One such language is the one used by the “Freemen on the land“, a fringe quasi-conspiracy-theory group that assigns special meaning to certain common legal phrases. 3) Finally, for many applied mathematicians, computation is a means to an end (e.g., solve the problem, better, faster) as opposed to an equal component of the mathematical problem, one to be studied rigorously for its own sake.

“Applications are whatever I enjoy doing, I enjoy math, therefore all math is applied math!”, An overly narrow definition, i.e. (But I recommend getting really good at programming, whichever path you follow.).  What if the whole universe is just a Turing simulation, the whole world is just a theory? So if you're a result-oriented person, maybe you should indeed move to applied math.  So it turns out that math is a language–people use the vocabulary of math to express ideas. So pure math is not just about 'gaining knowledge' or studying 'interesting stuff'.

(a) I think you are refering to a small fraction of TCS (and even a small fraction of algorithms research) whose concerns overlap with that of Applied math. There are two workshops on Fourier analysis, a topic that is central to applied and computational mathematics, at a conference ostensibly on the Foundations of Computer Science!

Monte carlo simulations certainly need random number generators! What is the point of pursuing theoretical math?

Enter your email address to follow this blog and receive notifications of new posts by email. That’s applied. I am just now starting to see students (and faculty) interested in broader areas such as machine learning, bio-informatics (or more discrete parts of mathematical biology), computer vision, image processing of all sorts, and “data stuff.” TCS is far ahead of math in respect to being outward-looking. Some are amenable to convex relaxation and others aren’t. What about crypto, complexity theory (e.g. See this interesting question for some examples of unintended practical uses of theoretical mathematics.  This is why “it follows easily” in a math paper often signals the hardest, most tedious nonsense you can imagine–the idea of “easy” as we use it in English doesn’t enter into that phrase.

I am just now starting to see students (and faculty) interested in broader areas such as machine learning, bio-informatics (or more discrete parts of mathematical biology), computer vision, image processing of all sorts, and “data stuff.”

 This is particularly a problem if the languages have a high degree of overlap and it isn’t clear in what sense the word is being used. The workshop that Piotr Indyk, Dina Katabi, and I are organizing at FOCS this year is a good example and I encourage anyone interested in learning more about these areas to come. As you say, language and therefore meaning comes prior to grammar, and when one speaks of “pure mathematicians” he/she means those who do mathematics for no reason other than to do so, whereas “applied mathematicians” are doing mathematics entirely for the sake of solving a problem *outside of “mathematics proper”*. I know of others who are also trying this (eg Tim Roughgarden and Ashish Goel at Stanford).  Find me a hipster that thinks graph theory is too “mainstream.”  Didn’t think so.  What if my “theoretical” physics problem leads to a real-life physics breakthrough? or the number of iterations (for any iterative algorithm). The fields you mention above all make sense, but I guess you'll have to pick and focus on one. reply from potential phd advisor. What is the consistency strength of this large cardinal? I took a quick look at the course webpage and assignment and it looks like a fantastic course! So what is a language, anyway?  Sometimes we borrow these words incorrectly: see hoi polloi.

Consider theoretical art versus applied art.

I also agree with you. Courses to take during pure math masters to keep data science and applied work as a possibility. I must respectfully disagree. So here’s the thing:  there is no applied math and no theoretical math. So, my complaint might surprise you, because I’m annoyed that math education doesn’t provide any motivation for what I’m doing. (Edit: If you wish to skip the prologue, you may go straight to the questions in the last few paragraphs.).

 The very idea of such a dichotomy is logically inconsistent, and self-referentially so. Theoretical physics (seems obvious, but worth asking).

Tools like matlab and scipy make implementations and experimentation much easier. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. what exactly is useful.

The speakers span a range of areas from TCS, applied math, and electrical engineering!

I think that Michael Mitzenmacher’s response to Boaz’s post is similar to the points in the last point when he says “I think the larger issue is the slow but (over long periods) not really subtle shift of the TCS community away from algorithmic work and practical applications.” Although, I am not sure either model is better. “If my motivation for doing mathematics…”. But that is just my opinion. fit, but forming that bridge and, more importantly, convincing both sides that they need a bridge rather than a simple function call or a simple hardware circuit, is a hard thing to do and not one the TCS community has been successful at. [1] Like every good rule, this one has an exception.  I’m kidding, I’m kidding. Applied mathematics focus on the mathematical methods used in real life applications in engineering, sciences, economics, finance, and many more subjects. To be honest, my favorite bit has probably been discrete (I am also interested in computer science, if it helps), closely followed by multivariable calculus and linear algebra. So this brings us to the problem: what is an application?  Within the language of mathematics, axioms, functions, numbers, vectors, and pretty much everything else take on a separate, distinct meaning from their ordinary English meaning. "To come back to Earth...it can be five times the force of gravity" - video editor's mistake? What are the similarities and differences in topics between applied and theoretical, especially regarding the ones I mentioned before?  Most people agree it has a lot of applications now.

 Well, obviously, we turn to the dictionary, and read that an application is “a formal request to an authority for something: an application for leave“.  Only a mathematician would think to try to compress the philosophy of every mathematician in the whole world into a bistable formalism, to organize every math department of every university into this formalism, to stake claims to fields and journals and tenure based on this formalism.  Grammar is a formalism that helps, but the language is true and real and defines the grammar, not the other way around.



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