# Mathematical Sciences Research Institute

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# Seminar

Polytropes and Tropical Eigenspaces November 26, 2012 (02:10 PM PST - 03:00 PM PST)
Location: UC Berkeley
Speaker(s) Ngoc Tran
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The UC Berkeley Combinatorics Seminar
Fall 2012, Monday 2:10pm - 3pm, Evans Hall 939, CCN 54496
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams

Title: Polytropes and Tropical Eigenspaces
Speaker: Ngoc Tran

Abstract: The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a fan partition of $\{R}^{n \times n}$, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of $\R^{n \times n}$ corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels and lead to a new combinatorial classification of polytropes and tropical eigenspaces.