Part I: Properties of Shortest Length Curves inside Semi-algebraic Sets

The first part of the paper concerns an analytical stratification question of real algebraic and semi-algebraic sets. An algebraic set is defined by finitely many polynomial equations and the definition of the more general semi-algebraic set may also entail polynomial inequalities.

In 1957 Whitney \cite{W} gave a stratification of real algebraic sets, it partitions a real algebraic set into partial algebraic manifolds. In 1975 Hironaka \cite{H} reproved that a real algebraic set is triangulable and also generalized it to sub-analytic sets, following the idea of Lojasiewicz's \cite{L} triangulation of semi-analytic sets in 1964. During the same year, Hardt \cite{H2} also proved the triangulation result for sub-analytic sets by inventing another method. Since any semi-algebraic set is also semi-analytic, thus is sub-analytic, both Hironaka and Hardt's results showed that any semi-algebraic set is homeomorphic to the polyhedron of some simplicial complex.

Following their examples and wondering how geometry looks like locally for a semi-algebraic set, Part I of the paper tries to come up with a stratification, in particular a cell decomposition, such that it satisfies the following analytical property. Given an arbitrary semi-closed and connected semi-algebraic set X in R2, any two points in X may be joined by a continuous path γ of shortest length. We will show that there exists a semi-algebraic cell decomposition A of X such that for each A∈A, each component of γ∩A is either a singleton or a real analytic geodesic segment in A; furthermore, γ∩A has at most finitely many components.

An application of this property is that given any semi-algebraic sets Y⊂X⊂R2, any shortest length curve in X intersects Y at most finitely many components. We try to generalize to higher-dimensional semi-algebraic sets and the question is still open. The other analogous open question concerns the regularity of a rectifiable current minimizing mass in a semi-algebraic set.

Part II: Problems related to an Erd{"o}s conjecture concerning lattice cubes.

The second part of the paper studies a problem of Erd"{o}s concerning lattice cubes. Given an n×n×n lattice cube, we want to find the maximum number of vertices one can select so that no eight corners of a rectangular box are chosen simultaneously. Erd"{o}s conjectured that it has a sharp upper bound, which is O(n11/4), but no example that large has been found yet. We start approaching this question for small n using the method of exhaustion, and we find that as n increases, the method becomes cumbersome and one reason is that the condition cannot be easily decoded into workable algebraic conditions. Next, we study an equivalent two-dimensional version of this problem and look for patterns that might be useful for generalizing to the three-dimensional case. Since an n×n grid is also an n×n matrix, we rephrase and generalize the original question to: what is the minimum number α(k,n) of vertices one can put in an n×n matrix with entries 0 and 1, such that every k×k minor contains at least one entry of 1, for 1≤k≤n? We discover some interesting formulae and asymptotic patterns that shed new light on the question. Then we examine many examples that succeed for O(n8/3) but fail for O(n11/4). Last we describe a new method which has the hope to prove that O(n11/4) is the sharp upper bound for the maximum number. In the end, we extend from the discrete questions to nondiscrete questions, which become very interesting to study, because we will see topological manifolds, and linear groups.