2021 Ph.D Thesis Defenses
Title: Properties of Shortest Length Curves inside Semi-algebraic Sets and Problems related to an Erdös Conjecture concerning Lattice Cubes
Thesis Advisor: Robert Hardt
Part I: Properties of Shortest Length Curves inside Semi-algebraic Sets
This part concerns an analytical stratification question of real algebraic and semi- algebraic sets. In 1957 Whitney  gave a stratification of real algebraic sets, it partitions a real algebraic set into partial algebraic manifolds. In 1975 Hironaka  reproved that a real algebraic set is triangulable and also generalized it to sub-analytic sets, following the idea of Lojasiewicz’s  triangulation of semi-analytic sets in 1964. During the same year, Hardt  also proved the triangulation result for sub-analytic sets by inventing an- other 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 any arbitrary 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.
Part II: Problems related to an Erd¨os conjecture concerning lattice cubes.
Part II of the paper studies a problem of Erd¨os 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¨os conjectured that it has a sharp upper bound, which is O(N 11/4), 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 there is not necessarily a unique maximal set of vertices (counting all possible symmetries). Next, we study an equivalent two-dimensional version of this problem looking 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 formulas 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 try to prove by contradiction that Erd¨os’ conjecture is wrong, and that O(n8/3) is the sharp upper bound for the maximum number. Here we work not only with discrete cases using combinatorics, but also with continuous cases, such as topological manifolds, using linear groups.