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 ...
- 32.5k
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 ...
- 114k
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 ...
- 91.8k
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, ...
- 188k
26
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\...
25
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 ...
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, ...
- 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 ...
- 188k
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 ...
- 12.3k
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
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 ...
- 887
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
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 ...
- 1,072
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 ...
- 32.1k
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 ...
- 28.6k
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 ...
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 ...
11
votes
What are applications of asymptotic freeness of random matrices?
Here are some applications of free probability of random matrices:
Neural networks: The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral ...
- 188k
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 ...
- 151k
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 ...
- 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 ...
Only top scored, non community-wiki answers of a minimum length are eligible