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114
votes
29answers
36k views

Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, ...
69
votes
17answers
6k views

Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
51
votes
10answers
6k views

Applications of mathematics

All of us have probably been exposed to questions such as: "What are the applications of group theory...". This is not the subject of this MO question. Here is a little newspaper article that I found ...
41
votes
28answers
20k views

Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
31
votes
12answers
3k views

Recent Applications of Mathematics

What are the recent and new applications of Mathematics in other Sciences ? Let me try to be more precise about the question: By "recent" I mean the last 15 years. By "new" I want to exclude the ...
26
votes
18answers
6k views

Interesting applications (in pure mathematics) of first-year calculus

What interesting applications are there for theorems or other results studied in first-year calculus courses? A good example for such an application would be using a calculus theorem to prove a ...
23
votes
1answer
537 views

Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...
21
votes
17answers
6k views

Applications of Brouwer's fixed point theorem

I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have: ...
19
votes
6answers
2k views

Applications of group theory to math. biology (pharmacology) ?

Are there applications of group theory (take it broadly: representation theory, Lie algs., q-groups, whatever ... ) to math. biology ? I am in particular interested about applications to pharmacology ...
18
votes
11answers
3k views

Algebraic geometry used “externally” (in problems without obvious algebraic structure).

This is a request for a list of examples of problems (or other mathematical situations) that are not initially of algebro-geometric nature, but can be solved or understood by using algebraic geometry. ...
16
votes
8answers
2k views

Where have you used computer programming in your career as an (applied/pure) mathematician?

For background: I'm working on a book to help mathematicians learn how to program. However, I need to see some examples from people in the field that have done different kinds of things than I have. ...
15
votes
10answers
3k views

Applications of infinite Ramsey's Theorem (on N)?

Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) ...
15
votes
2answers
812 views

Applications of topological and diferentiable stacks

What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...
14
votes
11answers
3k views

Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
14
votes
3answers
2k views

Algebra and Cancer Research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research ? It is well-known that mathematical modeling and computational biology are effective tools in cancer ...
13
votes
11answers
3k views

Mathematics and cancer research?

What are applications of mathematics in cancer research? My answer. Unfortunately I heard quite small about math, but I heard something about applications of physics. And let me put this story here, ...
13
votes
7answers
3k views

Nice applications of the spectral theorem?

Most books and courses on linear algebra or functional analysis present at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many ...
13
votes
3answers
861 views

Applications of and motivation for von Neumann's mean ergodic theorem

I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic ...
13
votes
5answers
812 views

Examples of research on how people perceive mathematical objects

What examples are there on research related to human perception and mathematical objects? For example, the shape of a beer glass influences drinking habits, since people are bad at integrating. ...
13
votes
2answers
596 views

Archaeogenetics

This question is meant to be applied to recover historic information from genetic data. The following model, is (probably) the simplest possible which takes recombinations into account. First, let ...
11
votes
4answers
1k views

Robotics, Cryptography, and Genetics applications of Grothendieck's work? [closed]

I was reading about the passing of Alexander Grothendieck, and something caught my interest: Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal ...
11
votes
3answers
845 views

Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool. The basis is Mamikon's theorem. The area of a tangent sweep is equal to the ...
11
votes
1answer
396 views

Application for Morse-Smale systems

All my life I do topological classification of Morse-Smale systems (flows and cascades). Today, fundamental science is not in fashion, to receive grants require an application. Can you please tell ...
11
votes
0answers
1k views

How connected are you? [closed]

I apologize if this question seems frivolous, but the motivation for it is quite serious. When I encounter the endless topic of the 'relevance' of mathematics, I am rather fond of referring to a ...
9
votes
4answers
1k views

Role of applications in modern mathematics [closed]

Older days scientists were universalists and philosophy, physics and mathematics were a part the same question - understanding the world. Nowadays one may get feeling that the role of applications ...
9
votes
1answer
419 views

Examples of applications of the Freyd-Mitchell embedding theorem.

The Freyd-Mitchell embedding theorem states the following: Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor ...
8
votes
4answers
804 views

Applications of Hilbert's metric

Among the fascinating constructions in mathematics is the Hilbert metric on a bounded convex subset of ${\mathbb R}^n$. Where, within mathematics, is it used ? I know at least a proof of the ...
8
votes
1answer
2k views

Defining “average rank” when not every ranking covers the whole set

Here's a mathematical modeling problem I came across while working on a hobby project. I have a website that presents each visitor with a list of movie titles. The user has to rank them from most to ...
7
votes
4answers
2k views

Applications of Math: Theory vs. Practice

I have a problem: I learned about a lot of the applications of mathematics from academics. Neither they nor I have had much contact with the "real world" to go and see for ourselves how mathematics ...
7
votes
2answers
2k views

Question on “publication List” for applying to post-doctoral jobs

1) Many Mathematics departments ask to send a "list of publications" while applying for research postdoctoral jobs. My question is: how important is it to post my papers in arXiv. I know, posting on ...
7
votes
1answer
535 views

Any nice examples of small cancellation theory appearing in applied mathematics?

Are there any nice discussions of applications of small cancellation theory, or other cases of the word problem, in applied mathematics or algorithms for seemingly non-group theoretic problems? I ...
6
votes
4answers
2k views

Applications of commutative algebra

Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think ...
6
votes
6answers
2k views

Applications and Natural Occurrences of Prime Numbers

I'm fascinated by prime numbers, and over the years, I've found multiple applications and natural occurrences for them. But can anyone suggest some alternatives that aren't in my list? Applications ...
6
votes
1answer
273 views

Costa's minimal surface and the structure of lungs

Seeing this image of Costa's minimal surface        (MathWorld image) made me wonder if the fine-grained structure of the human lung is somehow composed of pieces of ...
6
votes
3answers
405 views

Signal Processing reference for pure mathematician

Before giving a more detailed question below, the basic one is: can anyone recommend a good signal-processing reference which would be maximally readable by a pure mathematician (who nevertheless ...
6
votes
4answers
1k views

Robust black box function minimization with extremely expensive cost function

There is an enormous amount of information about the common applied math problem of minimizing a function.. software packages, hundreds of books, research, etc. But I still have not found a good ...
6
votes
1answer
233 views

Minimizing |FT(X)|_{\infty} by permutation of X_i - question on Fourier transform related to engineering problem (peak factor of OFDM system)

Consider vector X =( X_1 ... X_N), consider the discrete Fourier transform $Y=F(X)$. I am interested to minimize $|Y|_{\infty}$, by permutation of numbers X_i, how to do it ? Here $|Y|_{\infty}$ is ...
5
votes
2answers
2k views

Can I relate the L1 norm of a function to its Fourier expansion?

I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
5
votes
2answers
587 views

Applications of the knot theory to biology/pharmacology ?

What are the applications of the knot theory to biology/pharmacology ? I guess there should be some, since proteins are quite long and probably some of their properties are related whether they are ...
4
votes
7answers
2k views

Is Riemannian integration sufficient in physics?

Are there any applications in physics or engineering which require the Lebesgue integral and cannot be treated by Riemannian integration
4
votes
1answer
541 views

Non-trivial consequences of Lob's theorem

Informally, Löb's theorem (Wikipedia, PlanetMath) shows that: a mathematical system cannot assert its own soundness without becoming inconsistent [Yudkowsky] In symbols: if $PA\vdash$ ...
4
votes
1answer
186 views

Rectifying texture from image

I have a camera matrix $P$ which defines a projective transformation $\mathbb{P}^3 \rightarrow \mathbb{P}^2$. In the former space there is a plane $[ x|\pi^Tx=0 ]$. The image of the plane under $P$ ...
4
votes
1answer
499 views

Any applications integrable systems (pde,ode, q-,…) to math. biology (pharmakinetics, pharmadynamics) ?

Question Are there any relations/applications of integrable system theory (take it as broadly as one can: ODE, PDE, quantum, box-ball,...) to mathematical biology (in particular pharmacokinetics, ...
4
votes
0answers
251 views

Applications of moduli of curves theory

Are there some applications of moduli of curves theory? I was wondering if moduli of curves theory is used (or could be used) for doing research in applied mathematics. I am doing my PhD in algebraic ...
3
votes
3answers
1k views

Why are divisible abelian groups important?

I just quote wikipedia: "Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups." I am asking for detail ...
3
votes
2answers
2k views

On mentioning recommenders' names in cover letter for postdoctoral applications

If I want to apply for a postdoctoral job, can I mention the name of my recommenders in my cover letter just to bolster my application, particularly when I am sure that the people who will read my ...
3
votes
2answers
214 views

Physical and real life interpretation of the concept of regularity used in differential equations?

I guess the title kind of speaks for my questions: I'm curious to know what could be the physical interpretation or real life application of the concept of regularity that arises in PDE: take for ...
3
votes
2answers
2k views

Applications of algebraic geometry/commutative algebra to biology/pharmacology ?

Are there applications of algebraic geometry/commutative algebra to biology/pharmacology ? It might be that some Groebner basis technique is used somewhere ? I know there are some applications to ...
3
votes
2answers
456 views

Applications of propositional dynamic logic

Propositional dynamic logic (PDL) is an example of a (multi)modal logic with a structure on the set of modalities. In particular, the set of its modalities is indexed by "programs" and one can use ...
3
votes
1answer
260 views

Distances on generalizations of the symmetric group

I'm a computer vision student, and I'm looking for some symmetric group literature guidance. I'm going to provide some context, and finally ask two questions. The Cayley distance and other distances ...