Buy Online, Pick up in Store is currently unavailable, but this item may be available for in-store purchase. Temporarily Out of Stock Online Please check back later for updated availability. Overview This self-contained treatment of Morse Theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory over the reals. The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.
Picard-Lefschetz theory. This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The problem of dealing with the Poisson noise and measurement errors is addressed in Section 6 , where we make use of the persistence theory to remove spurious topological features from the Morse complex.
Practically, the filamentary network and associated voids, walls, etc.
Finally, in Section 7 , we address technical questions such as dealing with boundary conditions, smoothing the identified voids, walls and filaments, and important implementation problems before concluding in Section 8. Importantly, let us emphasize that within this framework, the mathematical theories that we use are fundamentally discrete and readily apply to the measured raw data; the unique supplementary but critical step consists in defining heuristically a consistent labelling of the segments, triangle and tetrahedron of the Delaunay tessellation with regards to the DTFE densities computed at the sampling points see Section 5.
Morse Theory and Floer Homology | alconnielenless.tk
This warrants that all the well-known and extensively studied mathematical properties of the Morse complex are ensured by construction at the mesh level and that the corresponding cosmological structures therefore correspond to well-defined mathematical objects with known mathematical properties.
It also provides a consistent way of reconnecting the corresponding network after the removal of insignificant non-persistent pairs of critical points. Note that a reference is given in the last two pages, in which most mathematical terminologies introduced in Sections 2—4 is defined in relatively simple terms. In that paper, we show the potential of this approach by applying it to typical cosmological data sets: a large-scale dark matter cosmological N -body simulation and the 7th Data Release DR7 of the SDSS galaxy catalogue Abazajian et al.
Mathematically speaking, the Morse theory is concerned with smooth scalar functions say, height of a mountain or the temperature in a room defined over generic manifolds. We will therefore restrict the present discussion to such geometries for the sake of simplicity. The Morse theory provides a way to capture the intricate relation between the geometrical and topological properties of a function. What one means by the geometrical property is basically any property unaffected by rigid motions such as translations or rotations.
If h is the altitude function of a mountain landscape, for instance, the altitude of the highest peak or its total surface are geometrical properties. Topology, on the other hand, captures how points are connected to each other with notions such as that of neighbourhood. Topological properties are invariant under smooth continuous transformations. Sometimes topology is coined to be rubber geometry. If we stick to the landscape analogy and define a mountain as a set of points that can be reached from its summit by going down the slope i.
Indeed, in winter, when covered with snow, or during summer, after the snow melted, the altitude map slightly changes, but the underlying mountain can still be easily identified as the same mountain. For the same reasons, a crest linking two mountains or a valley, for instance, are also topological properties of the landscape.
The relation between geometry and topology is intricate and while modifying the topology certainly requires a modification of the geometry, the reverse is not generally true. For instance, the shape of a mountain may only slightly change with season, but more drastic events such as the explosion of a volcano i. The gradient defines a preferential direction at every point the direction of the steepest ascent except where it vanishes i.
Morse Theory on Complex Grassmannians Construction from the classical outset to Morse-Bott theory
It is said to be of order k if the Hessian matrix has exactly k negative eigenvalues. Intuitively, in 2D, the top of a mountain is the maximum order 2 , a pass is a saddle point order 1 and the bottom of a valley is the minimum order 0. The top left-hand panel of Fig. On this picture, the blue, green and red circles stand for the critical points of order 0 minima , 1 saddle points and 2 maxima , respectively.
Note that according to Definition 2. This condition is essential to the Morse theory: a function f which obeys the Morse theory must necessarily satisfy this constraint. Conversely, such functions are called Morse functions.
A 2D density field with its gradient top left-hand panel , its descending 2-manifolds top right-hand panel , its ascending 2-manifolds bottom left-hand panel and its Morse—Smale complex bottom right-hand panel, see the black and white network. On the central left-hand part of the bottom right-hand panel, an arc i. A Morse function is a smooth function whose critical points are non-degenerate. We will assume from now on that f is a Morse function. The pink curves in the top left-hand panel of Fig. The integral lines of a Morse function actually always have critical points as the origin and destination.
Let us consider the case of an integral line passing through a base point P. One can show that such integral line obeys certain properties:. The integral lines of a Morse function f defined over and passing through a given point P is obtained by following the gradient and minus the gradient from P. It obeys the following properties:.
The set of all the integral lines covers all of and each point P of space belongs to exactly one integral line. An integral line with the base point a critical point P is reduced to that point P. The combination of the first and second properties is particularly interesting, as it allows to classify each point of space according to the origin or destination of its unique integral line.
Such classification defines distinct regions of space called ascending and descending manifolds. The descending k -manifold defines a region of space with dimension k , the set of points reached by integral lines with destination P. There exist exactly d different classes of ascending and descending manifolds, classified according to the order of the critical point at their origin or destination. Note that an ascending or descending d -manifold of a Morse function always spans a domain of dimension d [i. The middle panels of Fig.
The notions of ascending and descending manifolds are actually at the core of the Morse theory and the set of the descending or ascending manifolds is usually called the Morse complex. The Morse complex of a Morse function f is the set of its ascending or descending manifolds. The notion of the Morse complex can actually be extended by following Smale and adding one more condition to a Morse function.
In other words, two ascending and descending manifolds should not be tangent and they should always penetrate each other where they cross i.
Such a non-null intersection is called a Morse—Smale n -cell:. A 1-cell is generally called an arc, a 2-cell is a quad and a 3-cell a crystal. The bottom right-hand panel of Fig.
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The purple region, for instance, is the quad defined by the intersection of the red descending 2-manifold and the blue ascending 2-manifold: all integral lines within this region have the minimum on its boundary as the origin and the maximum as the destination see also the upper right-hand and lower left-hand panels. Similarly, the yellow curve defines an arc at the intersection of the blue ascending 2-manifold and the green descending 1-manifold, as only one integral line has the minimum and saddle point at its extremities as the origin and destination.
The set of all n -cells defines the Morse—Smale complex. The Morse—Smale complex of a Morse—Smale function f is the set of all the n -cells of f. In the same figure, the Morse—Smale complex is described by the critical points and the black and white curves. Basically, the critical points are its 0-cells, the set of black or white curves linking two critical points are its arcs 1-cells , and the regions bounded by a black and a white border are its quads 2-cells.
In the 3D case we will consider in the next sections, the Morse—Smale complex is also composed of 3-cells the so-called crystals.
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Note that the notion of an n -cell is very interesting as it defines a natural partition of space induced by the flow of the gradient, literally dividing it into a so-called cell complex a generalization of the concept of a simplicial complex presented in Section 3. We do not give further details here though as only the concept of the arc is really needed for our purpose; the arcs really define how critical points are connected to each other by integral lines.
Actually, and although this is not formally correct, the reader may find it simpler to only consider the nodes critical points and arcs of the Morse—Smale complex, each arc connecting the critical points at their extremities, two critical points being potentially connected only if their orders differ by 1 i. For instance, the arcs connecting maxima to saddle points are subsets of the ascending 1-manifolds and they enclose the information on how each filament represented by its saddle point connects exactly two maxima.
Note that the geometry of an arc is determined by the integral lines whose origin and destination are the two critical points the arc connects.
climorbasre.tk In this figure, the dark matter density distribution in a cosmological simulation is displayed in the top panel, together with its ascensing 3-, 2- and 1-manifolds on the second, third and fourth panels from the top, respectively. The ascending 3-manifolds associated to minima clearly trace the underdense regions, usually denominated voids. The type 1 critical points trace the geometry of the walls through their ascending 2-manifolds and the filaments are traced by the ascending 1-manifolds, associated to critical points of type 1.
As stated as the beginning of this section, a Morse complex actually establishes the link between the geometrical where are the critical points? The purpose of this paper is to construct, from the particles, a discrete Morse function which closely resembles 5 the sampled density which in fact matches it at the vertex of the tessellation and which will therefore warrant all the corresponding discrete topological features. Even though the idea of applying the Morse theory directly to the analysis of the cosmic web is quite appealing a priori, the task is actually not straightforward in practice.
Indeed, the Morse theory is defined for a Morse function, which is basically a smooth and at least twice differentiable real-valued function satisfying the Morse criterion Definition 2. Whether it is because they result from fundamentally discrete processes, as in the case of the galaxy distribution, or obtained through sampling, as for numerical simulation or observational data, typical astrophysical data sets typically do not comply with those criteria in general.
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In contrast, the discrete Morse theory, first introduced by Forman b , , is a very powerful theory which manages to capture the essence of the smooth Morse theory while still being readily applicable to discrete or sampled data commonly available to scientists. It is basically a combinatorial adaptation of the Morse theory that applies to intrinsically discrete functions defined over simplicial complexes.
Let us start by defining the basic building block of such spaces, the simplex. A k -simplex is the simplest possible geometrical figure of dimension k or simply speaking the k -dimensional analogue of a triangle: a 0-simplex, for instance, is a point, a 1-simplex a segment, a 2-simplex a triangle, a 3-simplex a tetrahedron, etc.
More formally:. In other words, it is the set of points within the smallest possible solid with vertices i. In general, when k and l only differ by 1, a face is called a facet and a coface is called a cofacet.