5 ms·

Initially dismissed, representation theory is now central to much of mathematics

32 points by dnetesn 6y ago

btilly 6y ago
I have no idea how nautil manages to so consistently talk about mathematical concepts in a way that is so hard to make heads or tails of. Here is a simpler explanation. A group is a collection of elements with a multiplication such that multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative. The classic example is the set of all permutations of a set using function composition as multiplication. "Multiplication" is associative because (f o (g o h))(x) and ((f o g) o h)(x) works out to be f(g(h(x))) for any x. The identity permutation is "leave everything where it was". The inverse of a permutation is simply "map everything in reverse". They spend time talking about S3. That is just the permutations of 3 objects. (Which they visualized as the corners of a triangle.) Another classic example is the set of invertible nxn matrices using multiplication as multiplication. A representation is just a function from a group to some set of invertible matrices such that F(x * y) = F(x) * F(y). Note that a representation does NOT have to be one-to-one - you can always just map everything to the identity matrix. OK, so representations exist. Why would we care? The answer is that each representation of a group shows something about the structure of that group. And says it in a language that we have a lot of tools to work with. Admittedly, the identity representation doesn't say much useful. But the others do. And representations have provided ways to reduce a lot of hard problems to easier ones that we have a better chance to solve. There are a lot of examples, but the most widely known example of a hard problem that used this idea as part of the solution was Andrew Wiles' solution of Fermat's Last Theorem.
- emsal 6y ago
  This is really concise, thanks! Removes a lot of the mysticism that is all too present in mathematics writing for laypeople (this explanation is just as understandable).
  - imustbeevil 6y ago
    Weird, I felt the exact opposite. In the article I can use the pictures to understand exactly what all of these mathematical terms mean. No one who graduated college with a degree in English is going to know what "multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative." means, and that's the first sentence...
    - memexy 6y ago
      What phrasing would have been better in your opinion?
      - deleted 6y ago
        [deleted]
      - imustbeevil 6y ago
        > For example, reflecting the triangle and then rotating it 120 degrees reorders the vertices the same way as if you’d merely performed a different reflection. With pictures, from the article.
      - pathseeker 6y ago
        You have to have fully internalized those concepts already for that explanation to be useful. So it's good for people who already have a somewhat solid mathematical foundation and working knowledge. For it to be useful to a layperson, you'll need to provide concrete examples and more of an explanation of each concept. Or, if you want to help people who are visual learners, something like a triangle.
    - btilly 6y ago
      Not nobody. I know high school kids who would understand that. Furthermore I submit that someone who understands the phrases "linear algebra", "matrices", and "the trace of a matrix" is likely to understand that sentence. Anyone who knows what a "Lie group" is, DEFINITELY understands that sentence. (All of those phrases occur in the article without real explanation.) As for what that sentence means, it means that given any two elements we can "multiply" them. Multiplication is associative, meaning that x * (y * z) is always the same as (x * y) * z. There is an identity simply means that an element we can call 1 exists such that 1 * x is always x. There is a multiplicative inverse means that for every x there is some element we can call x^(-1) such that x * x^(-1) = 1. By not commutative I mean that x * y doesn't have to be the same as y * x. (An example of non-commutative is that if you swap the first two items in a list then the first and third you get a different result than if you swap the first and third then swap the first two items.) But still, you have answered my actual question. What that site attempts to do is, at length, build some intuition about a topic, so that it can talk about it. By contrast my explanation just reminds someone who understands the fundamentals of the topic of the concepts so that we can start talking. So someone like you has a chance to understand something. Someone like me has to work to keep track of where they are, only to say, "You're just talking about group homeomorphisms."
- llamaz 6y ago
  I don't know anything about representation theory, but the way you describe it I thought "how's this different from a homomorphism." All elements of a group (e.g. any x, y in your example), are invertible by the definition of a group. The wikipedia sentence definition makes more sense to me: "a "representation" means a homomorphism from the group to the automorphism group of an object" This seems natural, since if you take _any_ algebraic structure (e.g. a ring), then its automorphism group is a group. Which is a reason why group theory is worth studying - since _any_ algebraic structure can be studied using group theory by considering its automorphism group. (Also, an automorphism group is just the set of symmetries, so group theory can be described as the study of symmetry).
- wutbrodo 6y ago
  > I have no idea how nautil manages to so consistently talk about mathematical concepts in a way that is so hard to make heads or tails of Because they're writing for laymen. I have a degree in math and work in a subfield of CS that relies on math a lot, so I have a lot of practice explaining concepts to some of the smartest people I know without a math background. The tradeoff for your concise expression is that it uses jargon that adds cognitive overhead for people unfamiliar with college-level or higher math, even if they've got a high quantitative intelligence. A non-exhaustive list that would trip up, say, my friend with a PhD in pathology: 1) associative and commutative are not commonly-know terms 2) "A group is a collection of elements 'with' multiplication" is underdetermined. What does it mean to be "with"? The notion that multiplication isn't commutative is unintuitive without understanding that the definition of multiplication is flexible here, defined based on the type of the element in the group. 3) Using o as a composition operator is something that a layman is guaranteed never to have come across 4) What does invertible mean? It's not trivially obvious to a layman that some matrices A have no matrix B such that AB = I. Etc etc etc. Note that each of these gaps and confusing points (for laymen) compound; I'll occasionally read a paper in a field I'm not an expert in (like economics), and the being bogged down every other sentence by confusion is a severe impediment to understanding how these pieces are composed into something novel. Hell, even when I understand each piece, sometimes the most foreign concepts haven't marinated long enough for me to use them as building blocks yet. Whats your complaint here? "I have a math degree, how dare nautilus target anyone but me with their articles?"
  - tsimionescu 6y ago
    This may be very dependent on your education system, but commutativity and associativity are terms I was taught around fifth grade, while function composition, groups and matrix inverses were high school maths. I've lost contact with many of these after finishing my university, but they are still pretty common terms around me.
    - btilly 6y ago
      Where and when I went to school, commutativity and associativity were taught in elementary school, and again in high school because we had all forgotten it. Matrix math was something I learned in university. However that was a long time ago and standards have changed. http://www.corestandards.org/Math/Content/HSN/VM/C/9/ http://www.corestandards.org/Math/Content/HSN/VM/C/9/ indicates that matrix math is now considered high school material across the USA.
ad404b8a372f2b9 6y ago
One thing I've been having trouble with as I'm learning math is to coalesce new concepts into a broader context. When learning things I usually assume there's a tree of knowledge that I can travel back up, but with math it seems like a tangled graph of concepts which can be grouped into many different permutations to form different theories, also it's not entirely clear at what level of abstraction one should stop digging. I'm beyond lost.
- zomglings 6y ago
  Just want to offer some help. If you are interested in math, don't get disheartened about how tangled things seem. Even if you can't find any external references to a broader context, you will find that, given enough time and effort, a broader context emerges in your mind. Dig until you get tired or bored - because you could really keep digging forever. If you ever feel lost, read the masters - Euler and Dirichlet are two favorites. Here is a great book: https://www.amazon.com/Euler-Master-Dolciani-Mathematical-Expositions/dp/0883853280 https://www.amazon.com/Euler-Master-Dolciani-Mathematical-Ex... If you get tired of that, read about the greats. MacTutor has a great archive. I especially like their biographies: https://mathshistory.st-andrews.ac.uk/ https://mathshistory.st-andrews.ac.uk/ The most important thing (only my opnion) is to not lose the sense of wonder at the beauty of the universe as seen mathematically. I don't know what your particular situation is - are you in university or school and studying math or learning math as you juggle a career? Any more specific advice depends on your circumstances and your objectives, but I really hope I was able to help somewhat.
  - zomglings 6y ago
    Re: masters. Also Shelah: https://arxiv.org/abs/math/0211398 https://arxiv.org/abs/math/0211398
cwzwarich 6y ago
I'm not so sure about the historical accuracy of the claim that representation theory was initially dismissed. They quote Burnside being skeptical of representation theory in 1897 in the introduction of his book Theory of Groups of Finite Order, but soon after he this he reformulated Frobenius' work on characters in a group/representation-theoretic framework. Burnside used representation-theoretic methods to prove his eponymous theorem in 1904, and representation theory appears in the second 1911 edition of this book. A lot of people don't realize that the concepts of abstract groups, vector spaces, linear transformations, etc. were not particularly obvious to people, and a lot of results were developed in specific contexts that were later realized to be more general. After the development of abstract algebra, these subjects morphed into something much closer to the standard algebra curriculum we see today.
Koshkin 6y ago
Also discussed at https://news.ycombinator.com/item?id=23473919 https://news.ycombinator.com/item?id=23473919

32 points by dnetesn 6y ago

I have no idea how nautil manages to so consistently talk about mathematical concepts in a way that is so hard to make heads or tails of. Here is a simpler explanation. A group is a collection of elements with a multiplication such that multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative. The classic example is the set of all permutations of a set using function composition as multiplication. "Multiplication" is associative because (f o (g o h))(x) and ((f o g) o h)(x) works out to be f(g(h(x))) for any x. The identity permutation is "leave everything where it was". The inverse of a permutation is simply "map everything in reverse". They spend time talking about S3. That is just the permutations of 3 objects. (Which they visualized as the corners of a triangle.) Another classic example is the set of invertible nxn matrices using multiplication as multiplication. A representation is just a function from a group to some set of invertible matrices such that F(x * y) = F(x) * F(y). Note that a representation does NOT have to be one-to-one - you can always just map everything to the identity matrix. OK, so representations exist. Why would we care? The answer is that each representation of a group shows something about the structure of that group. And says it in a language that we have a lot of tools to work with. Admittedly, the identity representation doesn't say much useful. But the others do. And representations have provided ways to reduce a lot of hard problems to easier ones that we have a better chance to solve. There are a lot of examples, but the most widely known example of a hard problem that used this idea as part of the solution was Andrew Wiles' solution of Fermat's Last Theorem.

emsal 6y ago
This is really concise, thanks! Removes a lot of the mysticism that is all too present in mathematics writing for laypeople (this explanation is just as understandable).
- imustbeevil 6y ago
  Weird, I felt the exact opposite. In the article I can use the pictures to understand exactly what all of these mathematical terms mean. No one who graduated college with a degree in English is going to know what "multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative." means, and that's the first sentence...
  - memexy 6y ago
    What phrasing would have been better in your opinion?
    - deleted 6y ago
      [deleted]
    - imustbeevil 6y ago
      > For example, reflecting the triangle and then rotating it 120 degrees reorders the vertices the same way as if you’d merely performed a different reflection. With pictures, from the article.
    - pathseeker 6y ago
      You have to have fully internalized those concepts already for that explanation to be useful. So it's good for people who already have a somewhat solid mathematical foundation and working knowledge. For it to be useful to a layperson, you'll need to provide concrete examples and more of an explanation of each concept. Or, if you want to help people who are visual learners, something like a triangle.
  - btilly 6y ago
    Not nobody. I know high school kids who would understand that. Furthermore I submit that someone who understands the phrases "linear algebra", "matrices", and "the trace of a matrix" is likely to understand that sentence. Anyone who knows what a "Lie group" is, DEFINITELY understands that sentence. (All of those phrases occur in the article without real explanation.) As for what that sentence means, it means that given any two elements we can "multiply" them. Multiplication is associative, meaning that x * (y * z) is always the same as (x * y) * z. There is an identity simply means that an element we can call 1 exists such that 1 * x is always x. There is a multiplicative inverse means that for every x there is some element we can call x^(-1) such that x * x^(-1) = 1. By not commutative I mean that x * y doesn't have to be the same as y * x. (An example of non-commutative is that if you swap the first two items in a list then the first and third you get a different result than if you swap the first and third then swap the first two items.) But still, you have answered my actual question. What that site attempts to do is, at length, build some intuition about a topic, so that it can talk about it. By contrast my explanation just reminds someone who understands the fundamentals of the topic of the concepts so that we can start talking. So someone like you has a chance to understand something. Someone like me has to work to keep track of where they are, only to say, "You're just talking about group homeomorphisms."
llamaz 6y ago
I don't know anything about representation theory, but the way you describe it I thought "how's this different from a homomorphism." All elements of a group (e.g. any x, y in your example), are invertible by the definition of a group. The wikipedia sentence definition makes more sense to me: "a "representation" means a homomorphism from the group to the automorphism group of an object" This seems natural, since if you take _any_ algebraic structure (e.g. a ring), then its automorphism group is a group. Which is a reason why group theory is worth studying - since _any_ algebraic structure can be studied using group theory by considering its automorphism group. (Also, an automorphism group is just the set of symmetries, so group theory can be described as the study of symmetry).
wutbrodo 6y ago
> I have no idea how nautil manages to so consistently talk about mathematical concepts in a way that is so hard to make heads or tails of Because they're writing for laymen. I have a degree in math and work in a subfield of CS that relies on math a lot, so I have a lot of practice explaining concepts to some of the smartest people I know without a math background. The tradeoff for your concise expression is that it uses jargon that adds cognitive overhead for people unfamiliar with college-level or higher math, even if they've got a high quantitative intelligence. A non-exhaustive list that would trip up, say, my friend with a PhD in pathology: 1) associative and commutative are not commonly-know terms 2) "A group is a collection of elements 'with' multiplication" is underdetermined. What does it mean to be "with"? The notion that multiplication isn't commutative is unintuitive without understanding that the definition of multiplication is flexible here, defined based on the type of the element in the group. 3) Using o as a composition operator is something that a layman is guaranteed never to have come across 4) What does invertible mean? It's not trivially obvious to a layman that some matrices A have no matrix B such that AB = I. Etc etc etc. Note that each of these gaps and confusing points (for laymen) compound; I'll occasionally read a paper in a field I'm not an expert in (like economics), and the being bogged down every other sentence by confusion is a severe impediment to understanding how these pieces are composed into something novel. Hell, even when I understand each piece, sometimes the most foreign concepts haven't marinated long enough for me to use them as building blocks yet. Whats your complaint here? "I have a math degree, how dare nautilus target anyone but me with their articles?"
- tsimionescu 6y ago
  This may be very dependent on your education system, but commutativity and associativity are terms I was taught around fifth grade, while function composition, groups and matrix inverses were high school maths. I've lost contact with many of these after finishing my university, but they are still pretty common terms around me.
  - btilly 6y ago
    Where and when I went to school, commutativity and associativity were taught in elementary school, and again in high school because we had all forgotten it. Matrix math was something I learned in university. However that was a long time ago and standards have changed. http://www.corestandards.org/Math/Content/HSN/VM/C/9/ http://www.corestandards.org/Math/Content/HSN/VM/C/9/ indicates that matrix math is now considered high school material across the USA.

ad404b8a372f2b9 6y ago

One thing I've been having trouble with as I'm learning math is to coalesce new concepts into a broader context. When learning things I usually assume there's a tree of knowledge that I can travel back up, but with math it seems like a tangled graph of concepts which can be grouped into many different permutations to form different theories, also it's not entirely clear at what level of abstraction one should stop digging. I'm beyond lost.

zomglings 6y ago
Just want to offer some help. If you are interested in math, don't get disheartened about how tangled things seem. Even if you can't find any external references to a broader context, you will find that, given enough time and effort, a broader context emerges in your mind. Dig until you get tired or bored - because you could really keep digging forever. If you ever feel lost, read the masters - Euler and Dirichlet are two favorites. Here is a great book: https://www.amazon.com/Euler-Master-Dolciani-Mathematical-Expositions/dp/0883853280 https://www.amazon.com/Euler-Master-Dolciani-Mathematical-Ex... If you get tired of that, read about the greats. MacTutor has a great archive. I especially like their biographies: https://mathshistory.st-andrews.ac.uk/ https://mathshistory.st-andrews.ac.uk/ The most important thing (only my opnion) is to not lose the sense of wonder at the beauty of the universe as seen mathematically. I don't know what your particular situation is - are you in university or school and studying math or learning math as you juggle a career? Any more specific advice depends on your circumstances and your objectives, but I really hope I was able to help somewhat.
- zomglings 6y ago
  Re: masters. Also Shelah: https://arxiv.org/abs/math/0211398 https://arxiv.org/abs/math/0211398

cwzwarich 6y ago

I'm not so sure about the historical accuracy of the claim that representation theory was initially dismissed. They quote Burnside being skeptical of representation theory in 1897 in the introduction of his book Theory of Groups of Finite Order, but soon after he this he reformulated Frobenius' work on characters in a group/representation-theoretic framework. Burnside used representation-theoretic methods to prove his eponymous theorem in 1904, and representation theory appears in the second 1911 edition of this book. A lot of people don't realize that the concepts of abstract groups, vector spaces, linear transformations, etc. were not particularly obvious to people, and a lot of results were developed in specific contexts that were later realized to be more general. After the development of abstract algebra, these subjects morphed into something much closer to the standard algebra curriculum we see today.

Koshkin 6y ago

Also discussed at https://news.ycombinator.com/item?id=23473919 https://news.ycombinator.com/item?id=23473919