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DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

Mathematics is the study of any kind of order that the human mind can recognize.
There are many branches of mathematics, and they are interrelated. The usual classification breaks mathematics into algebra, topology, and analysis. The various branches of geometry are the included under one of those headings or geometry is added as a fourth branch. Combinatorics and logic are additional smaller branches.
Algebra is the study of algebraic structures such as groups, rings, modules, fields and vector spaces and includes the study of solutions of polynomial equations. It has largely adopted geometric methods and much active research involves algebraic geometry.
Topology, sometimes called “rubber sheet geometry” is fundamentally the study of continuous functions defined on general sets which have a topological structure. Topology is broken into pointset or general topology and algebraic topology. Point set topology is fundamental to all of modern mathematics, while algebraic topology adds algebraic methods that are able to address questions efficiently that are intractable by settheoretic methods. Most modern research involves algebraic topology.
Analysis is a large branch of mathematics that is basically “on beyond calculus”. It includes functional analysis, which itself includes Fourier analysis, Hilbert spaces, Banach spaces, Banach algebras, ordinary and partial differential equations, and the representation theory of Lie groups. Complex analysis of one and several variables is also included.
Geometry includes algebraic geometry and differential geometry, both of which use large quantities of analysis and topology and have deep applications to partial differential equations and back again to topology. The recent solution by Pereleman of the Poincare conjecture, a longstanding problem in algebraic topology was based on sophisticated techniques from differential geometry.
Combinatorics is the mathematics of counting, and includes the subject of graph theory, the most widely known theorem being the Four Color Theorem.
Logic is the axiomatic study of set theory and the foundations of mathematics. The work of Kurt Godel demonstrated the limits of logic and the fundamental Zermelo Frankel axioms. Paul Cohen subsequently showed that the "obvioius" Axiom of Choice is independent of those axioms.
Contrary to popular impression the focus of mathematics is not on solving more and more complicated algebraic equations, “solving” integrals or other exercises in symbol pushing. The focus is on understanding and elucidating that order that is recognized.
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Last edited by DrRocket on Fri Jul 06, 2012 1:09 am, edited 1 time in total.





15uliane


Original Member
Joined: Wed Aug 10, 2011 9:09 pm Posts: 110 Location: Boston

I thought you said once that geometry is part of topology?





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

15uliane wrote: I thought you said once that geometry is part of topology? Not generally. Like all of mathematics there are interrelationships. There are topological methods and considerations in geometry. Geometry was used to solve the Poincare conjecture, which was the most famous outstanding problem in topology. Classifications should be taken rather loosely. The more mathematics you understand, the more the boundaries become blurry. Mathematics is a huge subject.
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Azriel


Original Member
Joined: Fri Aug 05, 2011 9:59 pm Posts: 56

What exactly does a manifold represent?
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DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

Azriel wrote: What exactly does a manifold represent? A manifold is a topological space that is "locally Euclidean". The surface of a sphere (like the globe) is a 2dimensional manifold since in small patches it can be modeled as part of a plane. What is represented by a manifold depends on the specific instance. In general relativity spacetime is taken to be a special type of 4manifold, one with a Lorentzian metric.
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GiantEvil


Original Member
Joined: Sat Aug 06, 2011 10:19 am Posts: 786

DrRocket wrote: Azriel wrote: What exactly does a manifold represent? A manifold is a topological space that is "locally Euclidean". The surface of a sphere (like the globe) is a 2dimensional manifold since in small patches it can be modeled as part of a plane. What is represented by a manifold depends on the specific instance. In general relativity spacetime is taken to be a special type of 4manifold, one with a Lorentzian metric. I understand about points on a sphere being modeled as a tangential Euclidean 2space, and that the sphere considered globally is embedded in a 3space. If spacetime is the whole enchilada(not embedded), then locally it's still a 4space?
_________________ It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. W. K. Clifford





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

GiantEvil wrote: I understand about points on a sphere being modeled as a tangential Euclidean 2space, and that the sphere considered globally is embedded in a 3space. If spacetime is the whole enchilada(not embedded), then locally it's still a 4space? The embedding of a 2sphere in 3space is unnecessary. It is jus the wat that you typically encounter a balloon. It is best to just forget about the embedding. Spacetime is locally Minkowski 4space. The lack of an embedding is irrelevant. The whole point of modern manifold theory is that manifolds can be studied independently of any embedding. There are theorems telling you that an embedding in some suitably gargantuan dimension can be found in most circumstances  but it doesn't help much, and this has nothing to do with the physical universe being embedded in anything larger.
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iceaura


Original Member
Joined: Thu Sep 08, 2011 8:05 pm Posts: 391

One view of mathematics is as the creation of virtual sensory organs  software emulation of brain/body hardware setup.
That seems to me a useful viewpoint when considering its use in the physical sciences, say  when matters outside the ranges of our senses need description or investigation.





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

iceaura wrote: One view of mathematics is as the creation of virtual sensory organs  software emulation of brain/body hardware setup.
That seems to me a useful viewpoint when considering its use in the physical sciences, say  when matters outside the ranges of our senses need description or investigation. That is in some respects a fair characterization of how mathematics is used in the physical sciences. But it is rather different from how mathematics is viewed by mathematicians who create mathematics. No surprise there. Similarly engineers who use physics, chemistry, and biology view those subjects a bit differently than do chemists, physicists, and biologists. In physics, in particular, you bring up an important point. Esseantially everything at the forefront of modern physics is outside the range of our senses, and even direct observation by instrumentation. Our understanding is therefore inexricably intertwined with our mathematical models.
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GiantEvil


Original Member
Joined: Sat Aug 06, 2011 10:19 am Posts: 786

_________________ It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. W. K. Clifford





messy


Joined: Thu Sep 29, 2016 6:16 pm Posts: 13

its like an Interpretational dance of monomials i think MODNOTE: Image removed by request from posterthis is mathematics in my humble opinion
Last edited by iNow on Sun Oct 02, 2016 12:40 pm, edited 2 times in total. removed image per request of messy





