The foundations of analysis, - Topological ideas

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This subject offers challenging theoretical problems matched with opportunities for important applications and is perfect for an enterprising PhD student. This is not a complete list of our interests, and we always welcome new collaborations. If you would like to work with us, or study for a PhD, then please contact us. The group welcomes applications for postgraduate studies.

Foundations of Quantum Mechanics: Topology of a vector space

Mathematical Sciences. Our research. Group Overview We are interested in the following specific topics: Theoretical foundations of multiparameter persistent homology: Persistent homology has emerged as a key computational tool in topological data analysis, where it provides numerical characteristics of the shape of data at a range of scales.

Rebuilding the foundations

It is also a very interesting new homology theory, with many possible applications within mathematics. Our group is studying the formal properties of this new theory, its links to other homology theories and other structures in algebraic topology. We are in particular investigating symmetries, group actions, and localisation. A significant part of our work is devoted to a deeper understanding of stability and instability of persistent homology. Mapper can be regarded as a form of flexible hierarchical clustering, which combines combinatorial information of the data with its statistical properties.

We are also interested in the stability properties of mapper and its interactions with other topological methods. Topology and neural networks: Neural networks are a very successful part of Machine Learning and a central methodology in the development of Artificial Intelligence AI. The theoretical foundations of this methodology are not so well understood and again topology can be of great help here. We are developing topological methods to understand topological complexity of the data which will help suggest the correct architecture of the network.

This leads to a remarkable geometric characterization of the class of rational complex functions: they are the differentiable functions on the sphere.


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One similarly finds that the elliptic functions complex functions that are periodic in two directions are the differentiable functions on the torus. Functions of three, four, … variables are naturally studied with reference to spaces of three, four, … dimensions, but these are not necessarily the ordinary Euclidean spaces. The idea of differentiable functions on the sphere or torus was generalized to differentiable functions on manifolds topological spaces of arbitrary dimension. Riemann surfaces, for example, are two-dimensional manifolds.

Manifolds can be complicated, but it turned out that their geometry, and the nature of the functions on them, is largely controlled by their topology , the rather coarse properties invariant under one-to-one continuous mappings. In particular, Riemann observed that the topology of a Riemann surface is determined by its genus, the number of closed curves that can be drawn on the surface without splitting it into separate pieces. For example, the genus of a sphere is zero and the genus of a torus is one. Thus, a single integer controls whether the functions on the surface are rational, elliptic, or something else.

The topology of higher-dimensional manifolds is subtle, and it became a major field of 20th-century mathematics. The concepts of topology, by virtue of their coarse and qualitative nature, are capable of detecting order where the concepts of geometry and analysis can see only chaos. The moral of these developments is perhaps the following: It may be possible and desirable to eliminate geometry from the foundations of analysis, but geometry still remains present as a higher-level concept.

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Continuity can be arithmetized, but the theory of continuity involves topology, which is part of geometry. Thus, the ancient complementarity between arithmetic and geometry remains the essence of analysis. Article Media.


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Load Previous Page. Rebuilding the foundations Arithmetization of analysis Before the 19th century, analysis rested on makeshift foundations of arithmetic and geometry , supporting the discrete and continuous sides of the subject, respectively.

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Learn More in these related Britannica articles:. In the next century this program would continue to develop in close association with physics, more particularly mechanics and theoretical astronomy. The extensive use of analytic methods, the incorporation of applied subjects, and….

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