Are fractals the new frontier in mathematics?

[giann_tee]

I'm not sure what to do with some questions I have in mind. Would it be ok to ask for something like fractals, or dynamical systems (sounds big) or mathematical physics as a major forum category?

 

[Moonbear]

Rearrangement of the math forum titles has been recently suggested (just look down a few threads from this one). For now, you can choose General Math, or whichever of the other topics seems like the best fit based on your best judgement of the content of your post. As long as it isn't homework, you won't get "dinged" for posting it in the wrong math forum if the math mentors think it fits better somewhere else and move it.

 

[Chris Hillman]

I [post=1491724]proposed[/post] the following reorganized roster of Math subforums

• Calculus and Miscellaneous
• Combinatorics, Graphs, and Number Theory
• Differential Equations and Dynamical Systems
• Linear and Modern Algebra
• Logic and Foundations
• Manifolds and Geometry
• Probability, Information Theory, and Statistics
• Topology and Analysis

The OPs query provides a good example of why it would be useful to have a conveniently found list of possible topics. In this case,

• fractal quantities arise in "chaotic dynamics" and in "holomorphic dynamics (Julia sets and all that--- dynamical systems meets analysis),
• quantities such as entropy arise in dynamics and IT as the Hausdorff dimension of a fractal set (e.g. the set of "plausible sequences" for a given Markov measure), and Hausdorff dimension is defined in terms of Hausdorff measure, which belongs to measure theory, which is part of modern analysis; plausible sequences involves a probability measure, which also is part of probability and information theory,
• quantities such as "correlation dimension" which arise in less rigorous parts of chaotic dynamics can be thought of as approximations to Hausdorff dimension of certain sets

Thus, I'd say that questions about fractals could fit into "PIT&S" or "T&A", but a vague question about fractals might seem odd the proposed "DE&DS" forum, which would probably mostly contain questions about differential equations, not analysis. In any case, if my scheme were adopted, the question would most likely wind up in "C&M", which would work.

 

[JasonRox]

I don't think we should add a forum. We have 7 forums already and I believe that's sufficient.

It's just a matter of renaming it.

I think it's essential that we have a General Math forum too. It allows for random math talk. A Calculus and Misc. forum wouldn't cut it. It would just end up being all Calculus.

 

[giann_tee]

Thanks everyone for considering all of this. After re-reading my post I realize that this may be more important than whether I get an answer to my queries.

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DYNAMICAL SYSTEMS should be a category of physics. Thats pretty big - its anything from:

advanced knowledge of types of motion, mechanics, oscillators, stability, fractures, cumulative effects, structures, constructing against earthquakes, sound and air flow, aerodynamics, turbulence, vehicles...

materials, media, granular media, vibrations, vibrated grains, avalanche, soil and agriculture, geography in some areas, explosives, weapons...

fluids, weird fluids like soaps...

energy efficiency, heat shields, thermodynamics of biomolecules, biological systems, ecosystems, ecology...

chaotic currents, signal processing, quality of transmission, types of networks...

distribution of energy, grids, solar storms...

building computer models, simple algorithms, precision, weather software, climate change...

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Fractal geometry is great, but that's in the tools departments of math: definitions, software, purpose.

Dynamical systems are anything - its a type of behavior in the 21st century, something new. If you think about it, you can make any sort of home made kitchen experiment ranging from mechanics, granular media, ice crystals... run trials on computer no matter how wrong. On the other hand all other types of discussions involve unsettling inability to perform anything of that kind of a cheap experiments. Categories like "standard model" are just in the clouds.


We all know what mathematical physics meant in the past - lots of vectors, statistics, exact solutions, certain kind of mentality but this is about a nice new notion about how nature is filled with these motions all around and its just getting the right recognition. Not a single parameter about mechanics is a "straight line", "perfect sphere", "ideal elastic", etc. Maybe that's just engineering but then I really can't get any realistic working knowledge out of the standard course.

What I meant to ask is this. I'm reading a graduation work on the department of math and computing related to fractals. There is a sort of an unfinished chit-chat on fabulous demonstrations of fractals before it all goes into equations. Its about a pendulum placed above a table with several magnets. Nothing special there except that with a suitable arrangement you can't calculate a single step in where the pendulum will be.

So, what is the value of Maxwell's electromagnetism if you can't confirm not even for 10 seconds that it's equations work? Is that the law in question ruling the motion of pendulum at all? If 99% of all that happens belongs to a dynamical chaos and if within that "chaos" we see patterns, then are they a physical phenomenon resulted from a Principle and Law or is it a computational artifact and a sort of illusion as we're simply entertained? Does it fit as an effect of evolving entropy?

So you see it doesn't fit into mechanics nor math. I learned that 99% of all differential equations in science can't be solved analytically. Thats' great, less work for me. In case of dynamical systems that's a 99% plus rather immediately than not and in a form of a negative photo provides some argument to study the problem.

Well anyhow, don't listen to me, its either a part of internet experience or not so whatever follows... I just got carried away by the impression.

:-))

 

[Chris Hillman]

I don't think we should add a forum. We have 7 forums already and I believe that's sufficient.

It's just a matter of renaming it.

I was discussing a proposed reorganization of the Math subforums at PF; see [post=1491724]this[/post], which would indeed largely amount to renaming existing subforums. Jason, can you repeat your comment in that thread? The discussion there has been somewhat tentative and I tried to make clear that IMO it's not something to be done in haste. I suggested seeking input from Matt Grime and mathwonk but have been too lazy to PM them so they probably don't yet know about the thread.

DYNAMICAL SYSTEMS should be a category of physics.

This term is generally regarded as referring to a field within mathematics. I am glad to see that you are very enthusiastic about the physical applications, but believe it or not, despite the breadth of possible applications which you mentioned, from my POV (mathematics) you have only touched upon that (relatively!) small part dealing with so-called "chaotic dynamical systems", from a less rigorous stance. See for example the well-known book by Ott, Sauer, and Yorke, Coping with Chaos. For a rigorous approach to fractal dimensions which arise in the study of dynamical systems, see Yakov B. Pesin, Dimension Theory in Dynamical Systems, University of Chicago Press, 1997.

But there is much more to dynamical systems! For example, I wrote a diss on a kind of generalized Penrose tiling (introduced by de Bruijn), which belongs to the subject of symbolic dynamics; see Kitchens, Symbolic Dynamics, Springer, 1998. Historically, the term "dynamical systems" is a recent invention (of Steve Smale) and originally had a more narrow meaning referring to systems of differential equations; see Pierre Tu, Dynamical Systems, Springer, 1994, for an introduction to this part of the subject. You would probably also be interested in the classic monograph by Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, which gets deeper into bifurcations.

For an excellent nontechnical overview which well illustrates the breadth of both the mathematical theory employed in this field, and the wealth of applications, see the two volume "picture book" by Jackson, Perspectives on Nonlinear Dynamics. For a more conventional undergraduate textbook, try Robert C. Hilborn, Chaos and Nonlinear Dynamics, Oxford University Press, 1994. For a graduate textbook try Anatole Katok and Boris Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. All three books emphasize the breadth and depth of the subject.

Fractal geometry is great, but that's in the tools departments of math: definitions, software, purpose.

Not even the less rigorous parts of chaotic dynamics could be described as software

See the book by Pesin for a standard definition-theorem-proof approach to fractal dimensions.

The "theory of dynamical system" is embraces many tightly interconnected mathematical theories, with applications to a huge range of subjects including but not limited to physics.

 

[JasonRox]

I was discussing a proposed reorganization of the Math subforums at PF; see [post=1491724]this[/post], which would indeed largely amount to renaming existing subforums. Jason, can you repeat your comment in that thread? The discussion there has been somewhat tentative and I tried to make clear that IMO it's not something to be done in haste. I suggested seeking input from Matt Grime and mathwonk but have been too lazy to PM them so they probably don't yet know about the thread.

It's not only about the organization of mathematics and how subjects in mathematics fit together. You also have to keep in mind that this a forum and you want people posting in the forums.

With more math forums as there are now, I wouldn't be surprised that 2 of them would become so inactive that they will essentially be dead.

General Math is an active forum, therefore it should remain. Calculus is also active and will always be so there are no worries to lose that. But what you're proposing is to take two very active forums in the math section and lump them together. I completely disagree on that. In the context of keeping as many active forums as possible, that is a bad idea.

 

[giann_tee]

1. Ott, Sauer, and Yorke,
Coping with Chaos

2. Yakov B. Pesin,
Dimension Theory in Dynamical Systems,
University of Chicago Press, 1997

3. Kitchens,
Symbolic Dynamics,
Springer, 1998

4. Pierre Tu,
Dynamical Systems,
Springer, 1994

5. Guckenheimer and Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

6. Jackson,
Perspectives on Nonlinear Dynamics

7. Robert C. Hilborn,
Chaos and Nonlinear Dynamics,
Oxford University Press, 1994

8. Anatole Katok and Boris Hasselblatt,
Introduction to the Modern Theory of Dynamical Systems,
Cambridge University Press, 1995

Thats a very positive response, higher than three in general.

 

[J77]

This is a nice thread -- it should probably be moved to the DS forum

imo, Ref.5 above is very good. It'd add to it:

For a broad look at the basics:

Steven H. Strogatz -- Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry.

Stephen Wiggins -- Introduction to Applied Nonlinear Dynamical Systems and Chaos

R. Seydel -- Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos

More advanced:

Yu. Kuznetsov -- Elements of Applied Bifurcation Theory

Leading to more specialised topics, like:

Arnold et al -- Singularity theory

 

[giann_tee]

I started exploring fractals with Fractint book (associated with a small dos software).

Gleick James, "Chaos" (popular)

Barnsley, available for a dollar

Benoit Mandelbrot, "Fractal Geometry of Nature"

Keneth Falconer - practically unreadable math textbook :-)

Programming Fractals in Pascal (and C), (old)

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Collecting
1. great textbooks
2. free online textbooks
could be a sticky! :-))

I know its a bit difficult to have alive forum categories because the way they branch is not that "organic" but more like a table. Whatever the table there will always be loose ends without activity while some bell shaped curve might fill in the middle with popular homework questions.