Persistent homology for functionals (Q6608787)

This paper investigates the connection between persistent homology -- a tool widely used in topological data analysis -- and functionals in mathematical analysis. The authors generalize classical Morse inequalities using persistent homology diagrams and propose topological conditions under which persistence diagrams can be reliably constructed for sublevel set filtrations of functionals. Applications are shown in the setting of minimal surface theory, particularly revisiting Morse and Tompkins' proof of the Unstable Minimal Surface Theorem [\textit{M. Morse} and \textit{C. Tompkins}, Ann. Math. (2) 40, 443--472 (1939; Zbl 0021.03405)].\N\N\begin{enumerate}\N\item \textbf{Persistent Homology and Persistence Modules:}\N\begin{itemize}\N\item \textit{Persistent homology} analyzes the evolution of topological features as a parameter varies.\NThis is captured via \textit{persistence modules}, which are functors from real numbers to vector spaces, equipped with linear maps describing how the topology evolves.\N\item A \textit{persistence diagram} is an invariant summarizing these features, including their creation (birth) and destruction (death).\N\end{itemize}\N\N\item \textbf{\( q \)-Tameness:}\N\begin{itemize}\N\item A persistence module is \textit{\( q \)-tame} if the maps between vector spaces associated with different parameter values have finite rank. \( q \)-tameness ensures the existence of well-defined persistence diagrams.\N\end{itemize}\N\N\item \textbf{(Definition 4.1, 4.2) Local Homological Smallness (LHS):}\N\begin{itemize}\N\item For sublevel set filtrations of a real-valued function, the LHS condition implies that for any local neighborhood and subintervals, the inclusion map in homology induces maps of finite rank. The paper shows that the LHS condition guarantees \( q \)-tameness for compact sublevel sets (Theorem 4.4).\N\end{itemize}\N\N\item \textbf{Generalized Morse Inequalities:}\N\begin{itemize}\N\item Traditional Morse inequalities relate the critical points of a smooth function to the topology of its domain. More precisely, the number of critical points with index \(q\) is not smaller than the \(q\)-th Betti number.\NThe authors extend this framework by linking persistence diagrams to these inequalities, enabling applications to non-smooth or infinite-dimensional contexts (Theorem 3.3).\N\end{itemize}\N\N\N\item \textbf{(Section 5) Applications in Minimal Surface Theory:}\N\begin{itemize}\N\item The authors reinterpret the Unstable Minimal Surface Theorem, showing that Morse and Tompkins' functional topology can be placed within the framework of persistent homology.\N\end{itemize}\N\N\N\item \textbf{Key Theorems:}\N\begin{enumerate}\N\N\N\item \textbf{Theorem 3.3: Generalized Morse Inequalities}\N\begin{itemize}\N\item For a graded \( q \)-tame persistence module \( M \) with finite cap numbers \( c^\epsilon_d \) and essential dimensions \( p_d \), the following inequality holds for all dimensions \( n \):\N\[\N\sum_{d=0}^n (-1)^{n-d} \left( c^\epsilon_d - p_d \right) \geq 0.\N\]\NThis extends the classical Morse inequalities to persistence diagrams.\N\end{itemize}\N\N\item \textbf{Theorem 4.4: \( q \)-Tameness and LHS Condition}\N\begin{itemize}\N\item If the sublevel set filtration of a function \( f : X \to \mathbb{R} \) is compact and satisfies the LHS condition, then its persistent homology is \( q \)-tame. Consequently, \( f \) admits a persistence diagram.\N\end{itemize}\N\N\item \textbf{Corollary 4.12: Continuity-Based Weakening of LHS. (Example 4.13)}\N\begin{itemize}\N\item A weaker local-connectivity condition suffices to replace LHS if the filtration is defined by a continuous functional, broadening the applicability of the \( q \)-tameness results.\N\end{itemize}\N\N\item \textbf{Corollary 5.18: Counterexample for Morse's Local Connectivity}\N\begin{itemize}\N\item The local connectivity conditions originally proposed by Morse are shown insufficient for \( q \)-tameness in certain contexts, demonstrated via a counterexample.\N\end{itemize}\N\N\item \textbf{Proposition 5.20: Application to the Douglas Functional}\N\begin{itemize}\N\item The Douglas functional, motivated by Plateau's Problem, satisfies the compactness and LHS conditions, allowing the authors to demonstrate the \( q \)-tameness of its persistence homology.\N\end{itemize}\N\N\N\item \textbf{Theorem 5.8 (Unstable Minimal Surface Theorem):}\N\begin{itemize}\N\item \textit{Statement:} There exist critical points of the Douglas functional that are not local minima. These critical points correspond to unstable minimal surfaces, providing solutions to Plateau's Problem in certain non-minimal configurations.\N\end{itemize}\N\end{enumerate}\N\N\N\end{enumerate}