🗺️Morse Theory Unit 3 – Hessians and Non–Degenerate Critical Points

Key Concepts and Definitions

• Hessian matrix represents the second-order partial derivatives of a scalar-valued function
• Non-degenerate critical point occurs when the Hessian matrix is non-singular at that point
  • Implies the function has a unique tangent plane at the critical point
• Morse function is a smooth function whose critical points are all non-degenerate
• Index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian matrix
• Morse lemma states that near a non-degenerate critical point, a Morse function can be written as a quadratic form
• Morse inequality relates the number of critical points of a Morse function to the topology of the manifold
• Morse theory studies the relationship between the critical points of a function and the topology of the manifold on which it is defined

Historical Context and Development

• Marston Morse introduced the concept of non-degenerate critical points in the 1920s
• Morse's work built upon earlier contributions by Poincaré and Birkhoff in dynamical systems and topology
• In the 1930s, Morse developed the theory further, establishing the Morse inequalities and the Morse lemma
• Morse theory gained prominence in the 1950s and 1960s, with contributions from Smale, Milnor, and others
  • Smale's work on the h-cobordism theorem and the generalized Poincaré conjecture relied heavily on Morse theory
• In the 1980s and 1990s, Morse theory found applications in physics, particularly in the study of instantons and quantum field theory
• Recent developments include the study of Morse-Bott functions, which allow for degenerate critical points, and the use of Morse theory in data analysis and visualization

The Hessian Matrix: Construction and Properties

• The Hessian matrix $H(f)$ of a function $f: \mathbb{R}^n \to \mathbb{R}$ is defined as: \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix} $$
• The Hessian matrix is symmetric, meaning $\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}$
• At a critical point, the gradient of $f$ vanishes, and the Hessian matrix determines the local behavior of the function
• The eigenvalues of the Hessian matrix at a critical point determine whether the point is a local minimum, maximum, or saddle point
  • If all eigenvalues are positive, the critical point is a local minimum
  • If all eigenvalues are negative, the critical point is a local maximum
  • If there are both positive and negative eigenvalues, the critical point is a saddle point
• The determinant of the Hessian matrix is non-zero at a non-degenerate critical point

Non-Degenerate Critical Points Explained

• A critical point $p$ of a function $f$ is non-degenerate if the Hessian matrix $H(f)(p)$ is non-singular
  • Equivalently, all eigenvalues of $H(f)(p)$ are non-zero
• Non-degenerate critical points are isolated, meaning there exists a neighborhood around the point containing no other critical points
• The Morse lemma states that near a non-degenerate critical point $p$, a Morse function $f$ can be written as: $f(x) = f(p) \pm x_1^2 \pm x_2^2 \pm \cdots \pm x_n^2$
  • The number of minus signs is equal to the index of the critical point
• The index of a non-degenerate critical point determines the local topology of the level sets of the function
  • Index 0 corresponds to a local minimum, index $n$ to a local maximum, and other indices to saddle points
• Morse functions, whose critical points are all non-degenerate, provide a way to study the topology of a manifold through the critical points of a function defined on it

Relationship to Morse Theory

• Non-degenerate critical points are a fundamental concept in Morse theory
• Morse functions, which have only non-degenerate critical points, are the main objects of study in Morse theory
• The Morse lemma provides a standard form for a Morse function near a non-degenerate critical point
  • This allows for the local study of the function and its level sets
• The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
  • Specifically, the weak Morse inequalities state that the number of critical points of index $k$ is greater than or equal to the $k$-th Betti number
  • The strong Morse inequalities provide a more refined relationship involving alternating sums
• Morse theory can be used to prove the Poincaré-Hopf theorem, relating the Euler characteristic of a manifold to the indices of the zeros of a vector field
• The concept of non-degenerate critical points and the Morse lemma can be generalized to infinite-dimensional settings, such as Hilbert spaces, leading to infinite-dimensional Morse theory

Applications in Differential Topology

• Morse theory, built upon the concept of non-degenerate critical points, has numerous applications in differential topology
• The Morse inequalities provide lower bounds for the Betti numbers of a manifold in terms of the critical points of a Morse function
  • This can be used to prove the existence of certain topological features, such as non-contractible loops or higher-dimensional holes
• Morse theory can be used to study the topology of sublevel sets of a Morse function
  • The sublevel set $M^a = \{x \in M : f(x) \leq a\}$ changes its topology only when $a$ passes through a critical value
  • The change in topology is determined by the index of the critical point
• The handle decomposition of a manifold can be obtained using Morse theory
  • Each critical point of index $k$ corresponds to the attachment of a $k$-handle
  • This provides a way to build up the manifold from simple pieces
• Morse theory has been applied to the study of the topology of complex algebraic varieties
  • The Lefschetz hyperplane theorem can be proved using Morse theory
• In symplectic topology, Morse theory is used to study the Floer homology of Lagrangian submanifolds
  • The critical points of the action functional correspond to intersection points of Lagrangian submanifolds

Computational Techniques and Examples

• Computational methods can be used to find and classify critical points of a function
• The Newton-Raphson method is an iterative algorithm for finding critical points
  • It uses the first and second derivatives of the function to converge to a critical point
  • The Hessian matrix is used to determine the type of the critical point (minimum, maximum, or saddle)
• Gradient descent and its variants (e.g., stochastic gradient descent) can be used to find local minima of a function
  • These methods follow the negative gradient of the function to converge to a minimum
  • However, they may get stuck in saddle points or local minima that are not global minima
• Example: Consider the function $f(x, y) = x^2 - y^2$
  • The critical points are $(0, 0)$, which is a saddle point (index 1)
  • The Hessian matrix at $(0, 0)$ is $\begin{bmatrix} 2 & 0 \\ 0 & -2 \end{bmatrix}$, which has eigenvalues 2 and -2
• Example: The height function on a torus has four critical points: a maximum, a minimum, and two saddle points
  • The Morse inequalities imply that the torus has non-trivial first and second homology groups
• Software packages like Mathematica and MATLAB have built-in functions for finding critical points and computing Hessian matrices

Advanced Topics and Current Research

• Morse-Bott theory is a generalization of Morse theory that allows for degenerate critical points
  • A Morse-Bott function is a smooth function whose critical set consists of non-degenerate critical submanifolds
  • The Morse-Bott lemma provides a normal form for the function near a critical submanifold
• Floer theory is an infinite-dimensional analogue of Morse theory used in symplectic topology
  • Floer homology groups are defined using the critical points of an action functional on the loop space of a symplectic manifold
  • The Floer complex is generated by the critical points, with the differential given by counting pseudo-holomorphic curves
• Morse theory has been applied to the study of the topology of energy landscapes in chemical physics
  • The critical points of the potential energy function correspond to stable and unstable configurations of a molecule
  • The topology of the energy landscape can provide insights into reaction pathways and conformational changes
• Discrete Morse theory is a combinatorial analogue of Morse theory for cell complexes
  • A discrete Morse function assigns a real number to each cell, satisfying certain conditions
  • The critical cells of a discrete Morse function generate a chain complex that computes the homology of the cell complex
• Research continues on the application of Morse theory to data analysis and machine learning
  • Morse theory can be used to study the topology of high-dimensional data sets and to guide the construction of neural networks
  • The topology of the loss landscape of a neural network can provide insights into the optimization process and the generalization performance of the network