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Say I am a mathematician who doesn't know any chemistry but would like to learn it. What books should I read?

Or say I want to learn about Einstein's theory of relativity, but I don't even know much basic physics. What sources should I read?

I am looking for texts that teach subjects that are not mathematics, but I do not want to read through standard high school, undergraduate (and beyond) material. I am looking for recommendations of sources that teach a scientific theory from a basic level, but not from a basic mathematical level. Strong preference would be to concise, terse texts that are foundational but totally rigorous.

Not sure if these exist, but I often wish they did.

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    $\begingroup$ A (personal?) finance book for mathematicians is LONG overdue. $\endgroup$ – user347489 Nov 20 at 6:55
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    $\begingroup$ This is one of the most useful MO questions ever. I just bought two of the books mentioned below, on topics I have long wanted to know more about. $\endgroup$ – Yly Nov 20 at 22:34
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    $\begingroup$ Questions of this type come up here repeatedly, often about physics. I'm a physicist, and they strike me as misguided at best. At worst they come off as arrogant. By the same token that there is no royal road to geometry, there is no royal road to physics. Physics, like mathematics, is a large and varied field. If I didn't know basics like freshman calculus or the complex number system, I wouldn't demand to be introduced to all of mathematics without the drudgery of wading through elementary textbooks. $\endgroup$ – Ben Crowell Nov 21 at 15:19
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    $\begingroup$ @BenCrowell I don't think they're implying that they want to skip the elementary parts of [new subject], rather they want to learn [new subject] from a source that presumes prior knowledge of math, so that they can learn more efficiently without having mathematical things over explained to them, meaning they can focus more on the subject itself. Or maybe with the assumption of prior knowledge in maths, the book does not need to artifically limit its scope. Maybe an analogy with different subjects in math might help. Someone here once asked for a book on geometry for analysts. (pt 1/2) $\endgroup$ – James Baxter Nov 22 at 7:46
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    $\begingroup$ ... They found that many texts took lots of care to explain sobolev functions, etc - things that were already familiar; while speeding through geometry background that would be "basic" for a geometer but needed more explanation for the uninitiated. I think the situation here is similar. (pt 2/2) $\endgroup$ – James Baxter Nov 22 at 7:47

22 Answers 22

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This is something I'm trying to learn from, a (text)book on music, written by a mathematician, and for a mathematically literate reader, called Music: A Mathematical Offering.

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    $\begingroup$ I'm taking zoom piano lessons during the pandemic, and a mathematical colleagues of mine suggested this. Packed with fascinating information of all sorts (the structure of the ear, various tonal scales and the mathematical compromises behind them, diophantine issues, and some pretty heavy duty PDEs, it's highly uneven but as the author suggests the best way to read it is to dip in here or there. And the electronic version is free! $\endgroup$ – Douglas Lind Nov 20 at 14:33
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There are three that I can think:

Brian Hall, Quantum Theory for Mathematicians.

and

Sachs & Wu, General Relativity for Mathematicians

Also

Saunders Mac Lane, Categories for the Working Mathematician

All three are excellent and are very readable.

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    $\begingroup$ So you imply that category theory is not mathematics? ;D $\endgroup$ – მამუკა ჯიბლაძე Nov 20 at 6:12
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    $\begingroup$ I took the liberty to correct Hall's name from 'Halls' to Hall. I hope you don't mind. $\endgroup$ – M.G. Nov 20 at 7:01
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    $\begingroup$ @მამუკაჯიბლაძე That title implies only that category theorists are not working mathematicians. ;D $\endgroup$ – Federico Poloni Nov 20 at 7:46
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    $\begingroup$ @M.G. The same should apply to MacLane then $\endgroup$ – მამუკა ჯიბლაძე Nov 20 at 8:13
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    $\begingroup$ @მამუკაჯიბლაძე: you're right ofc! I didn't see it the first time for whatever reasons. Fixed now. $\endgroup$ – M.G. Nov 20 at 8:24
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Michael Spivak's book, Physics for Mathematicians, Mechanics I, would definitely fit the bill. The goal is to discuss the foundations in a way that lays clear the underlying physical principles but doesn't simultaneously have to teach the underlying ideas of calculus.

Quoting from the (as usual, entertaining) introduction, ..."Ah, so you're going to be writing about symplectic structures, or something of that sort. And I would have to say, No, I'm not trying to write a book about mathematics for mathematicians, I'm trying to write a book about physics for mathematicians...."

Reading the first chapter, I felt like the book was written exactly for me!

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    $\begingroup$ Is there a second volume? $\endgroup$ – lalala Nov 21 at 19:28
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    $\begingroup$ @lalala That's a good question; I couldn't find any reference to it. See math.stackexchange.com/questions/322885/…. You could write to the publisher: publishorperishmath@gmail.com; if you learn something then let us know! $\endgroup$ – Danny Ruberman Nov 21 at 22:22
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    $\begingroup$ This book is great, its the only book I know of that is about physics and not about the mathematics behind physics as every other physics book mentioned here. Sadly (having read many physics books written by physicists) it is also the only physics book understandable to me. $\endgroup$ – Jannik Pitt Nov 22 at 12:29
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There are a lot of good physics books for mathematicians. My personal favorite is Mathematical Foundations of Quantum Mechanics by Mackey.

Let me also plug Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics by Wald, especially for anyone who's had some exposure to C*-algebras. It's very readable. Since the quantum fields are free, the treatment is rigorous, but since the underlying space is curved the theory is not trivial. The book culminates in an account of Hawking radiation.

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    $\begingroup$ This looks excellent, thank you for the response $\endgroup$ – Josh Nov 20 at 5:45
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    $\begingroup$ @Josh I forget who said this (maybe Mackey?): the best translations from A to B are done by native speakers of B. I've certainly found that physics books written by mathematicians are generally easier to understand than physics books written by physicists for mathematicians. $\endgroup$ – Nik Weaver Nov 20 at 16:44
  • $\begingroup$ @NikWeaver, on the other hand, to learn language A, I'd rather read a book written in A than one translated into my native language. Alas, that's my general impression about most of those "physics for mathematicians" books and how they compare to just normal physics textbooks. $\endgroup$ – Kostya_I Nov 22 at 10:33
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    $\begingroup$ @Kostya_I: I think the OP is asking "what's a good translation of War and Peace", not "how do I learn Russian so I can read it in the original". $\endgroup$ – Nik Weaver Nov 22 at 14:19
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    $\begingroup$ @NikWeaver, fair enough $\endgroup$ – Kostya_I Nov 22 at 20:51
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Russian for the Mathematician helps you learn basic skills for that language. Not only do they use words like "number" instead of "house", say, to teach declensions, they also use real-life mathematical sample texts and offer an overall more math-inclined approach to language learning.

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  • $\begingroup$ =D I love this so much... and not for the proper reasons. =D $\endgroup$ – An old man in the sea. Nov 22 at 18:08
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Like another answer says, there are lots of good physics books. My personal favorite is Quantum Field Theory: A Tourist Guide for Mathematicians by Gerald Folland

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    $\begingroup$ Folland is known for the excellence of his writing, but I found this book hard to read. Also we should mention that a book in QFT is not the right place to start learning physics! You really want to master elementary quantum mechanics before jumping into field theory. $\endgroup$ – Nik Weaver Nov 20 at 12:48
  • $\begingroup$ @NikWeaver I disagree that there is a right or wrong way to start learning physics. Starting with QFT would be weird (and very hard), but may provide one with a unique perspective which would be quite interesting. $\endgroup$ – Eigentime Nov 20 at 16:05
  • $\begingroup$ @Eigentime that is an interesting suggestion. Still, I would say "very hard" is an understatement ... $\endgroup$ – Nik Weaver Nov 20 at 16:41
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Programming for Mathematicians by Raymond Seroul.

I recommend reading the highly entertaining amazon review by Ian Jakovenko. He refers to the book as "Euclid's Elements for Cybernauts."

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    $\begingroup$ "I recommend reading the highly entertaining amazon review by Ian Jakovenko" - He says "Mathematicians: do NOT read this book!" though :D $\endgroup$ – Qfwfq Nov 21 at 23:56
  • $\begingroup$ On the topic of programming, I would tentatively suggest also "Functional Programming in Scala" by Chiusano and Bjarnason as an introduction to functional programming that has an approach/style that mathematicians may like. $\endgroup$ – Stiofáin Fordham Nov 22 at 11:25
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The IMHO most useful such book hasn't been mentioned yet:

The 85 Ways to Tie a Tie is a book by Thomas Fink and Yong Mao about the history of the knotted neckcloth, the modern necktie, and how to tie both. It is based on two mathematics papers published by the authors in Nature and Physica

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Here is one you may like:

Economics for Mathematicians by J.W.S. Cassels, London Mathematical Society. I am curious what economists have to say about it. Link to Mathscinet review here.

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    $\begingroup$ The link requires a log in $\endgroup$ – Argyll Nov 20 at 23:12
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    $\begingroup$ Economist here: I think the book is a bit outdated. $\endgroup$ – Michael Greinecker Nov 21 at 0:09
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    $\begingroup$ @MichaelGreinecker It is very old, essentially a reproduction of typewritten notes; it was old in 1984 when I remember Cassels actually teaching the course. And it only looks at a few selected topics. But it is telling that there are hundreds of different books called "Mathematics for Economists" with regular new editions to exploit economics students, while only this "Economics for Mathematicians". $\endgroup$ – Henry Nov 21 at 17:47
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    $\begingroup$ @BigbearZzz I think the part on macroeconomics is plainly outdated, though this is not my field. The microeconomic part might be useful, but is uneven. Standard graduate microecnomis books such as "Microeconomic Theory" by Mas-Colell, Whinston, Green" and the narrower but deeper "Microeconomic Foundations I" by Kreps are already written in a definition-theorem-proof+commentary way. $\endgroup$ – Michael Greinecker Nov 25 at 17:33
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    $\begingroup$ Possibly follow up with Topological Methods in Walrasian Economics by E. Dierker $\endgroup$ – Phil Harmsworth Nov 26 at 2:50
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Most of the good physics books have already been mentioned, so I'll add one about machine learning.

Understanding Machine Learning: From Theory to Algorithms, by Shai Shalev-Shwartz and Shai Ben-David.

Fully rigorous, and explains what the mathematical challenges are in machine learning.

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The answers show that there are many books on physics, especially on quantum mechanics written by mathematicians and for mathematicians. Let me add my favorite one:

L. Faddeev and and O. Yakubovskii, Lectures on quantum mechanics for mathematics students. (Russian original 1980, English translation: AMS, 2009).

Another classical book is

V. Arnold, Mathematical methods of classical mechanics.

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  • $\begingroup$ As a continuation/extension of Arnold, I'd recommend Abraham and Marsden's Foundations of Mechanics (2nd edition). $\endgroup$ – Phil Harmsworth Nov 22 at 7:12
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    $\begingroup$ There is also a continuation of Faddeev-Yakubovskii, it is written by Takhtajan, Quantum mechanics for mathematicians. $\endgroup$ – Alexandre Eremenko Nov 22 at 14:33
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In response to the (personal) finance book for mathematicians, I would suggest looking towards investment and risk management books as they tend to be written by mathematicians, so any related financial market and economic material will be relatively concise, which is the issue I think mathematicians have with more regular finance texts.

I felt Investment Science by David Luenberger was a great find when I was starting out though it seems to be harder to get hold of now.

On the risk management front, PRMIA is a professional risk management organisation that offers digestible blocks of material that cover background finance material, a brief bridge of the mathematical foundations of risk, and more detail beyond:

https://www.prmia.org/Public/Public/Resources/PRM_Handbooks.aspx

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Well, it's not exactly science, but The Mathematics of Juggling by Burkhard Polster is written by mathematician and is for mathematicians. I may add that you can enjoy the book, even you can't juggle.

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  • $\begingroup$ Note that the book does not teach you how to juggle; it is in essence a math book on an application of combinatorics. $\endgroup$ – Federico Poloni Nov 27 at 9:25
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The Geometry of Musical Rhythm by Godfried Toussaint:

The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? is the first book to provide a systematic and accessible computational geometric analysis of the musical rhythms of the world. It explains how the study of the mathematical properties of musical rhythm generates common mathematical problems that arise in a variety of seemingly disparate fields.

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I can recommend Mathematical Linguistics by Andras Kornai.

From the preface:

The book is accessible to anyone with sufficient general mathematical maturity (graduate or advanced undergraduate). No prior knowledge of linguistics or languages is assumed on the part of the reader. The book offers a single entry point to the central methods and concepts of linguistics that are made largely inaccessible to the mathematician, computer scientist, or engineer by the surprisingly adversarial style of argumentation (see Section 1.2), the apparent lack of adequate definitions (see Section 1.3), and the proliferation of unmotivated notation and formalism (see Section 1.4) all too often encountered in research papers and monographs in the humanities. Those interested in linguistics can learn a great deal more about the subject here than what is covered in introductory courses just from reading through the book and consulting the references cited.

Edit. See also Ian Chiswell's A Course in Formal Languages, Automata and Groups.

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    $\begingroup$ It surprises me slightly that mathematicians get so offended by the failure of other disciplines to stick to their own subject-specific conventions. For example, I was at a talk by a mathematician and they complained that they had to read awful papers by physicists with, shudder, no proofs. $\endgroup$ – Hollis Williams Nov 22 at 11:33
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A brand new book that might be of interest is Data Science for Mathematicians, edited by Nathan Carter (who also wrote Visual Group Theory). It assumes the audience is a mathematician (at, say, the graduate student level), then gives high level treatments of:

  • programming with data,
  • linear algebra (and its applications to data analytics),
  • basic statistics,
  • clustering,
  • operations research,
  • dimensionality reduction,
  • machine learning,
  • deep learning, and
  • topological data analysis

I should disclose that I wrote one of the chapters, but don't have any financial stake in the book. I recommend it because I think it's great, and will help mathematicians who want to embrace data science in their research, teaching, or as an alternative career.

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    $\begingroup$ Visual Group Theory? I heard that the new edition is Group Theory.NET $\endgroup$ – Asaf Karagila Nov 27 at 9:31
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I like Leonard Susskind's The Theoretical Minimum series a lot. I've read the Quantum Mechanics one and started the Classical Mechanics one. The series also covers Relativity, Statistical Mechanics, and Cosmology per Wikipedia, and there are also lectures available online.

I also like Feynman's QED: The Strange Theory of Light and Matter on Quantum Electrodynamics - it sounds very intimidating but it's actually very approachable and a very good introduction, knowing nothing going in.

Re: finance, Mandelbrot's The Misbehavior of Markets is quite interesting, and Flash Boys by Michael Lewis, while not specifically mathematical, gives a lot of insight into how modern digital markets work [or don't, depending on who's asking].

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    $\begingroup$ Feynman's QED is really great! Note however that the explicit goal is to minimize the mathematical prerequisites as much as possible so it's not really aimed at mathematicians. $\endgroup$ – Asvin Nov 21 at 19:18
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    $\begingroup$ I wouldn't describe Feynman's QED as being aimed at mathematicians, quite the opposite. $\endgroup$ – Hollis Williams Nov 21 at 22:59
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There are many sources on general relativity for mathematicians (see, for example, the lecture notes of Schoen and the textbook Geometric Relativity by Dan Lee).

It's been a while since I read any chemistry books, but I remember Atkins' textbook on physical chemistry being fairly readable for a mathematician.

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There is an interesting book that teaches dance concepts from a mathematical viewpoint:

Dance: Mathematical inquiry in the liberal arts

Its online version is free. I took a glance at the book and well, I must say I got stuck for an hour or two!

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Robin Giles, Mathematical Foundations of Thermodynamics. From the preface:

This monograph is an attempt to give an account of the foundations of thermodynamics which is more than usually rigorous, not only in its logical structure but also in the "rules of interpretation" in which physical meaning is assigned to the theoretical terms.

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When I read Cormen, Leiserson, Rivest's book on Data Structures and Algorithms, it was eye-opening to me to see how (mathematically) rigorous computer science could be.

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I think in general what you are looking for doesn't tend to exist (with exceptions, such as those in other answers). As background, I am a theoretical physicist with what I suspect is a stronger mathematics background than most, but I think theoretical physics (and probably computer science) is the only field where you could really expect to find books that satisfy:

Strong preference would be to concise, terse texts that are foundational but totally rigorous.

To find stuff like that you'd probably have to look at a very small selection of research papers (some of which I'll list at the end of this answer). However since these are research papers they tend to be focused on the subfield and probably assume some familiarity already (kind of defeating the purpose to learn from them).

I would say that the lack of these textbooks is not a bad thing, and that's because mathematics tends to be different to basically any other science. In mathematics you can start with your axioms and prove what you would find if you did experiments with that mathematical structure. While in basically every other field you start with knowing the results of experiments and have to try to deduce which axioms you should start with. There are probably exceptions to this but I think in general this argument holds.

To give a bit of a feel why this is different something I remember hearing about the theory of (I think) topologically ordered phases a bit over 5 years ago: when you read papers there are maybe 3 different ways that people define these phases, which aren't equivalent but kind of do the same thing in the end. While this might sound like there should be 3 different definitions used and introduced at the beginning of each paper, the end goal is to discuss and explain real experiments and so we only want and we only expect a single theory to exist. The different ways people have defined things the definition becomes a "I know it when I see it" situation and everyone working on the topic, even if using different definitions, should expect to see the same kinds of behaviours.

The other thing to keep in mind is that if you have a rigourous theory that uses sophisticated mathematics compared to a simple theory that only requires arithmetic, if the simple theory matches the experimental data better then the rigourous theory then that's the one that should be kept since the goal is to explain these experiments. Also if you can explain the same set of experiments with either simple or sophisticated mathematics with the only difference being rigour, unless your sophisticated mathematics approach is more powerful in explaining or making predictions for other experiments that the simple approach cannot, then the simple approach will remain dominate just because of the cognitive overhead associated with using more sophisticated mathematics.

The final thing is to think about is where would you start if you wanted to give a foundational description of genetics? we could start with central dogma of biology which is that DNA -> RNA -> proteins, explain what all these things are and build up how proteins effect the behaviour of cells and eventually an animal along with expressions of phenotypes (what it looks like). From this construction you could derive evolution. The problem with this approach is that we can't do it yet, there are heaps of things we don't understand about proteins and protein folding (which is important for how a protein changes cell behaviour), and when do you introduce the ability to inhibit/enhance different gene expressions (protein production) via the existence of certain proteins. This is a more complicated topic that if introduced too early distracts the reader from the core idea of DNA gives proteins which give looks/behaviour.

Instead it makes more sense to start with talking about the looks and behaviours you see and saying it is conjectured to have been caused by ... which is conjectured to be caused by ... and so on. This also means that new research, which is more likely to be at the molecular scale than animal scale, is only a correction to the end of your book, rather then the very first sentence. This would be different to a maths text where new research won't overturn the axioms but would be more likely to add new knowledge about the structure that arises from these axioms.

We could discuss the looks and behaviour in terms of other mathematical structures to give them a rigourous definition, but WHY? You add cognative overhead to those who don't know this mathematical structure while not adding any explanatory power. Furthermore for those who understand the structure you're possibly misleading them into thinking the choice of this structure is deep while preventing them from learning how to think about the field as everyone else does.

So longer then I initially expected (and missing points I was initially thinking of) but

tl dr: mathematics is unique in that in generally derives results from axioms rather than axioms from results like most other fields, this makes it very difficult to have concise terse texts that are foundational and totally rigourous (with exception of some theoretical physics and computer science).

Possible papers: These might require too much knowledge about the field, however maybe the mathematics in them might make it easier to read and be able to pull out the terms and facts that these experts seems to care about for future papers.

  1. Knot theory in understanding proteins J. Math. Biol. (2012) 65:1187–1213 DOI 10.1007/s00285-011-0488-3

Actually targeted at mathematicians trying to get them to work in the field and has some references to biology which might help to just gain a bit of familarity/knowledge of what's important before reading other texts.

  1. String-net condensation: A physical mechanism for topological phases PHYSICAL REVIEW B 71, 045110 2005

Probably not accessible without some knowledge of quantum physics, but argues about using the objects of fusion categories to describe phases of matter.

  1. CRICK, F., WATSON, J. Structure of Small Viruses. Nature 177, 473–475 (1956). https://doi.org/10.1038/177473a0

I think this was the paper that used group theory to explain the structure of a virus shell. There is a later paper that I think extended it to Kac-Moody algebras and quasi-crystals for an abnormal class of viruses but I can't quickly find it.

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  • $\begingroup$ Thank you for the insightful response! $\endgroup$ – Josh Nov 23 at 5:58

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