
arXiv:1602.04383, 108 pages.
C. Lai,
Affine Quantum Symmetric Pairs: Multiplication Formulas and Canonical Bases,
PhD dissertation (2016), 124 pages.
Z. Fan, C. Lai, Y. Li, L. Luo and W. Wang,
Affine Hecke algebras and quantum symmetric pairs,
Mem. Amer. Math. Soc. (to appear),
arXiv:1609.06199v1, 75 pages.
C. Lai and L. Luo,
An elementary construction of monomial bases of quantum affine gl_n, J. Lond. Math. Soc. 96 (2017), 15--27.
Z. Fan, C. Lai, Y. Li, L. Luo, W. Wang and H. Watanabe,
Quantum Schur duality of affine type C with three parameters,
Math. Res. Letters, 27 (2020), 79 --114.
C. Lai and L. Luo,
Schur algebras and quantum symmetric pairs with unequal parameters, Int. Math. Res. Not. (to appear)
arXiv:1808.00938, 34 pages.
C. Lai, D. K. Nakano and Z. Xiang,
On q-Schur algebras corresponding to Hecke algebras of type B, Transform. Groups (to appear),
arXiv:1902.07682, 29 pages.
M. Im, C. Lai and A. Wilbert,
Irreducible components of two-row Springer fibers and Nakajima quiver varieties,
arXiv:1910.03010, 37 pages, preprint.
M. Im, C. Lai and A. Wilbert,
Springer fibers via quiver varieties using Maffei-Nakajima isomorphism,
arXiv:2009.08778, 23 pages, preprint.
Teaching
Academia Sinica
Fall 2020, Representation Theory: Mini course on Springer Fibers and Quiver Varieties
Notes
week 1, week 2, week 3, week 4, week 5.
University of Georgia
Spring 2020, Accelerated Calculus III for Engineering Students (Math 2500 - 25083/38466)
Spring 2020, Calculus I for Science and Engineering (Math 2250 - 24973)
Fall 2019, Lie Algebras (Math 8080 - 42370)
Spring 2019, Calculus I for Science and Engineering (Math 2250 - 24969/24976)
Fall 2018, Topics in Algebra - Quantum Groups (Math 8030 - 38839)
Spring 2018, Calculus III for Science and Engineering (Math 2270 - 25075)
Fall 2017, Calculus I for Science and Engineering (Math 2250 - 15618)
Fall 2017, Calculus III for Science and Engineering (Math 2270 - 25509)