Totally Splittable Polytopes

稳定婚姻多面体是盒完全对偶积分的

The Stable Marriage Polytope is Box-Totally Dual Integral

本文证明稳定婚姻多面体是盒完全对偶积分的，结合已有结论推出Rothblum的稳定婚姻系统为盒完全对偶积分，丰富了组合优化中多面体对偶积分性质的研究。

单模多面体与通过塞缪尔分解定理的多面体完全单模矩阵列数界

Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem

本文通过塞缪尔分解定理证明了完全单模矩阵列数的精确上界，该结果改进了赫勒的经典界，并展示了单模多面体与顶点构成仿射格基的性质相关联。

完全非负的Peterson变种与强主导权重多面体

Totally nonnegative Peterson variety and strongly dominant weight polytope

本文研究了Peterson变种的完全非负部分，证明其与强主导权重多面体的同胚关系，并验证了Rietsch的猜想。

Totally nonnegative Grassmannian and Grassmann polytopes

These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes.

全对称自补平面分拆矩阵及相关多胞体

Totally symmetric self-complementary plane partition matrices and related polytopes

本文建立从全对称自补平面分拆到magog矩阵的双射，研究其计数性质，并分析相关多胞体的极点、不等式、顶点与面。

Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

We present a geometric approach towards derandomizing the Isolation Lemma by Mulmuley, Vazirani, and Vazirani. In particular, our approach produces a quasi-polynomial family of weights, where each weight is an integer and quasi-polynomially bounded, that can isolate a vertex in any 0/1 polytope for which each face lies in an affine space defined by a totally unimodular matrix. This includes the polytopes given by totally unimodular constraints and generalizes the recent derandomization of the Isolation Lemma for bipartite perfect matching and matroid intersection. We prove our result by associ…