June 21, 2024

Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Distinguished Lecture Series
October 15-17, 2013 at 3:30 p.m.
Fields Institute, Room 230

Maxim Kontsevich
Institut des Hautes Etudes Scientifiques

October 15, 2013
What is tropical mathematics?

In tropical mathematics the usual laws of algebra are changed, the subtraction is forbidden, the division is always permitted, and 1+1 is equal to 1. Analogs of usual geometric shapes like lines, circles etc. are replaced by new figures composed of pieces of lines. I will try to explain basics of tropical algebra and geometry, its relation with more traditional domains, and its role in mirror symmetry which is a remarkable duality originally discovered in string theory about 20 years ago.

October 16, 2013
Quivers, cluster varieties and integrable systems

I'll describe a new approach to cluster varieties and mutations based on
scattering diagrams and wall-crossing formalism. The central role here is played by certain canonical transformation (formal change of coordinates) associated with arbitrary quiver. Also, a complex algebraic integrable system under some mild conditions produces a quiver, and the associated canonical transformation is a birational map.

October 17, 2013
Fukaya category meets Bridgeland stability

Bridgeland's notion of stability in triangulated categories is believed to be a mathematical encoding of D-branes in string theory. I'll argue (using physics picture) that partially degenerating categories with stability should be described as a mixture between symplectic geometry and pure algebra. Spectral networks of Gaiotto, Moore and Neitzke appear as an example.


Speakers in the Distinguished and Coxeter Lecture Series have made outstanding contributions to their field of mathematics. The Lecture Sereies consists of three one-hour lectures.

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