Topology: Point-Set And Geometric

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A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one areconsidered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T-geometric one. T-geometry is the simplest geometric conception (essentially, only finite point sets are investigated) and simultaneously, it is the most general one. Itis insensitive to the space continuity and has a new property: the nondegeneracy. Fitting the T-geometry metric with the metric tensor of Riemannian geometry, we can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in T-geometry, but it is absent in the Riemanniangeometry. In T-geometry, any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. T-geometric conception appears to be more consistent logically, than the Riemannian one.

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one form or another as the fundamental constituents of geometry, physics, vector graphics, and many other fields.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line, this is easily confirmed under modern expansions of Euclidean geometry, and had grave consequences at the time of its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's axiomatization of points was neither complete nor definitive, as he occasionally assumed facts that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points, but in spite of this, modern expansions of the system have since removed these assumptions.

Combinatorially inspired configuration spaces, such as arrangements of points, lines, hyperplanes, polytopes, and the like, provide intricate material and ongoing challenge for topological and geometric techniques. The latter have often gone through a process of adjustment towards their discrete, stratified objects, as in the case of discrete Morse theory or application of Fourier analysis. Notably, the recent solution of the log-concavity conjecture for matroids by Adiprasito, Huh and Katz was achieved by developing Hodge theory for combinatorial geometries which opens up most exciting perspectives on further applications. The construction of higher dimensional expanders is yet another promising direction. Inspired by the rich theory of graph expanders and drawing on the techniques of combinatorial algebraic topology, the goal is to design high-dimensional simplicial complexes with strong connectivity properties. New applications abound, like the impact of discrete geometry on social choice and mathematical economics through balancing theorems and equilibrium configurations.

This workshop is about building bridges - providing intricate, combinatorially inspired spaces to the topologist and geometer, and versatile geometric tools to the combinatorialist. Computational and algorithmic aspects as well as experimental evidence are crucial for this purpose.

Comprehensive mathematics: Explore topics such as combinatorics, generating functions and partitions, graph theory, probability, combinational game theory, Galois theory, linear algebra, prime and factorization algorithms, congruencies and quadratic reciprocity, geometry of numbers, Euclidean and non-Euclidean geometries, geometric transformations, algebraic geometry, point-set topology, and knot theory.

The objective of research in TDA is to develop tools to detect and visualize these kinds of geometric features, and to develop methodology for quantifying the statistical significance of such features in randomly sampled data. Because much of the data arising in scientific applications lives in high-dimensional spaces, the focus is on developing tools suitable for studying geometric features in high-dimensional data.

In topology, we distinguish between several different kinds of holes. A hole at the center of a donut is an example of a first kind of hole; the hollow space inside an inflated, tied ballon is an example of a second kind of hole. In geometric objects in more than three dimensions, we may also encounter other kinds of holes that cannot appear in objects in our three-dimensional world.

As intuitive as the notion of a hole is, there is quite a lot to say about holes, mathematically speaking. In the last century, topologists have put great effort into the study of holes, and have developed a rich theory with fundamental connections to most other areas of modern mathematics. One feature of this theory is a well-developed set of formal tools for computing the number of holes of different kinds in a geometric object. TDA aims to put this set of tools to use in the study of data. Computations of the number of holes in a geometric object can be done automatically on a computer, even when the object lives in a high-dimensional space and cannot be visualized directly.

Besides the number of holes in an object, another (very simple) property of a geometric object that is preserved under bending, twisting, and stretching is the number of components (i.e. separate pieces) making up the object. For example, a plus sign + is made up of one component, an equals sign = is made up of two components, and a division sign is made up of three components. Deforming any of these symbols without tearing does not change the number of components in the symbol. We regard the problem of computing the number of components that make up a geometric object as part of topology. In fact, in a formal sense, this problem turns out to be closely related to the problem of computing the number of holes in a geometric object, and topologists think of these two problems as two sides of the same coin.

As mentioned above, topology offers tools for computing numbers of holes and components in a geometric object; we would like to apply these tools to our study of data. However, a data set X of n points in space has n components and no holes at all, so directly computing the numbers of holes and components of X will not tell us anything interesting about the geometric features of X.

Thus, for nice data sets X, we can get insight into the geometric features of X by studying the topological properties of T(X, Î´). The same strategy also works for studying the geometric features of a data set sitting in a high-dimensional space, in which case the data cannot be visualized directly.

Most data sets we encounter in practice are not as nice as those of Figures 1 and 3, and though the primitive TDA strategy we have described does extend to data in high-dimensional spaces, for typical data sets X in any dimension, the strategy has several critical shortcomings. For one, the topological properties of T(X, Î´) can depend in a very sensitive way on the choice of Î´, and a priori it is not clear what the correct choice of Î´ should be, or if a correct choice of Î´ exists at all, in any sense. Also, the topological properties of T(X, Î´) are not at all robust to noise in X, so that this strategy will not work for studying the geometric features of noisy data sets, such as those in Figures 2 and 4. Moreover, this approach to TDA is not good at distinguishing small geometric features in the data from large ones.

In work published in 2011 by Monica Nicolau, Gunnar Carlsson, and Arnold Levine (Professor Emeritus in the School of Natural Sciences),5 insight offered by TDA into the geometric features of data led the authors to the discovery of a new subtype of breast cancer.

To begin their analysis of the data, the researchers mapped the data from the 24,479-dimensional space into a 262-dimensional space in a way that preserved aspects of the geometric structure of the data relevant to cancer, while eliminating aspects of that structure not relevant to cancer.

The researchers then studied the geometric features of the data in 262-dimensional space using a TDA tool called Mapper.6 They discovered a three-tendril structure in the data loosely analogous to that in the data of Figure 5. In addition, they found that one of these tendrils decomposes further, in a sense, into three clusters. One of these three clusters, they observed, corresponds to a distinct subtype of breast cancer tumor that had hitherto not been identified. This subtype, which the authors named c-MYB+, comprises 7.5 percent of the data set (22 tumors). Tumors belonging to the c-MYB+ subtype are genetically quite different than normal tissue, yet patients whose tumors belonged to this subtype had excellent outcomes: their cancers never metastasized, and their survival rate was 100 percent.

Geometry was one of the earliest branches of mathematics to assume identity as a separate discipline, largely thanks to the efforts of Euclid (fl. c. 300 b.c.) in his work the Elements. Euclid remained the source for all matters geometric well into the nineteenth century, although some by that time had come to worry about the specific form that the axioms (the assumptions from which Euclid started) took. In general, the material of the Elements covered topics like the areas of plane figures and the volumes of solids as well as the more abstract domains of geometry. T