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Embedding structures with distortion

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If you have a question about this talk, please contact Gesa Dünnweber.

There is a wide variety of structures that are equipped with a distance. A familiar example from school mathematics is three-dimensional Euclidean space: here the distance is the length of the straight line segment joining two points which can be computed using Pythagoras’s theorem. One might also be interested in the distance between fingerprints: a police investigation might look for fingerprints whose distance is small to a given one thereby narrowing down the list of potential suspects. In Biology one might wish to measure evolutionary distance between species.

Some of these structures have additional features. For example in Euclidean space there is vector addition and scalar multiplication. There is a recent, very active and beautiful area of mathematics that at its core is concerned with the following question. Given an arbitrary structure with a distance, can we embed this in some way into another structure with a distance which has an additional vector structure like Euclidean space. Positive solutions to such questions have consequences for large data, algorithms, compressed sensing, etc. From a pure mathematics point of view what is particularly attractive about this field is the meeting and interaction of many different branches of mathematics: analysis, probability, geometry, combinatorics, etc.

In this talk we will explore the mathematics and some of the applications of this field. We will also cover some specific problems and even give some proofs. Much of this should be accessible to Part IA students.

This talk is part of the The Archimedeans series.

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