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46 votes
Accepted

Is algebraic geometry constructive?

If you forget about all the layers of abstraction, algebraic geometry is, ultimately (and very roughly speaking), the study of polynomial equations in several variables, and of the geometric objects ...
Gro-Tsen's user avatar
  • 30.7k
40 votes

What practical applications does set theory have?

Set theory is an extremely convenient language for being able to rigorously define and manipulate various "completed infinities" - not only just infinite sets such as the natural numbers or real ...
Terry Tao's user avatar
  • 112k
30 votes

Applications of complex exponential

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from ...
Alexandre Eremenko's user avatar
27 votes

Interesting and surprising applications of the Ising Model

An application of the Ising model in social sciences is to voter models: The dynamics of the Ising model tries to align neighbouring spins, similarly, perhaps, to humans deciding on their political, ...
Carlo Beenakker's user avatar
24 votes

Applications of mathematics in clinical setting

Machine learning is on its way to provide the type of personalized health care referred to by the OP. In June of this year the FDA (US Food and Drug Administration) has proposed a regulatory framework ...
24 votes

What are some nice uses of ultraproducts/ultrapowers?

One of my favourite applications is proving that $\sqrt2$ is irrational using ultraproducts. This requires knowing a nontrivial fact, that there are infinitely many prime numbers $p$ such that $x^2\...
21 votes

What are some nice uses of ultraproducts/ultrapowers?

Here are a few common uses that come to mind: Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by ...
19 votes

Is algebraic geometry constructive?

Yes it is! We have Gröbner basis algorithms that can answer the question of ideal membership and can be used to answer many other geometric questions. If you are interested in this further, Cox, ...
Watson Ladd's user avatar
  • 2,429
18 votes

Applications of complex exponential

Early applications of $e^{i\omega t}$ in the context of electromagnetism were understood as a mathematical device: the physical fields are real, and the complex exponential is a convenient method to ...
Carlo Beenakker's user avatar
17 votes

What "real life" problems can be solved using billiards?

Gregory Galperin invented billiard method of computing $\pi$, see Playing Pool With $\pi$ (The Number $\pi$ From A Billiard Point Of View) To calculate $\pi$, take two identical balls. Put one near a ...
Alexey Ustinov's user avatar
17 votes
Accepted

Applications of mathematics in clinical setting

An example of a simple mathematical/evolutionary game theory model used to determine treatment scheduling in clinical treatment of metastic and castrate resistant prostate cancer can be found at https:...
16 votes

Mathematicians learning from applications to other fields

If I may substitute "physics" for "engineering", one could argue that string theory is an example of a topic where mathematicians have interacted with a different field of research ...
15 votes

Research in applied algebraic geometry that essentially needs a background of modern algebraic geometry at Hartshorne's level

I will ignore the issue of what is "applicable" and what is only "potentially applicable", and the issue of whether something could be translated into classical language, and simply offer an example ...
Pop's user avatar
  • 867
15 votes

Listing applications of the SVD

I teach a course on Applied Linear Algebra, intended for Engineers, where the final project is always to give a presentation on an application of linear algebra in the student's field of study. Since ...
14 votes

Most striking applications of category theory?

David Spivak has found applications of category theory in many areas outside of pure mathematics, and many are recorded in his book “Category Theory for the Sciences.” He's also done important work ...
14 votes

Is algebraic geometry constructive?

Perhaps you will not consider this as a real-world application, but in recent years Theoretical Computer Science is using more and more Algebraic Geometry. For example, one main approach for attacking ...
Adam Sheffer's user avatar
  • 1,052
14 votes

Applications of mathematics in clinical setting

I'm going to conflate mathematics with statistics as Carlo Beenakker did. Then the earliest application that I know of is that Decision Trees were invented by Breiman et al. to analyze the issue of- ...
14 votes

Non-set-theoretic consequences of forcing axioms

Indeed there is a vast of applications, for example: Using Martin's axiom, Shelah showed that there is a non-free Whitehead group. The book `` Consequences of Martin's Axiom'' contains many other ...
Mohammad Golshani's user avatar
14 votes

Mathematicians learning from applications to other fields

I think the best example for interactions between engineers and mathematicians is FEM —Finite Element Method— and FEA —Finite Element Analysis—. The finite element method (FEM) for solving partial ...
13 votes

What are some nice uses of ultraproducts/ultrapowers?

Ultraproducts are useful in certain applications to combinatorics, with a famous example being Hrushovski's work on finite approximate groups, followed by the structure theorem of Breuillard, Green, ...
12 votes

Mathematicians learning from applications to other fields

Not sure if this is what you had in mind, but experiments on the random packing of tetrahedral dice were able to achieve denser packings than had been constructed by mathematicians at the time. Both ...
11 votes

Reference Request: Theoretical Mixing Times Research in Machine Learning / Artificial Intelligence (AI)

The question as asked is rather broad, because there are several works in ML/AI dedicated to mixing time analysis, as well as to detecting if mixing has happened. I would not draw too sharp a boundary ...
Suvrit's user avatar
  • 28.5k
11 votes

Interesting and surprising applications of the Ising Model

The Ising model defines a universality class, meaning lots of systems simplify to something that looks basically like a magnet. Renormalisation tells us that lots of systems share universal asymptotic ...
Dalton A R Sakthivadivel's user avatar
11 votes

What are some nice uses of ultraproducts/ultrapowers?

The paper Chromatic homotopy theory is asymptotically algebraic (by Barthel, Schlank and Stapleton) is an interesting example. In chromatic homotopy theory we study the category $\mathcal{L}(p,n)$ of ...
10 votes

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

To add a touch of very belated whimsy, cohomology has manifestations in art. Here are two: A Penrose triangle (an "impossible figure") can be viewed as a Čech $1$-cycle associated to an open covering ...
10 votes
Accepted

Applications of linear programming duality in combinatorics

How about Boosting1 and the Hardcore Lemma, as described in this paper? Trevisan, Luca, Madhur Tulsiani, and Salil Vadhan. "Regularity, boosting, and efficiently simulating every high-entropy ...
Joseph O'Rourke's user avatar
10 votes
Accepted

Persistent homology over the integers

As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded ...
Vidit Nanda's user avatar
  • 15.5k
10 votes

Applications of mathematics in clinical setting

I know of one mathematical system that was used before the widespread use of CT scans of brain in diagnosing stroke type. That of Scoring methods, which gave at those days a clinical decision that is ...
10 votes

Tools from other disciplines useful to mathematics research?

The William G. Pritchard Fluids Lab conducts experiments–actual, physical experiments with messy, wet fluids–in the Penn State Math Department. Perhaps this counts as drawing on laboratory techniques ...

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