I am a mathematician working on algebras and representation theory.
I am interested in cluster algebras and their relation to other research topics, including quantum groups, algebraic and geometric representation theory, categorification, algebraic geometry, tropical geometry and (higher) Teichmüller theory.
Publications / preprints
An analog of Leclerc's conjecture for bases of quantum cluster algebras, SIGMA 16 (2020), 122, DOI:10.3842/SIGMA.2020.122.
Dual canonical bases and quantum cluster algebras, preprint, arXiv:2003.13674.
Bases for upper cluster algebras and tropical points, Journal of the European Mathematical Society, to appear, arXiv:1902.09507.
Compare triangular bases in acyclic quantum cluster algebras, Transactions of the American Mathematical Society, Volume 372, Number 1 (2019), 485-501, DOI:10.1090/tran/7610 .
Triangular bases in quantum cluster algebras and monoidal categorification conjectures, 106 pages, Duke Mathematical Journal, Volume 166, Number 12 (2017), 2337-2442, DOI:10.1215/00127094-2017-0006.
Quantum groups via cyclic quiver varieties I, Compositio mathematica 152 (2016), no. 2, 299--326, DOI:10.1112/S0010437X15007551.
t-analog of q-characters, bases of quantum cluster algebras, and a correction technique, International Mathematics Research Notices 2014 (2014), no. 22, 6175--6232, DOI:10.1093/imrn/rnt115.
(with Yoshiyuki Kimura) Graded quiver varieties, quantum cluster algebras, and dual canonical basis, Advances in Mathematics 262 (2014), 261--312, DOI:10.1016/j.aim.2014.05.014.
(appendix by Bernhard Keller) Quantum cluster variables via Serre polynomials, Journal für die reine und angewandte Mathematik (Crelle's Journal) 668 (2012), 149--190, DOI:10.1515/CRELLE.2011.129.
Talk slides, notes, etc.
[Survey] Cluster algebras and their bases, proceedings of ICRA2020, to appear, arXiv:2108.09279.
32 pages. A brief introduction to cluster algebras and their important bases with examples and topological models. Introduce tropical properties, the (common) triangular basis and the generic basis, and their relation to representation theory, categorification.
Introduce cluster algebras and their important bases. Study tropical properties and describe all bases parametrized by the tropical points. Give some proof sketches.
Introduce cluster algebras. Define the (common) triangular basis and relate it to the dual canonical basis and categorification. Give some proof sketches.
Quantum cluster algebras: quantum groups and dual canonical basis, Sichuan, China, 2021.06
Quantum cluster algebras: good bases and surface examples, Sichuan, China, 2021.06
Some talk slides
git for the version control of text files.
Lyx as a tex editor.
TikzEdt or drawing diagrams in latex.
For calculation on cluster algebras: quiver mutation in Java.
Email: qin.fan.math AT gmail.com
Postal address: The School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240 China.