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 wallcrossing 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 Dbranes 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.
