Insights Groups, The Path from a Simple Concept to Mysterious Results

[fresh_42]

The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement that the order of two consecutive operations does not matter: $(1+2)+3=1+(2+3).$ That's it. The concept of a group is so simple that I still wonder why it wasn't part of my syllabus at school. And, yet, it covers such different sets like the integers, the hours that the big hand counts, the symmetries in a crystal, the Caesar cipher, or a light switch which is the basis of our electronic world. However, few requirements allow many additional, more specific refinements. In the case of groups, we arrive at strange-sounding results like the fact that the largest finite, and simple, sporadic group has
$$
808017424794512875886459904961710757005754368000000000
$$
many elements. This article is meant to shed some light on the betweens of a light switch and a group with more than $8\cdot 10^{53}$ elements that mathematicians dare to call simple. At least, they also call it the monster group, and the second largest finite, simple, sporadic group with its
$$
4154781481226426191177580544000000
$$
many elements baby monster group. And to be honest, even the simple fact that they found them is still a mystery to me.

This article explains fundamental concepts and only lists the mysterious results. It is meant as an introduction to group theory rather than a treatment of the many special areas into which group theory has branched out. Many statements especially in the sections about examples and structures can be verified by the readers if they wish to practice typical conclusions in group theory.

Continue reading...

 

[martinbn]

What do you mean when you say that the extended Riemann hypothesis is believed to be proven?

 

[fresh_42]

What do you mean when you say that the extended Riemann hypothesis is believed to be proven?

I meant that ERH is the interesting conjecture for cryptography and number theory, RH without ERH not so much.

 

[martinbn]

I meant that ERH is the interesting conjecture for cryptography and number theory, RH without ERH not so much.

Ok, but you wrote that it is believed to be proven. Is that what you meant? Where can we see the prove?

 

[pines-demon]

Most people consider group theory obscure but it is not group theory that they are referring when saying that. Group theory is easy to approach (it can be a good introduction on how to build mathematical proofs in general), the whole machinery that takes time and sweat is representation theory.

 

[fresh_42]

Ok, but you wrote that it is believed to be proven. Is that what you meant? Where can we see the prove?

Thanks. I have changed it to "... the extended Riemann hypothesis, the version that is really believed that has to be proven." I hope it is clearer now, although I'm not 100% sure whether it shouldn't have been "... the extended Riemann hypothesis, the version that is really believed that will have to be proven."

I stumbled upon the property of English that leaves out "that" in subclauses.

 

[fresh_42]

Most people consider group theory obscure but it is not group theory that they are referring when saying that. Group theory is easy to approach (it can be a good introduction on how to build mathematical proofs in general), the whole machinery that takes time and sweat is representation theory.

I think this view is too simplistic. Sure, groups are often defined via some representation, be it linear or not, so it is difficult in many cases to separate both. However, we have e.g. the theorem of Novikov-Boone-Britton: There exists a finitely presented group with an unsolvable word problem. Then we have the discrete Fourier transformation in cryptography, groups in the theory of elliptic curves, in number theory, or in topology, for instance, in knot theory. Group theory does not end with Sylow's theorems.

 

[WWGD]

I meant that ERH is the interesting conjecture for cryptography and number theory, RH without ERH not so much.

Criptography in its present form will be useless when Quantum Computers become common.

 

[pines-demon]

Criptography in its present form will be useless when Quantum Computers become common.

Standard cryptography only.

 

[fresh_42]

Criptography in its present form will be useless when Quantum Computers become common.

I'm not so sure. If factorization becomes faster, keys become larger, too. Error-correcting codes are also part of cryptology. Its principles based on the theory of cyclic groups won't change.

 

[WWGD]

Standard cryptography only.

RSA?

 

[pines-demon]

RSA?

Only that. People are already thinking in implementing post-quantum cryptography.

 

[WWGD]

Only that. People are already thinking in implementing post-quantum cryptography.

I hear hackers have downloaded (standardly-) encrypted comment to be decrypted when Quantum computers become available. Hopefully much of that will be outdated, useless by then.

 

[fresh_42]

It makes one wonder if the inverse is also true, if some of the most profound and mysterious objects in pure mathematics are not just abstract curiosities, but are themselves reflections of a hidden physical architecture. For example, it's known that the statistical distribution of the Riemann Zeta function's non-trivial zeros mirrors the energy level spacing in quantum chaotic systems.

The problem with this point of view is our measurement device, ourselves. We are evolutionarily determined to recognize patterns. We see faces on natural surfaces or in clouds, have countless images that betray our intuition, or what I just heard of yesterday, that Lowell, an MIT professor of astronomy, actually made claims about a population on Mars based on Schiaparelli's canali around 1900. And there have been quite a few attempts to prove the Riemann hypothesis with physical means. So, how can we decide whether there is a hidden architecture or only our bias to see patterns in randomness?

A common opinion among scientists is that a theory has to be nice if it reflects a final truth. Theories based on the different handling of many cases are considered incomplete and ugly. And I think it is true, although Hossenfelder probably disagrees with me. The quadratic potentials found by Newton and Coulomb, Maxwell's equations, Einstein's field equation, the classification of simple Lie algebras, Galois theory, and many more can all be considered as being beautiful. However, we created the terms that were necessary to phrase those theories in a nice way: spherical symmetry, divergence and rotation operators, curvature tensors, Killing-form, and normal subgroups. So what came first, chicken or egg, natural beauty or our aspiration to find a beautiful framework?

You may argue that this aspiration to not rest until we have found beauty is itself part of the hidden architecture, but that only leads us even deeper into the rabbit hole of philosophy. I don't think this question can be decided, neither by Hossenfelder nor me. Maybe we should read Kant's Critique of Pure Reason before we make up our minds.

 

[pines-demon]

The problem with this point of view is our measurement device, ourselves. We are evolutionarily determined to recognize patterns. We see faces on natural surfaces or in clouds, have countless images that betray our intuition, or what I just heard of yesterday, that Lowell, an MIT professor of astronomy, actually made claims about a population on Mars based on Schiaparelli's canali around 1900. And there have been quite a few attempts to prove the Riemann hypothesis with physical means. So, how can we decide whether there is a hidden architecture or only our bias to see patterns in randomness?

A common opinion among scientists is that a theory has to be nice if it reflects a final truth. Theories based on the different handling of many cases are considered incomplete and ugly. And I think it is true, although Hossenfelder probably disagrees with me. The quadratic potentials found by Newton and Coulomb, Maxwell's equations, Einstein's field equation, the classification of simple Lie algebras, Galois theory, and many more can all be considered as being beautiful. However, we created the terms that were necessary to phrase those theories in a nice way: spherical symmetry, divergence and rotation operators, curvature tensors, Killing-form, and normal subgroups. So what came first, chicken or egg, natural beauty or our aspiration to find a beautiful framework?

You may argue that this aspiration to not rest until we have found beauty is itself part of the hidden architecture, but that only leads us even deeper into the rabbit hole of philosophy. I don't think this question can be decided, neither by Hossenfelder nor me. Maybe we should read Kant's Critique of Pure Reason before we make up our minds.

I strongly disagree with most things Hossenfelder has ever said about quantum mechanics interpretations and non-physics stuff. That said, I think she does have a point against this common idea that "beauty has to guide physics". Physics can be "satisfying" and beautiful, but "beauty" is always in the eye of the beholder. Clearly many physics theories were invented not because they were beautiful but because they solved controversies or at least predicted results of an experiments. There are many examples of new theories being called "ugly" at their time, Dirac was repulsed by regularization/renormalization in QED, many people found quantum mechanics ad-hoc and are still looking for a classical sidestep, same with relativity or statistical physics. Most of this theories became physics because they were tested experimentally and solved existing issues. That's what makes them satisfying, mathematical formalization is what later gave them rigor, but I kind of feel that beauty came only after, when we are far from the novelty and we can understand why some piece of physics work.

 

[fresh_42]

That said, I think she does have a point against this common idea that "beauty has to guide physics". Physics can be "satisfying" and beautiful, but "beauty" is always in the eye of the beholder.

This would be an endless debate. I don't think that beauty is subjective; I think there are aesthetic scales on which beauty can be measured, and most scientists agree on what they call beautiful and what is not. As I have already sketched, there are various aspects to be considered: education, understanding, the linguistic scientific framework, and the philosophical dimension of what can be known at all. You are right that beauty often comes afterward. Galois theory is a nice example of this. Galois' original text is far from what Artin wrote in his book with the same title. But is this due to a better understanding, a better mathematical toolbox, or Artin's genius? Or is it inherently already part of the theory? Was David already in the stone, and Michelangelo only had to free him?

I know that my opinion about beauty as an absolute property is a minority opinion. Nevertheless, I do not see how any of those questions surrounding the topic could easily be answered, and even less so outside philosophy.

 

[elias001]

@fresh_42 Here is how abstract algebra get to be useful in real life. People date in groups in order to learn to play the fields, when they are ready, they propose with an expensive diamond ring. If one is of the female gender, they can switch to "...order to learn to not get played in the fields".

 

[Sarinle]

Continue reading...

After reading this, i have a better understanding of Tensor Algebra. It seems that Tensor Algebra is a book keeping system, we keep track of all possible multiplication routes between vectors. Later, we can then impose a specific restriction, taking a quotient algebra, to deal with certain unique systems.

Grassman, then seems to choose an algebra that represents the geometric quantities of Volumes.

In Quantum Mechanics, states are vectors in a Hilbert space, while observables form a vector space of Hermitian operators. By starting from the general tensor algebra and imposing the commutator bracket (with a factor of i to stay within the Hermitian set), this space acquires the structure of a Lie algebra.