There are cultural differences too.
Physicists tend to be interested in mathematical theories only to the extent that those theories formalize or elucidate some physical phenomenon. And mathematicians and physicists often disagree on what questions are interesting or important. Take the quantum theory of electromagnetism, for example, the field to which Dyson made his most notable contributions.
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In his remarks, Dyson identified this as a possible area where maths and physics needed to be reconciled. Still, maths will usually find a way. But a lesser-known phenomenon is what can be called the unreasonable effectiveness of physics in mathematics.
In particular, the roots of the interactions between physics and topology go deeper, and many of the major breakthroughs in topology of the past few decades were inspired by physics ideas. Solving equations from quantum physics, for example, has led topologists to discover surprising and exotic phenomena in 4D space. The path that mathematicians and physicists share has rarely run smooth.
But at least they are both heading in the same direction. They should keep talking — every now and then, with a little more conversation, they can leave science all shook up. For the best commenting experience, please login or register as a user and agree to our Community Guidelines.
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In essence, this means that they do not change under continuous deformation of the space and homotopy is a precise way of formulating the idea of continuous deformation. This module will concentrate on constructing the most basic family of such invariants, homology groups , and the applications of these homology groups.
All shook up over topology
The starting point will be simplicial complexes and simplicial homology. An n-simplex is the n-dimensional generalisation of a triangle in the plane. A simplicial complex is a topological space which can be decomposed as a union of simplices.
The simplicial homology depends on the way these simplices fit together to form the given space. Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex. For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes.
Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space. It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition.