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What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?

There is a lot of books on numerical methods, engineering math, but I do not know any good modern book, which emphasizes algorithmic complexity of the discussed problems.

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    $\begingroup$ Two things. First, I'm very interested in an answer to this question, but pertaining to the applications of more abstract mathematics (algebraic topology and geometry, galois theory, etc). Second, this looks like it's going for a sorted list, so should be community wiki. $\endgroup$
    – Charles Siegel
    Commented Dec 4, 2009 at 13:44
  • $\begingroup$ Are there actually applications of algebraic topology, algebraic geometry, and galois theory anywhere aside from theoretical physics? $\endgroup$
    – Harry Gindi
    Commented Dec 4, 2009 at 15:01
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    $\begingroup$ ALgebraic geometry is ubiquitous in cryptography and information theory. Curves play a major role in both. $\endgroup$
    – Steve Huntsman
    Commented Dec 4, 2009 at 15:20
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    $\begingroup$ I'm told that algebraic topology has lots of applications to sensor networks and target tracking as well, but I really know nothing about these things. I threw in Galois theory as something that I didn't know any examples of what they're applied to, but figured 'hey, it's abstract, but maybe someone will know' $\endgroup$
    – Charles Siegel
    Commented Dec 4, 2009 at 16:12
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    $\begingroup$ Yes, I've been to talks, and simplicial, singular and deRham cohomology were all used. $\endgroup$
    – Charles Siegel
    Commented Dec 4, 2009 at 22:11

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Since the question was tagged with "algorithms", I will give an algorithms recommendation. (You don't say specifically what type of problems you want to solve, but you do mention "algorithmic complexity.") For a book that was written to motivate the theory of algorithms from real-world problems, I would recommend Algorithm Design by Kleinberg and Tardos. It discusses many problem-solving methods. From the website for the book:

Algorithm Design introduces algorithms by looking at the real-world problems that motivate them. In a clear, straight-forward style, Kleinberg and Tardos teaches students to analyze and define problems for themselves and from this, to recognize which design principles are appropriate for a given situation. The text encourages a greater understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science.

Amazon link: http://www.amazon.com/Algorithm-Design-Jon-Kleinberg/dp/0321295358

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Geometric Fundamentals of Robotics by Selig applies algebraic and differential geometry to problems in robotics.

Computational Homology by Kaczynski et al has applications of homology to image processing and nonlinear dynamics.

Robert Ghrist, http://www.math.uiuc.edu/~ghrist/, applies topology to problems in engineering, including robotics and sensor networks.

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  • $\begingroup$ Yep, Rob is the one I was thinking of in my comment that fpqc took issue with. $\endgroup$
    – Charles Siegel
    Commented Dec 4, 2009 at 22:12
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This book by Erica Flapan relating chemistry and algebraic topology was of use to my wife when she has writing her undergrad thesis. It seems like it might qualify.

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Derivations of Applied Mathematicsis a book of applied mathematical proofs. If you have seen a mathematical result, if you want to know why the result is so, you can look for the proof here

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