Effective methods for spectrahedra in real and convex algebraic geometry

谱面体是半确定规划的可行集，同时也是一种非常通用的几何结构，它是连接（实）代数几何和凸优化的中心环节。尽管近年来在利用半定规划和面面体求解各种类型的实代数问题方面取得了巨大进展，但面面体的有效处理仍然是一个相当具有挑战性的问题。基于目前的技术水平，当前项目的目标是提高对光谱面体的理解和处理。特别是，我们将探索解决变形虫（代数变体的对数图像）的计算问题、实际代数几何中的谱面逼近和松弛问题、多项式的优化问题、谱面和完全正性问题以及处理谱面的稳定性问题的有效方法。

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.