# Home

### About me

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.

Introduction/Survey

[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

**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

### Organization

Conference

*Cluster Algebras and Related Topics*, 08.02-08.06, 2021.Online conference

*Cluster algebras 2020*, 08.17-08.28, 2020.CIRM conference

*Cluster algebras: twenty years on**, 03.19-03.23, 2018.*Reading seminar

*scattering diagrams and cluster algebras**, 2016*

### 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**

### Contact

Email: qin.fan.math AT gmail.com

Postal address: The School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240 China.