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