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Publications / preprints

• Applications of the freezing operators on cluster algebras, (preliminary version).

• Analogs of dual canonical bases for cluster algebras from Lie theory, (preliminary version). Slides

• Stability scattering diagrams and quiver coverings (with Qiyue Chen and Travis Mandel), preprint, arXiv:2306.04104. Slides

• Bracelet bases are theta bases (with Travis Mandel), 131 pages, preprint, arxiv:2301.11101.

• A refined multiplication formula for cluster characters (with Bernhard Keller and Pierre-Guy Plamondon), preprint, arxiv:2301.01059.

• The valuation pairing on an upper cluster algebra (with Peigen Cao and Bernhard Keller), Journal für die reine und angewandte Mathematik (Crelle's Journal),  806 (2024), 71–114, DOI:10.1515/crelle-2023-0080.

• Twist automorphisms and Poisson structures (with Yoshiyuki Kimura and Qiaoling Wei), SIGMA (Symmetry, Integrability and Geometry) 19 (2023), 105, DOI:10.3842/SIGMA.2023.105.

• An analog of Leclerc's conjecture for bases of quantum cluster algebras, SIGMA (Symmetry, Integrability and Geometry) 16 (2020), 122, DOI:10.3842/SIGMA.2020.122.

• Dual canonical bases and quantum cluster algebras, preprint, arXiv:2003.13674. Notes for the workshop CARTEA.

• Bases for upper cluster algebras and tropical points, Journal of the European Mathematical Society, Volume 26, Number 4 (2024), 1255-1312. DOI:10.4171/JEMS/1308.

• 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.

• Graded quiver varieties, quantum cluster algebras, and dual canonical basis (with Yoshiyuki Kimura), Advances in Mathematics 262 (2014), 261--312, DOI:10.1016/j.aim.2014.05.014.

• Quantum cluster variables via Serre polynomials (appendix by Bernhard Keller), Journal für die reine und angewandte Mathematik (Crelle's Journal) 668 (2012), 149--190, DOI:10.1515/CRELLE.2011.129.

Proceedings

• Cluster algebras and their bases, proceedings of ICRA2020, to appear, arXiv:2108.09279.

Scientific/Academic honors

• ICRA Award, 2020

• International Congress of Chinese Mathematicians (ICCM) 2019, Distinguished Paper

Talk slides, notes, etc.

• Introduction/Survey

  • [Survey] Categorification of cluster algebras and their bases

    • Cluster algebras are algebras with combinatorial structures. One fundamental problem in studying these algebras is looking for good bases. In particular, their generic bases (dual semi-canonical bases) arise from Calabi-Yau categories, while their triangular bases (Kazhdan-Lusztig type bases) arise from monoidal categories. We will present known results and some recent progress in this direction.

  • [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.

  • [Slides] An introduction to bases for upper cluster algebras and tropical points

    • Introduce cluster algebras and their important bases. Study tropical properties and describe all bases parametrized by the tropical points. Give some proof sketches.

  • [Slides] An introduction to triangular bases for cluster algebras

    • Introduce cluster algebras. Define the (common) triangular basis and relate it to the dual canonical basis and categorification. Give some proof sketches.

• Lectures

  • Notes: from dual canonical bases to triangular bases of quantum cluster algebras, workshop CARTEA, Seoul, Korea, 2023.06.

  • 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

  • Analogs of dual canonical bases for cluster algebras from Lie theory, Paris Algebra Seminar, 2023.10

  • Bases for strata of algebraic groups, Bases for Cluster Algebras in Honor of Bernard Leclerc, 2022.09

  • Categorification of Cluster Algebras and Their Bases, Algebra Days in Paris in Honor of Bernhard Keller, 2022.09

  • Bracelets are theta functions, Interdisciplinary applications of cluster algebras, 2021.11

  • Bases of cluster algebras, ICRA2020, 2020.11

  • Dual canonical bases and triangular bases of quantum cluster algebras, Online Cluster Algebra Seminar, 2020.09

  • Dual canonical bases and quantum cluster algebras, Paris Algebra Seminar, 2020.05

Organization

• Workshop Trends in Cluster Algebras, 09.20-09.22, 2022.

• Workshop Cluster Algebras and Related Topics, 08.02-08.06, 2021.

• Conference Cluster algebras 2020, 08.17-08.28, 2020.

• CIRM conference Cluster algebras: twenty years on, 03.19-03.23, 2018.

Useful tools

Free software

• 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.

Proprietary software

• Onenote for taking notes.

• Goodnote for handwritten presentation (iPad).

• Dropbox and Overleaf for collaboration.

• Zoom for online meetings.

Contact

• Email: qin.fan.math AT gmail.com

• Postal address: School of Mathematical Sciences, Beijing Normal University, No. 19 XinJieKouWai St., Beijing 100875, China