The applications tag has no usage guidance.

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### 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, ...

**83**

votes

**18**answers

8k 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 ...

**59**

votes

**10**answers

8k 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 ...

**49**

votes

**28**answers

32k 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 ...

**43**

votes

**15**answers

8k views

### How does the work of a pure mathematician impact society? [closed]

First, I will explain my situation.
In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.
I am in a really bad position, because ...

**36**

votes

**7**answers

3k views

### Deep Learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in ...

**33**

votes

**12**answers

4k 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 ...

**32**

votes

**19**answers

7k 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 ...

**29**

votes

**13**answers

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.
...

**29**

votes

**2**answers

989 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 ...

**23**

votes

**17**answers

7k 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:
...

**20**

votes

**11**answers

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. ...

**20**

votes

**6**answers

3k 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 ...

**19**

votes

**3**answers

951 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 ...

**16**

votes

**10**answers

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) ...

**16**

votes

**12**answers

4k 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, ...

**16**

votes

**11**answers

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 ...

**16**

votes

**3**answers

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 ...

**15**

votes

**7**answers

4k 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 ...

**15**

votes

**5**answers

1k 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.
...

**14**

votes

**3**answers

1k 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

**1**answer

480 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 ...

**13**

votes

**2**answers

605 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 ...

**12**

votes

**4**answers

2k views

### How is differential geometry used in immediate industrial applications and what are some source to know about it?

Intuitively it might be clear that differential geometry is a very applicable subject in engineering and industry. I'd like to know how some industries/companies use differential geometry. I'd guess ...

**12**

votes

**4**answers

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### 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

**3**answers

1k 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

**0**answers

2k 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 ...

**10**

votes

**4**answers

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 ...

**10**

votes

**0**answers

554 views

### Which journals publish applied mathematics with mostly pure mathematics content?

In the spirit of Which journals publish expository work? please advise:
What consistently high quality journals$^1$ today publish results that would otherwise go to a pure mathematics journal if ...

**9**

votes

**6**answers

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 ...

**9**

votes

**4**answers

1k 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 ...

**9**

votes

**1**answer

475 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 ...

**9**

votes

**4**answers

580 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 ...

**8**

votes

**2**answers

3k 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 ...

**8**

votes

**1**answer

3k 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 ...

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votes

**7**answers

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### 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

**7**

votes

**4**answers

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### 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 ...

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votes

**6**answers

3k 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 ...

**7**

votes

**1**answer

556 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

**4**answers

268 views

### Application and usage of representation of integers as sum of powers?

We know that there are many articles and manuscripts from the ancient to date talking about representation of integers as sum of squares, cubes etc. I would like to know what is it the usage and ...

**6**

votes

**1**answer

625 views

### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**6**

votes

**1**answer

326 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

**2**answers

837 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 ...

**6**

votes

**4**answers

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

**1**answer

64 views

### Least-squares solution of systems of Sylvester equations

The Sylvester equation $AX+XB=C$ has been studied quite a lot and there are known algorithms for solving it.
But has the situation where (an over-determined) system of equations $A_{i}X+XB_{i}=C_{i}$ ...

**6**

votes

**1**answer

238 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 ...

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votes

**2**answers

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### 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 ...

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votes

**3**answers

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### 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 ...

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votes

**2**answers

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### 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 ...

**4**

votes

**1**answer

784 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 $PA\vdash$ ...