Pictures appear throughout mathematical history. The goal of the Mathematical Picture Language Project at Harvard University is to reevaluate ways that one can use pictures, not only to gain mathematical insights, but also to prove mathematical theorems.

Arthur Jaffe and Zhengwei Liu began their collaboration by studying some problems in subfactor theory and quantum information. This led them to the discovery of the quon language, after which they realized that the quon language also sheds light on several other areas of mathematics. Here you can find links to this research or to articles about our project.

These events motivated the current project to use virtual and real mathematical concepts, simulated by pictures, as a tool to find new understanding ranging across operator algebras, subfactor theory, harmonic analysis, topology, representation theory, statistical physics, topological field theory, quantum field theory, and possibly other fields.

The Mathematical Picture Language Project was originally made possible by a generous grant from the Templeton Religion Trust, in a program under the auspices of W. Christopher Stewart, starting officially in Septmber 2017. This project has support starting May 2019 under a grant from the Army Research Office (ARO). Since June 2020, this project is also part of a Multi-University Research Project (MURI) under ARO.

Please join us on Tuesdays via Zoom for our Mathematical Picture Language seminar.

Prospective postdocs, please refer to the information on how to apply herePostdoctoral position applications and nominations are welcome for a postdoctoral research position with Professor Arthur Jaffe in the Mathematical Picture Language Project. Applicants should submit a curriculum vitae, statement of research interests, and a list of publications to Crystal Stillman at The applicant should also submit three letters of recommendation, sent by the recommenders directly to Ms. Stillman. Applications will be accepted on a rolling basis until the position has been filled. Questions can be addressed to

Harvard is an equal opportunity employer and all qualified applicants will receive consideration for employment without regard to race, color, sex, gender identity, sexual orientation, religion, creed, national origin, ancestry, age, protected veteran status, disability, genetic information, military service, pregnancy and pregnancy-related conditions, or other protected status.

For further information, please send an email to our project assistant, Crystal Stillman.