SPEAKER: Xavier Allamigeon (Paris) <br>

<br>

TITLE: Tropicalization of spectrahedra<br>

<br>

ABSTRACT: 


We introduce tropical spectrahedra, defined as the images by the 
nonarchimedean valuation of spectrahedra over the field of real Puiseux 
series. We provide an explicit characterization of generic tropical 
spectrahedra, involving principal tropical minors of size at most 2.
To do so, we study the images by the nonarchimedean valuation of 
semialgebraic sets. We prove in particular that, under a regularity 
assumption, the image by the valuation of a basic semialgebraic set is 
obtained by tropicalizing the inequalities which define it.
We finally show that the projections of tropical spectrahedra are 
precisely the sets of winning certificates of stochastic mean-payoff 
games, and discuss the applications of these results to semidefinite 
programming over nonarchimedean fields.

This is joint work with St&eacute;phane Gaubert and Mateusz Skomra.