# Category: ∑ Mathematics

• ## Solutions to exercises from Introduction to Analytic Number Theory by Apostol

As I am currently working my way through the exercises in Apostol’s book “Introduction to Analytic Number Theory”. Since I wasn’t able to find solutions to all exercises and since the book does not contain solutions, I googled for them.

And then I found this: https://gregoryhurst.com/solutions/. A PDF file containing solutions to most if not all exercises in the book.

😳

Since graduating, I decided to work out all solutions to keep my mind sharp and act as a refresher.

Greg Hurst, author of the solutions, in its preface

I was very happy to have found this work and I am very thankful to its author!

• ## Annotating Serre

I very much like the thought of annotating the book “A course in arithmetic” by Jean-Pierre Serre. It is a fascinating and dense math text book that I find very attractive. The book has also been recommended by Richard Borcherds as one of his favorite math books, and I very much like Borcherds’ online math lectures.

But on the other hand, “A course in arithmetic” is a very hard book. I tried to read parts of it and most of what I read seemed to be very hard to understand. As I wrote in the second sentence: It is a very dense book. And people with much better mathematical backgrounds have tried before me and struggled: https://www.reddit.com/r/math/comments/80of13/how_to_read_a_course_in_arithmetic/

When I read this reddit thread, I could very much sympathize with the original poster. But I was even more fascinated by one of the posts’ linking to this annotated version of the book: https://people.ucsc.edu/~weissman/Math222A/SerreAnn.pdf. It is admittedly not a complete annotation of the book but it is a nice start. It seems to be a project that started in a graduate course at the University of California, Santa Cruz.

Serre always gives the maximally elegant proof, putting in only what he considers the absolutely necessary steps. No motivation is given. This is the style popularized by Bourbaki. (A French mathematician informed me that they did this because didn’t want to contaminate the reader with their informal mental picture of what was going on. The reasoning was that their teachers would provide the intuition and “picture”).

But to appreciate this elegance and economy of presentation, you either have to be so mathematically gifted as to be able to immediately see why he would do things the way he does, or to work hard and gain experience. (Most of us have to go the latter route).

I think I would like to work through some parts of the book myself and see what annotations I come up with.

• ## Free lecture notes: Statistics and Machine Learning in Python

These lecture notes were sent to me by a friend and I thought why not add them to my collection. The notes are available as a browsable website (https://duchesnay.github.io/pystatsml/) as well as a PDF (https://raw.github.com/duchesnay/data/master/pdf/StatisticsMachineLearningPython.pdf). They also come with exercises, some of which even come with prepared jupyter notebooks.

I guess the “official” way to cite the book in a scientific publication is this way:

Edouard Duchesnay, Tommy Lofstedt, Feki Younes. Statistics and Machine Learning in Python. Engineering school. France. 2021. ⟨hal-03038776v3⟩

• ## Free math lecture notes: Irrational and transcendental numbers by Michel Waldschmidt

I only recently found out that it is an open question whether the Euler-Mascheroni constant is irrational or not and whether is transcendental or not. I found that quite surprising and looked for lecture notes on the topic. I found these lectures by Michel Waldschmidt with the title: “An introduction to irrationality and transcendence methods.”:

There is also a series of video lectures by the same author on Youtube:

The title image of this post is by Arthur Baelde on Wikimedia Commons. See https://commons.wikimedia.org/wiki/File:First_irrational_numbers.svg

• ## Opening up mathematics

### What do I mean by that?

Is there a way to make research mathematics a more collaborative, asynchronous and open endeavor? Is it possible to open it up to the public, to scale it up in some sense? So that, at least in principle, all mathematicians around the world could participate on a given problem, maybe even aspiring young mathematicians and hobbyists?

In the past, there has been one successful approach to this question (that I know of) that is called Polymath Projects. These were (up to now) 16 attempts to solve research projects in public, under the guidance of the proposing professional mathematician. As far as I can see, many of the proposing mathematicians used their personal blog to coordinate the progress on the projects. They would post a problem definition together with their current knowledge of the area and then everybody would be able to comment on the blog and contribute their ideas and knowledge. Every now and then, the blog owner would update everybody with status blog posts (and thereby also partition the comment history into more readable threads).

According to the Polymath Projects Wiki, it all started with this blog post by Timothy Gowers, in which he stated the following vision:

It seems to me that, at least in theory, a different model could work: different, that is, from the usual model of people working in isolation or collaborating with one or two others. Suppose one had a forum (in the non-technical sense, but quite possibly in the technical sense as well) for the online discussion of a particular problem. The idea would be that anybody who had anything whatsoever to say about the problem could chip in. And the ethos of the forum — in whatever form it took — would be that comments would mostly be kept short.

https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/

I have always liked the fail-fast-learn-fast attitude. the generosity of ideas and the collaborative aspect of this approach. Also, Gowers’ original blog post seems to be lowering the entry barrier to participating in such problems, thereby making it more fun to engage.

The title picture of this post, by the way, is from a study called “The polymath project: lessons from a successful online collaboration in mathematics” by Justin Cranshaw, A. Kittur. It was published in 2011 in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.

Recently, I wondered if this model could somehow be standardized and made more scalable? As it was done previously, one person would have to organize the progress centrally. What if it was done unsupervised(-ly?) and asynchronously?

And also: would it be possible to make the process of solving problems more diverse? For instance, some people might be interested in taking over expository tasks, like summarizing the state of a problem. Other people might be good at performing numerical computations if needed. Yet other people are good at using theorem proving assistants or formal computer tools to contribute something. In general, making the process more open, collaborative and diverse would be something I would very much look forward to.

### What is there already?

I have recently found and written about a list of open math problems in this blog post. But to solve such problems, one would need a sort of central website that would contain not only a list of open problems but furthermore some kind of scorecard for each problem and a forum for people to discuss their ideas and upload content. As an open platform, people should be able to enter new problems and add references and the state of the art to that problem. Everybody could then go and read the resume of the problem and try to make progress on a problem her or she is interested in.

While researching the list of lists of open problems, I stumbled upon a website called SciLag: https://www.scilag.net/. This seems, at first glance, to be a promising first step towards an implementation of such a standardization. They even have a “Problem Mining” section which is supposed to mine papers on the arXiv for potential open problems or conjectures (wonder how well that works yet). The existing open problems in their database are tagged with areas of mathematics but unfortunately, these tags are not clickable as of now.

A similar website is called http://www.openproblemgarden.org/, which seems to have a similar functionality, except for the problem mining part.

### Which problems do I see with this?

While this approach sounds very promising and fun to me, I can imagine (because I am not a professional mathematician) that there might be some inertia in the system and that the amount of onboarding to be done by people who want to join a problem might be large.

#### Resistance from the establishment

After all, most of the knowledge that I would love to see open-sourced is currently guarded by a group of professional mathematicians. They share their knowledge openly in conferences and (open and paywalled) publications. But they do still (mostly) share it at a very suboptimal state of the problem, namely when they think it has been solved. In some sense, this is too late and, more often than not, progress might have been faster (albeit less heroic) had they collaborated more.

As Gowers writes:

The next obvious question is this. Why would anyone agree to share their ideas? Surely we work on problems in order to be able to publish solutions and get credit for them. And what if the big collaboration resulted in a very good idea? Isn’t there a danger that somebody would manage to use the idea to solve the problem and rush to (individual) publication?

I can imagine that some people might be hesitant to share their ideas and progress on a problem in public, partly because of a fear of looking stupid or losing a race to credit. I am currently convinced that sharing ideas earlier would be better for mathematical progress overall, but maybe someone can convince that this is false. I fear that some people want to work only with people who they know and trust in order for them to be able to publish regular papers that give them reputation in the regular currency and not some fancy new (blockchain-ish) online blog/polymath reputation currency.

#### Too high of an entry barrier for non-professionals to make any progress

The other question that comes to my mind is whether or not it would even be worth the effort. Does it even make sense to invest the time to post a problem and its state of the art on a platform given that very few people in the world are even remotely able to contribute anything useful. And those few people might be familiar with the problem and its progress already anyway. So why bother?

As mentioned by Gowers in his original post, there is probably a class of problems for which it is worthwhile to invest the time. But I feel that there should be more than 16 such problems. And as mentioned before, there could also be a diversification of tasks in the sense of exposition, library search, idea generation, computation, etc. that might fit many people’s needs.

### A last remark

As a last remark, let me comment on the following quote from Gowers’ original article:

Now I don’t believe that this approach to problem solving is likely to be good for everything. For example, it seems highly unlikely that one could persuade lots of people to share good ideas about the Riemann hypothesis.

I don’t quite understand why that would be the case. I have heard (or read) at least one mathematician say that they would love to see a solution to the Riemann hypothesis in their lifetime or first thing after getting off from a time machine in some distant future. I think it would be to the benefit of many people to share even vague ideas on such prestigious problems openly. For almost all people it would be better to know the solution in their lifetime than to have a huge price for a single person attached to it. Most people would hopefully agree that they won’t be the single person who solves the problem anyway.

• ## A list of open research problems in mathematics

I recently did a search for open problems in mathematics. Here are a few links I’ve found (manly via this math tackexchange post: https://math.stackexchange.com/questions/1354028/database-of-unsolved-problems-in-mathematics):

SciLag is a free web-based platform for facilitating dynamic organisation of scientific problems at a research level. It represents an online service for scientists for sharing their knowledge about the forefront of research. The current version is set up for the Mathematical community only to test and refine the concept in action.
[…] The platform is designed to be home for scientific problems which are (were) open. For many of these open problems to be addressed, the science, mathematics in this case, needs new tools and ideas to be discovered. In other words, the current knowledge is lagging behind, hence SciLag for “Science Lagging behind”.

• ## Free math lecture notes: Number theory by Jared Weinstein

Great lecture notes on elementary number theory.

Very clear and gentle introduction to number theory by Jared Weinstein. These lecture notes really start off the ground and explain number theory in a gentle and clear way.

• ## Free math text book: Topology without tears by Sidney A. Morris￼

Link to the homepage of the book: https://www.topologywithouttears.net/

A PDF version of the book can be downloaded for free from the above website. It contains lots of exercises, but without solutions. I did go through chapters 1 and 2 and I found it clear and well readable with lots of examples.

• ## Collecting free online resources for learning math

Inspired by a recent video on self-study math videos and books by Aleph 0 and by the website libraryofbabel.info (which I found via this tweet) I decided to collect all online resources (text books, playlists, websites) for learning undergraduate and graduate math online that I know of. This collection will hopefully grow over the next weeks and months.

I decided to list each individual text book and Youtube playlist on a certain topic individually rather than only listing, for example, the Youtube channel that contains them or the author’s homepage, in case an author has more than one text book online. I think this makes the list more searchable and (probably?) of higher quality.

I hope that the list will be helpful to someone. Here’s the link to the blog category that I’ll use to file the posts: Math-resources