Mirror symmetry and cluster algebras

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• Mark Gross ( University California, San Diego)
• Tuesday 02 July 2013, 14:00-15:00
• CMS, MR12.

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Mirror symmetry is a subject which originated in string theory in 1990, and has since developed mathematically in many different directions. Roughly put, mirror symmetry gives a correspondence between different algebraic varieties, in which certain calculations on one variety (usually involving counting curves) coincides with a different type of calculation on the other (usually computing integrals of certain forms). Joint work with Bernd Siebert over the last 10 years has explored the underlying geometry of mirror symmetry, and the structures that have emerged in this work can now be applied to diverse situation.

In this talk I will discuss joint ongoing work with Keel, Kontsevich

and Hacking. I will make a connection between structures called scattering diagrams, or wall-crossing structures, which emerged in the study of mirror symmetry, and cluster algebras of Fomin and Zelevinsky, which were developed to model aspects of canonical bases of Lusztig and Kashiwara. In particular, a construction of an analogue of theta functions, motivated by the homological mirror symmetry conjecture, allows the construction of canonical bases of cluster algebras in many cases, as well as the proof of a number of conjectures concerning cluster algebras.

This talk is part of the Special DPMMS Colloquium series.

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