Effective methods for spectrahedra in real and convex algebraic geometry

谱面是半定规划的可行集，同时也被广泛认为是一种非常通用的几何结构，它形成了（真实的）代数几何和凸优化之间的中心环节。虽然近年来已经提供了巨大的进步，在使用半定规划和spectrahedra接近真正的代数问题的各种类型，有效地处理spectrahedra仍然是一个相当具有挑战性的问题。在现有技术的基础上，当前项目的目标是推进对光谱面体的理解和处理。特别是，我们将探索有效的方法来解决计算问题的变形虫（对数图像的代数簇），spectrahedra方法和松弛在真实的代数几何和优化的多项式，spectrahedra和完全的积极性，以及稳定性问题，在处理spectrahedra。

Spectrahedra are the feasible sets of semi definite programming and are meanwhile widely recognized as a very versatile geometric structure which forms a central link between (real) algebraic geometry and convex optimization. While recent years have provided tremendous progress in the use of semidefinite programming and spectrahedra in approaching real-algebraic problems of various types, effective handling of spectrahedra is still a rather challenging issue. Building upon the state of the art, the goal of the current project is to advance the understanding and handling of spectrahedra. In particular, we shall explore effective methods for tackling computational problems of amoebas (the logarithmic images of algebraic varieties), spectrahedral approaches and relaxations in real algebraic geometry and optimization of polynomials, spectrahedra and complete positivity, as well as stability issues in handling spectrahedra.