1. On the Atiyah—Singer index theorem
I shall explain a little about what the Atiyah—Singer index theorem is, why it is important, and what it is uful for. Here is a brief statement:
Theorem (M.F. Atiyah and I.M. Singer): Let P(f) = 0 be a system of
differential equations. Then
analytical index(P) = topological index(P) .
The word “theorem” (from the Greek “theorein”, to look at, cf. “theater”) means that this is a mathematically proved asrtion that is worthy of clor examination. The result was announced in 1963 and published in 1968.
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2. Introduction
Modern applications of mathematics usually start out with a mathematical model for a part of reality, and such a model is almost always described by a system of differential equations. To make u of the model one eks the solutions to this system of differential equations, but the can be almost impossi
ble to find. The critical new insight of Atiyah and Singer was that it is much easier to answer a slightly different question, namely: “How many solutions are there?” The Atiyah—Singer index theorem gives a good answer to this question, and the answer is expresd in terms of the shape of the region where the model takes place.
一家人简笔画It is a point here that it is not necessary to find the solutions of the system to get to know how many solutions there are. And converly, knowing the number of solutions can in fact make it easier to find the solutions.
A simple analogue can be to look at triangles and quadrangles in the plane. It can be complicated to find the angles in the corners of some of the figures, but sometime before Euclid someone realized that the sum of the angles in all the corners is always 180 degrees for a triangle, and 360 degrees for a quadrangle. The answer to this question is thus easily given, and depends only in a simple way on the shape of the figure, namely whether it has three or four vertices.
The study of functions, derivation and integration is called mathematical analysis (from Greek “analyein”, to break up). The study of the corresponding simple information about the shape of the region where the model takes place (was it a triangle or a quadrangle?) is called topology (from Gre
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ek “topos”, place). The names of the mathematical subfields give ri to the terms analytical index and topological index that we find in the index theorem.
3. Table of contents
Applications of mathematics频繁的反义词
牛的性格A picture of the mathematical world
Analysis
Topology
Mathematical models
Systems of differential equations
Analytical index
The Atiyah--Singer index theorem, revisited
A non-elementary example
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Historical remarks
Conclusion
4. Applications of mathematics
核桃饼干In the beginning, mathematics was ud to count (arithmetic), e.g. for bookkeeping, planning and trade, or to describe shapes (geometry), e.g. for measuring a plot of land, for cutting out fabric for a dress, or for building a bridge. Modern applications of mathematics are often concerned with modeling, and thereby predicting the development over time, of a complex, composite system, such as how oil and gas flow in porous rocks under the North Sea, how queues of text messages in a cellular network can best be resolved, or what the weather will be like this week-end.
Since Newton and Leibniz the mathematical models have almost always been described by a system of differential equations. To u mathematics for the intended application, one eks to find the solutions of this system, The Atiyah—Singer index theorem is a fundamental insight that says that we can find out how many solutions the system has esntially by just knowing some simple, flexible pieces of information about the shape of the region being modeled. Even if the index theorem is a purely mathematical result, which links together analysis and topology, it can thus be us
ed as a tool in almost all applications of mathematics.
5. A picture of the mathematical world
The subject of mathematics can coarly be divided into four areas: algebra, analysis, topology and logic. Mathematics is a diver language that can describe, discuss and model many different objects and problems, and the four areas tend to focus on different aspects of the objects. Nonetheless, there are no clear boundaries between the areas, and mathematics does also not live in isolation from other subjects.
economics physics sciences
computer science algebra analysis applied
mathematics
linguistics logic topology medicine
We shall here emphasize analysis and topology.
6. Analysis
In analysis an object is studied by first partitioning it into small pieces, and thereafter reasmbling them (synthesis). Emphasis is put on the limiting ca when the pieces become arbitrarily small, and simultaneously arbitrarily numerous. Keywords: differentiation, integration and calculus.
Area under a curve A sail?
7. Topology
In topology one studies how an object can have a shape, or a spatial aspect. In particular one emphasizes properties of the whole global shape, rather than the local appearance. If the shape is described by some notion of distance, then we usually talk about geometry.
M.Thistlethwaite: “Symmetric knot”
浙江产假G. Francis, J. Sullivan and S. Levy: “Spherical eversion”
8. Mathematical models
A mathematical model is an attempt to describe (part of) reality in mathematical language. One can also attempt to describe reality on ordinary language, but the mathematical language has the advantage that one can argue and reason with it in a completely preci and indisputable fashion. Therefore one can pursue a chain of thoughts in mathematical language through very many steps and still expect that the conclusion “about reality” is correct.
A mathematical model usually takes place in some spatial domain, or region, which we call X . This mathematical “space” can very well correspond both to space and time in the physical n. For example, in a meteorological model a point in X can correspond to a particular place in the atmosphere above the northern hemisphere at a particular time during the coming week. In a medical model for the electrical impuls that regulate the heart, a point in X can correspond to a small part of the body at a specific time in the cour of a ries of heartbeats.
A torso An electrocardiogram (ECG) The state of the model is described by a ries of numbers for each point of X , e.g. the temperature, air pressure, humidity, wind speed etc. at the particular place in the atmosphere and the particular time. Mathematically this state is described by a ries of functions f defined on the space X . In the medical model such a function can indicate the electrical field strength in the various regions of the body at the various times. The colored surfaces in the image on the left show regions with the same electrical field strength, near the beginning of a heartbeat.
9. Systems of differential equations
The physical laws that govern how the temperature, air pressure etc. (resp. the field strength) will change are well-known as long as one is only considering a small region in X , i.e., only looks as a small part of the atmosphere (resp. the body), for a short span of time. The laws can be expresd as a collection of equations, i.e., a system of equations.
The equations involve the functions f that describe the state of the model, but also the derivatives f'(x) = df/dx of the functions. The are new functions which express how that state changes, either from one place to another, or from one time to another.
The derived functions are also called differentials, and therefore such a collection of equations is called a system of differential equations. Such a system can be briefly expresd in the form P(f) = 0 . A ries of functions f on X that are such that all
