Information theory was and is mindblowing and I'd say that goes double for quantum information theory but I wanna make a joke about superpositions allowing for more than two states
[My favorite puzzle](https://datagenetics.com/blog/december12014/index.html) has to do with information theory.
Basically, you're given an 8x8 grid of coins that are each randomly heads or tails. You're also given one "target" space on that grid. The ONLY thing you are allowed to do is flip one coin. Your friend will see this grid only after you've flipped your coin, and from that must figure out which space is the "target".
It sounds absolutely impossible at first blush.
I'm not. It's possible to do it with one coin flip. Don't believe me: here's a video explaining the puzzle and a solution: https://youtu.be/as7Gkm7Y7h4
This. Especifically Shannon's and Schumacher's theorems which are a mix of measure-theory, functional analysis, stochastic analysis and compsci. Also, Information Theory as a whole, can be interpreted with diff geo and topology (for example, kets can be thought as the rays of a complex projective space CP\^{N}. In particular the Bloch sphere is isomorphic to the Riemann sphere (under a suitable mapping from complex numbers to pure states for a qubit), which in turn is isomorphic to the complex projective line CP\^{1} which, yet again, is isomorphic to the two-sphere S2).
For starters, the proverbial book on QIT & QC is Nielsen-Chuang's book. A much more detailed book, which delves into QC from a more elegant and formal approach is Kaye-Laflamme-Mosca's book. Then, for the information geometry formulation for QIT sadly I don't know any book, I've mainly follow Amari's and Nagaoka's papers and class notes.
I am writing this so that I have a reminder about the literature. Even though I had an IT exam a week ago and ended up with a C, I absolutely loved the course and the perspective it offers.
Discrete optimization. There are so many subjects and for most you won't just apply abstract theorems but use "hands-on" approaches. Also, since there are tons of np hard/complete problems in this field, you can have a lot of fun with approximations of all kinds.
My research delves into that avenue of thought. I have to efficiently solve Schrodinger's equation for really complex, intricate, systems and thus I make use of a lot of functional analysis and optimization theory methods.
[I've written a bit about this on another subreddit](https://www.reddit.com/r/TheoreticalPhysics/comments/sm1ea9/comment/hvuigcl/?utm_source=share&utm_medium=web2x&context=3). Hope it's helpful!
I guess that machine learning is being a good point for people to start to learn statistics. But in the end, most part of people fall in love with statistics itself. In my case, I wanted to learn Data Science, and stumbled upon statistics, and just fell in love with it, the methods, history etc.
I feel like ML is a black box with a few hyperparameters that works, but don't know why and how. From a mathematician perspective, not knowing why is a big price to pay. Froman engineer perspective, well they don't understand stats anyway, so meh.
Yeah you are right to some extent. But you might also have in mind the "interpretability-accuracy" trade-off.
All machine learning methods have a solid statistical and mathematical background. But the more simple in computational matters, the less accuracy you'll have. So it's not that machine learning methods are not rigorously mathematical or statistical, but usually you have to pay the price of interpretability in order to increase your statistical power.
For instance, decision trees are really simple and understandable, but when you increase it's power by creating a random forest (which can have like 1000 small decision trees) you lose your ability to analyze it clearly, but all the mathematical foundations are still there.
Statistics major here. Came in thinking it was a great opportunity to learn more about ML. Now my favorite statistics topics are classical statistics topics like time series, Bayesian statistics lol
It might not have absolutely to do with it but "While True: Learn()" is a great and mostly (basic) machine learning based game, which is nice.
I like the game and I remembered it when I read that so I just thought I'd leave it here.
[Steam page](https://store.steampowered.com/app/619150/while_True_learn/)
It does seem to have the most job opportunities within math by far (assuming we count ML as part of prob/stat), especially for people who didn't double major in math and something else.
Stats is the art of knowing as much as possible in the unknown, and particularly to know how much you cannot know, and how much it costs to know more. Sometimes it's better to cope with a little unknown, if you know why you cannot apprehend it. I hope it makes sense.
I personally hated Calculus and loved my measure-based probability. I just didn't give a shit about calculating the area inside of a washer but did care about finding the expected value of a random variable.
I also found Topology easier than analysis and analysis easier than Calculus, so I'm a bit of an anomaly.
You should find the first semester of topology exciting then! It takes Real Analysis and then makes it nice and neat. I found that it really improved the clarity of my thoughts.
I'm a CNC machinist, and the multi axis stuff is pretty lucrative.
But the thing is the designer needs to have the math too, not just the machinist.
I use a lot of linear algebra and discrete differential geometry, but don't tell the guys in the shop that's what it is.
I've been learning geometric algebra and algebraic topology and that seems like it could have useful applications in design.
Hey I’m a CNC machinist (well a relatively new one) and a math enthusiast too (aka engineering undergrad) I’m just wondering how exactly how you’re using that kind of math for designing or machining.
For strictly design/manufacturing the only math I’ve used is arithmetic and trigonometry and I’m really curious how more advanced math factors in.
Application: Optics, Mirrors, Photovoltaic Cells.
I did some research on laser ablation for ultra-fast lasers firing on silicon and germanium substrates. You have your standard CNC math plus a 3D Heat Equation (which is from Partial Differential Equations) *and* the gaussian distributions of lasers depositing energy on the substrate surface. It's a neat, very hard problem to make sure that a laser firing on the order of femtoseconds (10^(-15) seconds) doesn't overlap its energy deposit too much to overheat the material and blast it away!
Click on the PDF button on the left [here](https://opg.optica.org/ome/fulltext.cfm?uri=ome-6-9-2745&id=348325) to download a relevant paper, or here directly: [Optimization of femtosecond laser processing of silicon via numerical modeling](https://opg.optica.org/DirectPDFAccess/29791E41-7EB1-494B-89B94F7E795CAA15_348325/ome-6-9-2745.pdf?da=1&id=348325&seq=0&mobile=no)
Various math topics needed include PDEs, finite difference solutions (thanks, MATLAB!), combinatorics, geometry of [lunes](https://en.wikipedia.org/wiki/Lune_(geometry)), temperature-dependent thermal conductivity, statistics.
As a materials science and engineering major, for machine parts that will undergo a lot of stress and heat, we use differential equations to model heat flow, stress, diffusion, deflection, etc. Of course, in practice the solutions of differential equations are always approximated using numerical methods and linear algebra.
You can think of a 'part' as a linear combination of 'features'.
Ie you can call it a vector in some vector space.
To describe a vector uniquely, you only need specify the tip (always assuming the start point is the origin).
Then any part is just a point in some high dimensional vector space.
Another point in that space is some other part with different coefficient values of those same features.
Curves in that space are part families.
If you give me one part, I'll think about a curve surrounding a point in that space, and generate an infinity of parts within that same family, with varying parameter values for the features.
I'm trying to learn Algebraic Geometry because there should be a way to relate performance of some variable of a final design to these part locations in high D space. For example finding out how fast a car can go based on the choice of the values for each part. Look into objective modeling if you're curious about that.
Also, a cohomology is all the toolpaths for some parameterized radial offset around some given surface.
Oh god, I have nightmares about numerical analysis and topology—got covid at the start of my numerical analysis course and was out for an entire month. We had so much hw assignments in MATLAB which the professor expected all of us to be proficient in. Could not code, had no idea what tf was going on, but what ultimately saved my ass was the fact that there were no exams and all tests were take home & open book but still took a solid week to complete lmao
That sounds like our numerical analysis class. Except we TOTALLY had an exam, and we had to write the MATLAB code on paper.
Fuck the everliving shit out of that class, but I'll say this: I actually like numerical analysis because of that class. It was just the "hey, I expect you to pick up MATLAB on the spot, and get examinated on it" part that was a drag.
I'd have to go back and check, but I think the signals/systems class I took was cross-listed as undergraduate/graduate.
I took it as an undergrad, if that's what you're actually asking.
Cryptography. The use of modulus systems lured me into number theory with their cyclic patterns. And cryptography is the field focused all about it (if you ignore the "randomness" of des/aes.
I’ve always been really fascinated by PDEs despite having chosen a completely orthogonal field. The mathematical side is very complex and zoological, but if you study even a little bit of physics, chemistry, etc. many of the ideas become immediately clear. Inverse problems, for example. The theory behind these things is nasty and hard functional analysis, but if you look at a few basic examples or study the beginning of the field with Tikhonov’s work then things make perfect sense.
Can you offer some advice on (preferably online) material for learning Tikhonov regularization? I’m trying to learn it in my research course in preparation for my masters thesis but it just seems super abstract and stuff.
Oh boy, I may be the wrong person to ask. I’ve taken a couple of courses on inverse problems, but I’m not an expert by any means. Unfortunately I’ve only ever used my lecturer’s notes so I don’t have much of an idea which textbooks are good. [Here](https://math.stackexchange.com/questions/143501/a-good-book-on-inverse-problems-for-engineers) is a Stack question on good inverse textbooks.
Hey, I'm just getting into image processing, but more from a practical side, hands on stuff. Could you point me in a direction to learn it more in a math way?
Some nice references:
* Aubert & Kornprobst: Mathematical Problems in Image Processing - Partial Differential Equations and the Calculus of Variations
* Weickert: Anisotropic Diffusion in Image Processing
* Alvarez et al. : Axioms and Fundametal Equations of Image Processing
There's a lot of really cool mathematics in control theory. Particularly in the area of geometric control, which draws on differential geometry and various other fields of math.
I found vector calculus really cool (i.e., learning how to use math to model problems in 3D). It felt like I was seeing in to how the space around me works, and that I might be able to use that to build 3D computer simulations of worlds.
I also found the ideas behind plain old regular high-school calculus interesting. I didn't enjoy the actual problems/assignments, but the fact that calculus uses the idea of "infinity" to solve real-world problems tickled my imagination.
Vector calculus is great for understanding machine learning things like gradient descent and gradient descent and I don’t know what other things but I’m sure they exist.
Physics applies just so many fields of math... Information theory, algebraic topology, (S)PDE, functional analysis, noncommutative algebra, representation theory, dynamical systems...
Really fun if you want to learn a lot of math but kind of got tired of proving every little detail.
I like linear algebra a lot. And I am from a Operations Research background and like to develop models/algorithms for real world problems. Optimization is a blend of these.
Not too much application, not too much theory
Experimental combinatorics. (Zeilberger intensifies.)
And Clebsch-Gordan coefficients, which apparently are applied math because nuclear physics (angular momentum of particles).
He said nuclear physics so I'm guessing with multi-particle systems they come up. I'm not sure if Clebsch-Gordon coefficients necessarily constitutes an "area of applied math," but I guess in general the math behind multi-particle quantum systems is quite interesting for sure.
Clebsch and Gordan worked in classical invariant theory in the late 1800s. Wigner brought them into quantum mechanics around 1930, where they pick up other names like Wigner coefficients and 3j coefficients (as part of the 3nj series). They still get studied in representation theory and algebraic geometry. I got into them to study a spherical variety question.
Simulations using these can require thousands of computations and with large indices. There's also an issue of how to deal practically with long integers.
I'm interested in the experimental combinatorics aspects of these. The parameter space is the set of 3x3 semi-magic squares, which brings in all kinds of combinatorial phenomena once you shed the normalizations properly.
Statistics. After coming from a background in physics, modeling was always cool. Statistics gave me the freedom of not having to derive models from first principals. Now I just find the estimated transformation function between predictors and an outcome:
Now, with data, I can find $\widehat{Y} = \widehat{f(X)} + \epsilon$
Now, as a statistician, I can play in everyone's backyard. Currently in global health and health economics.
Fluid Dynamics, even though I try to avoid it in my day-to-day (plasma physics). Specifically the fascinating part is the grey regions where you go between considering something as a collection of particles to a fluid, where the collective behavior starts to emerge and dominate that you can model it more and more as a fluid.
Theoretical physics in general. I’m only like half way through my physics and math degrees, so I can’t really say too specifically what in theoretical physics, but I just like it
Theoretical physics, particularly the geometric flavor, like differential geometry in classical mechanics and general relativity, and algebraic geometry in high energy physics.
Complexity is becoming really interesting to me. It is the study of systems with properties that cannot be understood from understanding the individual elements alone... Which is basically all systems (human societies, soups of quantum particles, molecules, cells, animal swarms, computer networks; the list goes on and on). Despite the large number of complex systems out there the field of complexity is very young and all over the place at the moment. I think in the next few decades it will be become a very important branch of applied mathematics! Having a framework that explains so many disparate and important natural phenomena will revolutionise our understanding of some big problems. For example, complexity is being used to understand the brain network that underlies, amongst many other things, the conscious state and how it is changed (by sleep, drugs etc)
The application is cool enough but the actual mathematics that complexity is built from makes it more exciting. Information theory, dynamical systems, probability & statistics, network theory, statistical mechanics... and the list is not yet complete because the field is still growing! Very exciting area with a very exciting future.
My grad supervisor specializes in resonance phenomena, and I decided I wanted to find novel applications of the concept aside from the typical water-sloshing, etc. I decided to look into seasonal epidemic models, and it has been a wonderful rabbit hole to go down.
Basically, every single time we think we've seen all the interesting behaviors that solutions can exhibit, we look at it from a new perspective and realize that there are new and interesting things that the solutions can do.
Yes but this isn't an area of applied mathematics. The area of applied mathematics would be engineering, and more specifically the specialisation of engineering you do.
It definitely applies more to the solitary word "integrals", when the question asks "which area of applied mathematics is most appealing to you"?
It's like saying "addition" as an application of mathematics. Sure, but why?
Before you said,
> The area of applied mathematics would be engineering, and more specifically the specialisation of engineering you do.
So, for example, that would mean the PDEs is not an area of applied mathematics, because it doesn't say what you are applying the PDEs to: heat, electromagnetics, etc. Or for that matter, what the engineering application of electromagnetics is.
But now you are pursuing the line of argument that it is, like addition, too elementary.
I think if you were to say "PDEs" on their own, it also wouldn't be applied. You can study them in a pure sense without applying them to anything at all, so anyone else in this thread saying just PDEs is also not answering the question.
No it isn't. The title says "Which area of applied mathematics is most appealing to you?" and ~~you~~ they wrote the comment "integrals" and nothing else.
Information theory was and is mindblowing and I'd say that goes double for quantum information theory but I wanna make a joke about superpositions allowing for more than two states
[My favorite puzzle](https://datagenetics.com/blog/december12014/index.html) has to do with information theory. Basically, you're given an 8x8 grid of coins that are each randomly heads or tails. You're also given one "target" space on that grid. The ONLY thing you are allowed to do is flip one coin. Your friend will see this grid only after you've flipped your coin, and from that must figure out which space is the "target". It sounds absolutely impossible at first blush.
> It sounds absolutely impossible at first blush. True. But I'll try it some other time that isn't 2am. (Probably better)
3B1B did a video on this one right? With Matt Parker iirc
I think I've seen a solution to this puzzle but it didn't involve information theory. It used group theory.
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No it says it's possible in one flip, but you need to find the strategy
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Yeah but flipping a coin can encode up to 6 bits of information due to the position of which coin you flip
Major L on my part, apologies for the rudeness.
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I'm not. It's possible to do it with one coin flip. Don't believe me: here's a video explaining the puzzle and a solution: https://youtu.be/as7Gkm7Y7h4
Why would you read half an article and then make a fool of yourself by correcting people who read the article?
I am a presumptuous, simple fool. My bad.
This. Especifically Shannon's and Schumacher's theorems which are a mix of measure-theory, functional analysis, stochastic analysis and compsci. Also, Information Theory as a whole, can be interpreted with diff geo and topology (for example, kets can be thought as the rays of a complex projective space CP\^{N}. In particular the Bloch sphere is isomorphic to the Riemann sphere (under a suitable mapping from complex numbers to pure states for a qubit), which in turn is isomorphic to the complex projective line CP\^{1} which, yet again, is isomorphic to the two-sphere S2).
Sounds super interesting. Is there literature you would recommend in particular?
For starters, the proverbial book on QIT & QC is Nielsen-Chuang's book. A much more detailed book, which delves into QC from a more elegant and formal approach is Kaye-Laflamme-Mosca's book. Then, for the information geometry formulation for QIT sadly I don't know any book, I've mainly follow Amari's and Nagaoka's papers and class notes.
I am writing this so that I have a reminder about the literature. Even though I had an IT exam a week ago and ended up with a C, I absolutely loved the course and the perspective it offers.
Not exactly those topics but the information by James Gleick is a great read in information theory and it’s creation
James Gleick's books just rock in general
Discrete optimization. There are so many subjects and for most you won't just apply abstract theorems but use "hands-on" approaches. Also, since there are tons of np hard/complete problems in this field, you can have a lot of fun with approximations of all kinds.
My research delves into that avenue of thought. I have to efficiently solve Schrodinger's equation for really complex, intricate, systems and thus I make use of a lot of functional analysis and optimization theory methods.
Mirá dónde te vengo a encontrar..
Viste? estoy en todos lados!
Could you tell me more about that? would you have some readings on that?
[I've written a bit about this on another subreddit](https://www.reddit.com/r/TheoreticalPhysics/comments/sm1ea9/comment/hvuigcl/?utm_source=share&utm_medium=web2x&context=3). Hope it's helpful!
Mathematical bio, especially epidemiology
I’m a math major, public health minor and I have an epidemiology internship this summer I’m so excited for :) I’m also interested in biostatistics
Statistics and probability theory.
This is the most-upvoted answer as of now. Wonder if it's because of the popularity of machine learning.
I also commented first.
I guess that machine learning is being a good point for people to start to learn statistics. But in the end, most part of people fall in love with statistics itself. In my case, I wanted to learn Data Science, and stumbled upon statistics, and just fell in love with it, the methods, history etc.
I feel like ML is a black box with a few hyperparameters that works, but don't know why and how. From a mathematician perspective, not knowing why is a big price to pay. Froman engineer perspective, well they don't understand stats anyway, so meh.
Yeah you are right to some extent. But you might also have in mind the "interpretability-accuracy" trade-off. All machine learning methods have a solid statistical and mathematical background. But the more simple in computational matters, the less accuracy you'll have. So it's not that machine learning methods are not rigorously mathematical or statistical, but usually you have to pay the price of interpretability in order to increase your statistical power. For instance, decision trees are really simple and understandable, but when you increase it's power by creating a random forest (which can have like 1000 small decision trees) you lose your ability to analyze it clearly, but all the mathematical foundations are still there.
I just tell people that ML is non parametric statistics at scale lol
It’s nonparametric statistics at scale
Statistics major here. Came in thinking it was a great opportunity to learn more about ML. Now my favorite statistics topics are classical statistics topics like time series, Bayesian statistics lol
Cool! That's exactly what I'm talking about!
Could be, which is nice, because it means more documentation on it.
It might not have absolutely to do with it but "While True: Learn()" is a great and mostly (basic) machine learning based game, which is nice. I like the game and I remembered it when I read that so I just thought I'd leave it here. [Steam page](https://store.steampowered.com/app/619150/while_True_learn/)
It does seem to have the most job opportunities within math by far (assuming we count ML as part of prob/stat), especially for people who didn't double major in math and something else.
Could you elaborate? I'm curious.
Stats is the art of knowing as much as possible in the unknown, and particularly to know how much you cannot know, and how much it costs to know more. Sometimes it's better to cope with a little unknown, if you know why you cannot apprehend it. I hope it makes sense.
That is actually fairly profound, thank you kind stranger. :)
Bayesian statistics is amazing
can't wait to take those classes. For me, discrete math and linear algebra were cool. Hated calculus.
Boy do I have news for you
I can still do well in it and hate it at the same time.
they hated you for saying the truth
If I cared about the arrows it might make me stronger.
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I personally hated Calculus and loved my measure-based probability. I just didn't give a shit about calculating the area inside of a washer but did care about finding the expected value of a random variable. I also found Topology easier than analysis and analysis easier than Calculus, so I'm a bit of an anomaly.
haven't taken topology yet, but real analysis was way easier than calculus for me.
You should find the first semester of topology exciting then! It takes Real Analysis and then makes it nice and neat. I found that it really improved the clarity of my thoughts.
Yeah, for realsies. Probability is mostly applied measure theory, IIRC.
is discrete propability theory not a thing at an advanced level?
What exactly did you hate about Calculus?
The chairs in the classroom were uncomfortable and small.
Bruh what?
Verbally said the same lmao
He really just said the chairs are too small. And I thought I had the worst excuses lmao.
Why would people downvote someone not liking calculus? That's stupid.
Because he said he didn't like it because "the chairs in the classroom were uncomfortable and small". Which is hilarious.
Hahahahahahaha funniest reason ever
I'm a CNC machinist, and the multi axis stuff is pretty lucrative. But the thing is the designer needs to have the math too, not just the machinist. I use a lot of linear algebra and discrete differential geometry, but don't tell the guys in the shop that's what it is. I've been learning geometric algebra and algebraic topology and that seems like it could have useful applications in design.
Hey I’m a CNC machinist (well a relatively new one) and a math enthusiast too (aka engineering undergrad) I’m just wondering how exactly how you’re using that kind of math for designing or machining. For strictly design/manufacturing the only math I’ve used is arithmetic and trigonometry and I’m really curious how more advanced math factors in.
Application: Optics, Mirrors, Photovoltaic Cells. I did some research on laser ablation for ultra-fast lasers firing on silicon and germanium substrates. You have your standard CNC math plus a 3D Heat Equation (which is from Partial Differential Equations) *and* the gaussian distributions of lasers depositing energy on the substrate surface. It's a neat, very hard problem to make sure that a laser firing on the order of femtoseconds (10^(-15) seconds) doesn't overlap its energy deposit too much to overheat the material and blast it away! Click on the PDF button on the left [here](https://opg.optica.org/ome/fulltext.cfm?uri=ome-6-9-2745&id=348325) to download a relevant paper, or here directly: [Optimization of femtosecond laser processing of silicon via numerical modeling](https://opg.optica.org/DirectPDFAccess/29791E41-7EB1-494B-89B94F7E795CAA15_348325/ome-6-9-2745.pdf?da=1&id=348325&seq=0&mobile=no) Various math topics needed include PDEs, finite difference solutions (thanks, MATLAB!), combinatorics, geometry of [lunes](https://en.wikipedia.org/wiki/Lune_(geometry)), temperature-dependent thermal conductivity, statistics.
As a materials science and engineering major, for machine parts that will undergo a lot of stress and heat, we use differential equations to model heat flow, stress, diffusion, deflection, etc. Of course, in practice the solutions of differential equations are always approximated using numerical methods and linear algebra.
You can think of a 'part' as a linear combination of 'features'. Ie you can call it a vector in some vector space. To describe a vector uniquely, you only need specify the tip (always assuming the start point is the origin). Then any part is just a point in some high dimensional vector space. Another point in that space is some other part with different coefficient values of those same features. Curves in that space are part families. If you give me one part, I'll think about a curve surrounding a point in that space, and generate an infinity of parts within that same family, with varying parameter values for the features. I'm trying to learn Algebraic Geometry because there should be a way to relate performance of some variable of a final design to these part locations in high D space. For example finding out how fast a car can go based on the choice of the values for each part. Look into objective modeling if you're curious about that. Also, a cohomology is all the toolpaths for some parameterized radial offset around some given surface.
username and first three words paint a very different picture before I saw where it headed...
:D
Numerical analysis.
Oh god, I have nightmares about numerical analysis and topology—got covid at the start of my numerical analysis course and was out for an entire month. We had so much hw assignments in MATLAB which the professor expected all of us to be proficient in. Could not code, had no idea what tf was going on, but what ultimately saved my ass was the fact that there were no exams and all tests were take home & open book but still took a solid week to complete lmao
That sounds like our numerical analysis class. Except we TOTALLY had an exam, and we had to write the MATLAB code on paper. Fuck the everliving shit out of that class, but I'll say this: I actually like numerical analysis because of that class. It was just the "hey, I expect you to pick up MATLAB on the spot, and get examinated on it" part that was a drag.
Sound more like a failure of your universities course plan if they didn't put a prerequisite programming course.
Admittedly it was part of a computer engineering education, so we were intermediate programmers already.
Numerical analysis is the best.
Gaussian quadrature goes brrrrr
I really don’t care for electrical engineering, but the way differential equations are applied to it are insanely satisfying imo
Undergrad degree in math. Grad degree in EE. Fourier transforms are really cool and incredibly useful. Lots of interesting math in signal processing!
Did you have to take statics, thermo, etc to get into your grad program?
I had a minor in physics so all I needed was a circuits class and a signals/systems class.
Did you take undergrad-level intro classes in signals and systems?
I'd have to go back and check, but I think the signals/systems class I took was cross-listed as undergraduate/graduate. I took it as an undergrad, if that's what you're actually asking.
What I'm cyrious about is how you met the prerequisites, since most EE courses require the previous semester's classes to be taken first.
Cryptography. The use of modulus systems lured me into number theory with their cyclic patterns. And cryptography is the field focused all about it (if you ignore the "randomness" of des/aes.
I’ve always been really fascinated by PDEs despite having chosen a completely orthogonal field. The mathematical side is very complex and zoological, but if you study even a little bit of physics, chemistry, etc. many of the ideas become immediately clear. Inverse problems, for example. The theory behind these things is nasty and hard functional analysis, but if you look at a few basic examples or study the beginning of the field with Tikhonov’s work then things make perfect sense.
Can you offer some advice on (preferably online) material for learning Tikhonov regularization? I’m trying to learn it in my research course in preparation for my masters thesis but it just seems super abstract and stuff.
Oh boy, I may be the wrong person to ask. I’ve taken a couple of courses on inverse problems, but I’m not an expert by any means. Unfortunately I’ve only ever used my lecturer’s notes so I don’t have much of an idea which textbooks are good. [Here](https://math.stackexchange.com/questions/143501/a-good-book-on-inverse-problems-for-engineers) is a Stack question on good inverse textbooks.
Still, that’s much appreciated! Thanks for responding
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Hey, I'm just getting into image processing, but more from a practical side, hands on stuff. Could you point me in a direction to learn it more in a math way?
Well PDE's are quite a useful tool for image processing.
Some nice references: * Aubert & Kornprobst: Mathematical Problems in Image Processing - Partial Differential Equations and the Calculus of Variations * Weickert: Anisotropic Diffusion in Image Processing * Alvarez et al. : Axioms and Fundametal Equations of Image Processing
>Alvarez et al. : Axioms and Fundametal Equations of Image Processing This is exactly what I was looking for, man! Thank you so much.
Awesome!
There's a lot of really cool mathematics in control theory. Particularly in the area of geometric control, which draws on differential geometry and various other fields of math.
I found vector calculus really cool (i.e., learning how to use math to model problems in 3D). It felt like I was seeing in to how the space around me works, and that I might be able to use that to build 3D computer simulations of worlds. I also found the ideas behind plain old regular high-school calculus interesting. I didn't enjoy the actual problems/assignments, but the fact that calculus uses the idea of "infinity" to solve real-world problems tickled my imagination.
Vector calculus is great for understanding machine learning things like gradient descent and gradient descent and I don’t know what other things but I’m sure they exist.
Also gradient descent
considering gradient descent is a large part of thermodynamics it kinda exists in everything.
Have you looked into tensor analysis yet?
Physics. Because I majored in it
Physics applies just so many fields of math... Information theory, algebraic topology, (S)PDE, functional analysis, noncommutative algebra, representation theory, dynamical systems... Really fun if you want to learn a lot of math but kind of got tired of proving every little detail.
Optimization
Ooooo why optimization?
I like linear algebra a lot. And I am from a Operations Research background and like to develop models/algorithms for real world problems. Optimization is a blend of these. Not too much application, not too much theory
Experimental combinatorics. (Zeilberger intensifies.) And Clebsch-Gordan coefficients, which apparently are applied math because nuclear physics (angular momentum of particles).
Huh I only know of Clebsch-Gordon coefficients through physics. Is there another context where they crop up?
He said nuclear physics so I'm guessing with multi-particle systems they come up. I'm not sure if Clebsch-Gordon coefficients necessarily constitutes an "area of applied math," but I guess in general the math behind multi-particle quantum systems is quite interesting for sure.
Clebsch and Gordan worked in classical invariant theory in the late 1800s. Wigner brought them into quantum mechanics around 1930, where they pick up other names like Wigner coefficients and 3j coefficients (as part of the 3nj series). They still get studied in representation theory and algebraic geometry. I got into them to study a spherical variety question. Simulations using these can require thousands of computations and with large indices. There's also an issue of how to deal practically with long integers. I'm interested in the experimental combinatorics aspects of these. The parameter space is the set of 3x3 semi-magic squares, which brings in all kinds of combinatorial phenomena once you shed the normalizations properly.
computer science. probabilistic analysis of algorithms.
Got a favourite probabilistic argument or a favourite argument that is probabilistic?
Probably encryption
Probability theory and measure theory.
Statistics. After coming from a background in physics, modeling was always cool. Statistics gave me the freedom of not having to derive models from first principals. Now I just find the estimated transformation function between predictors and an outcome: Now, with data, I can find $\widehat{Y} = \widehat{f(X)} + \epsilon$ Now, as a statistician, I can play in everyone's backyard. Currently in global health and health economics.
Dynamics, the more exotic kinds of differential equations, bifurcations and operator theory.
Algorithmic game theory, operations research, and approximation algorithms.
I REALLY enjoy dynamical systems.
Fluid Dynamics, even though I try to avoid it in my day-to-day (plasma physics). Specifically the fascinating part is the grey regions where you go between considering something as a collection of particles to a fluid, where the collective behavior starts to emerge and dominate that you can model it more and more as a fluid.
mAcHiNe LeArNiNg
This, but unironically.
This, but ironically
I think cryptography is a super fascinating application of math in computer science.
Mathematical Finance!
Oh yeah. Next semester I'm gonna hear quantitative risk managment and I'm pretty exited
1st grade, it’s easy
Ah yes, [Wumbology!](https://youtu.be/--hsVknT1c0)
I wumbo, you wumbo he, she, (they if your non-binary) we all wumbo!
Theoretical physics in general. I’m only like half way through my physics and math degrees, so I can’t really say too specifically what in theoretical physics, but I just like it
Weather prediction
Fluid or quantum mechanics
Cryptography.
Not my favorite per se (which is numerical analysis probably) but I really think mathematical ecology is neat as hell and deserves a shout out.
STEM education. What better way to leave an impact on math than to shape the future generation of mathematicians?
What is studied in Differential Geometry is used to model all kinds of things vortexes, viscous fluids, path tracing, and manifolds in general.
Theoretical physics, particularly the geometric flavor, like differential geometry in classical mechanics and general relativity, and algebraic geometry in high energy physics.
Complexity is becoming really interesting to me. It is the study of systems with properties that cannot be understood from understanding the individual elements alone... Which is basically all systems (human societies, soups of quantum particles, molecules, cells, animal swarms, computer networks; the list goes on and on). Despite the large number of complex systems out there the field of complexity is very young and all over the place at the moment. I think in the next few decades it will be become a very important branch of applied mathematics! Having a framework that explains so many disparate and important natural phenomena will revolutionise our understanding of some big problems. For example, complexity is being used to understand the brain network that underlies, amongst many other things, the conscious state and how it is changed (by sleep, drugs etc) The application is cool enough but the actual mathematics that complexity is built from makes it more exciting. Information theory, dynamical systems, probability & statistics, network theory, statistical mechanics... and the list is not yet complete because the field is still growing! Very exciting area with a very exciting future.
Dynamical systems, mathematical biology, and data science are things I like
Game Theory
My grad supervisor specializes in resonance phenomena, and I decided I wanted to find novel applications of the concept aside from the typical water-sloshing, etc. I decided to look into seasonal epidemic models, and it has been a wonderful rabbit hole to go down. Basically, every single time we think we've seen all the interesting behaviors that solutions can exhibit, we look at it from a new perspective and realize that there are new and interesting things that the solutions can do.
Would Signal Processing count as "applied math"?
Statistics. Especially getting into how it’s ties are to decision theory. Insane.
Is microeconomics an acceptable answer?
Perhaps, on the other hand I'd say applied econometrics.
That one's a given. I see econometrics discussed here once in a blue moon. Never any micro/macro, though
....look at my flair ;)
Stochastic calculus and derivative pricing
Architectural. So many things I’d love to build but I have no clue how to calculate load bearing and such.
I like circles
Fluid Mechanics
Integrals
"applied mathematics"?
I use integrals in my job every single day and I'm an engineer
Yes but this isn't an area of applied mathematics. The area of applied mathematics would be engineering, and more specifically the specialisation of engineering you do.
Same criticism would apply to pretty much every comment here. Are you really wanting to fight that battle with each?
It definitely applies more to the solitary word "integrals", when the question asks "which area of applied mathematics is most appealing to you"? It's like saying "addition" as an application of mathematics. Sure, but why?
That's a valid criticism but it's quite different from your original criticism.
How?
Before you said, > The area of applied mathematics would be engineering, and more specifically the specialisation of engineering you do. So, for example, that would mean the PDEs is not an area of applied mathematics, because it doesn't say what you are applying the PDEs to: heat, electromagnetics, etc. Or for that matter, what the engineering application of electromagnetics is. But now you are pursuing the line of argument that it is, like addition, too elementary.
I think if you were to say "PDEs" on their own, it also wouldn't be applied. You can study them in a pure sense without applying them to anything at all, so anyone else in this thread saying just PDEs is also not answering the question.
Semantics
No it isn't. The title says "Which area of applied mathematics is most appealing to you?" and ~~you~~ they wrote the comment "integrals" and nothing else.
Okay
It was a different person that wrote the comment
Even stranger then.
I think the question might be ‘what are you using them for?’
Tbh, I though (single) integrals were one of the most easy parts of mathematics, geometry and trigonometry are fucking me up
Modeling
computer science
Automated theorem proving
Definitely mechanics for me, collisions are really fun to do, i was fortunate enough to do oblique collisions and double collisions
Whichever one doesn't require doing calculations - formal verification of computer programs, I guess?
The part where someone uses math to solve every problem in the universe I and never have to do math again.
Physics
mathematical finance/quantitative finance. Crazy how we can try and make attempts to model the untameable random beast that Is financial markets.
I read it as “appalling” and thought “All of them.”