In my opinion, this actually may be the best time in human history to learn math, or just about anything for that matter. If you are of sufficient means to have access to the internet, which unfortunately not all people are, you have access to an unprecedented amount of information. The problem is the overwhelming amount of information that is out there and how to find it. So I wanted to share some of my tricks to finding good math information, which I call "Math Mining", of course these can be used for any academic information, so maybe "Academic Mining" may be more apropos.

One of the reasons that the present time is a good time to learn math is due to the diversity of sources for information, Wikipedia is one of those sources, Wiki Surfing, as has been previously discussed, but that is just the tip of the math iceberg and the really cool thing is that there are many resources that can give you entirely new and fresh perspectives on things that may sometimes seem dull and obfuscated by more traditional approaches found in books, not that there aren't a lot of great books too. It really is an exciting time.

Many professors have their publications, notes and course resources freely available on the internet and some of these include full books in pdf or ps or html format. In fact that leads me to my little google trick, let's assume you want to find some information about a math subject, we'll use Linear Algebra as an example. Then if you use the Google search:

"Linear Algebra" inurl:pdf

You will get a lot of hits that are academic pages, these will be a mix of publications and course related material. Once you find a document, you can use that url to find more information. For example let's say that our above search leads us to the following (fictitious) url:

www.math.umars.edu/~hjfarnsworth/math420/fall2010/chapter10.pdf

After you click the link and get your reward, you should realize that this is a potential gateway to much more information. Now admittedly this might be seen as a moral gray area, because sometimes I get the feeling that some of these resources are not as openly exposed as they could be so it may be that the instructors do not want to openly share their work and they are practicing security through obscurity, but in my opinion if directory browsing is enabled and/or your documents are indexed by Google, then they're fair game, so if you are someone who this applies to, I suggest that you either share it openly it or lock it down. I encourage anyone who is sharing their work to do it freely and openly regardless of whether people are taking your classes. After all it's for the greater good. "The greater good." And if you openly share it then people like me can read it, learn it, know it and talk about how awesome you and your work are. It's a win-win.

## Hack the Site

### Hack #1 Url Suffix Removal

By removing "chapter10.pdf" yielding www.math.umars.edu/~hjfarnsworth/math420/fall2010/ will expose more resources if this directory has browsing enabled or if it has a default page. You can progressively remove directories to find one that is useful, and actually sometimes it is worth it to jump directly to www.math.umars.edu/~hjfarnsworth/ which will often be a professor home page which can yield links to publications, course pages with documents, and other potentially interesting information.

### Hack #2 File Name Enumeration

So you looked at chapter10.pdf and it's awesome but Hack #1 did not yield it or the related chapters. Due to the naming convention try: www.math.umars.edu/~hjfarnsworth/math420/fall2010/chapter09.pdf or www.math.umars.edu/~hjfarnsworth/math420/fall2010/chapter9.pdf, often this approach will yield other related documents.

### Hack #3 Invoke the Power of Google

Let's say the hack #1 didn't work and the resultant url had a random characteristic like:

www.math.umars.edu/~hjfarnsworth/math420/fall2010/larnd42-chap10.pdf

The following Google search will ferret out those pesky hard to find pdf's:

site:www.math.umars.edu/~hjfarnsworth/ inurl:pdf

Also you can use .ps and .ps.gz in place of .pdf for file type searches. If you feel that this is crossing some kind of moral line then don't do it, but I like to say all is fair in Love and Math.

I would like to give another example of this technique, I recently came across "Mapreduce & Hadoop Algorithms in Academic Papers (4th update - May 2011)" which linked to "Max-cover algorithm in map-reduce" which caught my interest, and of course the ACM is charging for it, but no worries, there is usually no need to pay them, actually I recommend boycotting them. I employed the above tricks but they didn't work, simply Googling one of the authors did (always pick the most unique name(s)):

"Flavio Chierichetti"

Pulled up his web site which had a free copy of the paper, now all I have to do is find the time to read it. Also the above techniques yielded the paper's "cliff notes".

Of course you can just look up someone by name, for example, you can find some of Donald Knuth's publications here.

In regards to academic publications there are two excellent repositories with a wealth of information these are Citeseer out of Penn State, this site can be a little flaky in terms of availability, at least that's been my experience in the past and the other is arXiv run by Cornell University. These mostly contain research oriented work but you can often find relevant information even for neophytes, actually a lot of advanced papers and books for that matter start out with introductory sections that can be worth looking at.

## Encyclopedic and other Miscellaneous Resources

Wikipedia, obviously, as previously mentioned. Also the oft controversial Stephen Wolfram provides an excellent resource called Wolfram Mathworld.

Project Euler is a site dedicated to collaboratively solving math oriented problems programmatically more about it can be found here.

Math on the Web by category here provides some interesting links, I believe this is run by the American Mathematical Society but I am not sure.

The National Institute of Standards and Technology site: NIST Digital Library of Mathematical Functions.

Also there is Mathoverflow which is a Stackoverflow type of question and answer community devoted to Math.

## Blogs

There are a number of blogs that blog about both math and programming related math. Actually if your primary interest is machine learning, I recommend Bradford Cross's Measuring Measures blog, it is hard to find things on his site and it was recently restyled with a magenta/maroon background which I now find a little bit harder to read. The relevant links here are: Learning About Network Theory, Learning About Statistical Learning, and Learning About Machine Learning, 2nd ed. Additionally Ravi Mohan did a follow-up: Learning about Machine Learning.

Good Math Bad Math by Mark Chu-Carroll has lots of good articles about math including some for beginners in various areas. Catonmat by Peteris Krumins has some nice entries with notes about the online MIT courses that he has worked through which currently covers Algorithms and Linear Algebra also mentioned above. The Unapologetic Mathematician has a lot of nice articles, this is a bit more advanced though. Math-Blog has a lot of articles as well. They tend to focus on more traditional areas of math. Math blog's abound and there are too many to mention, here's a few:

- The n-Category Café
- Michi's blog
- A Neighborhood of Infinity
- XOR's Hammer
- Gödel's Lost Letter and P=NP
- Terrance Tao's Blog
- Arsmathematica
- Steven Strogatz
- Devlin's Angle by Keith Devlin
- The programming/CS Theoretical Lambda The Ultimate

## Online Courses

The well known Khan Academy offers a number of courses including several math courses.

MIT Open Courseware has many online courses most notably for CS majors Introduction to Algorithms by the venerable Charles Leiserson and Erik Demaine videos here and Linear Algebra by Gilbert Strang.

On Stanford Engineering Everywhere the following might be of interest:

Artificial Intelligence | Machine Learning

Artificial Intelligence | Natural Language Processing

Linear Systems and Optimization | The Fourier Transform and its Applications

The Mechanical Universe is primarily dedicated to physics, but several math topics such as Calculus and Vectors are covered explicitly. It's also a nice series of lectures on the topic in spite of being a little dated in productions values.

## Other Online Videos

Two math documentaries are covered here are Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem and the overly dramatic but still interesting Dangerous Knowledge.

The story of Maths by Marcus du Sautoy.

Keith Devlin talks about Pascal and Fermat's coorespondance while working out probability in this intersting talk: Authors@Google: Keith Devlin.

Bob Franzosa - Introduction to Topology.

The Catsters videos on youtube cover various Category Theory related topics.

N J Wildberger's Algebraic Topology

Dan Spielman has a video discussing Expander Graphs.

Introduction to Game Theory by Benjamin Polak at Yale.

The site videolectures has many lectures in Computer Science and Math including:

- Basics of probability and statistics by Mikaela Keller.
- Probability, Information Theory and Bayesian Inference by Joaquin Quiñonero Candela.
- Introduction To Bayesian Inference by Christopher Bishop
- Information Theory by David MacKay
- Gaussian Process Basics by David MacKay,
- Statistical Learning Theory by John Shawe-Taylor
- Bayesian Learning by Zoubin Ghahramani.

If you find these videos too slow this might interest you.

## Math Software

There are many math related software packages and libraries three of which are covered in more detail here.

Math library Sage written in Python

The R project for Statistical Computing

Maxima, a Computer Algebra System

## Various Books and Academic Stuff

Here are a bunch of interesting courses and books that I have encountered during my searching which you might find interesting as well. These are presented in no particular order:

### Algorithims

Algorithms by S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani.

Jeff Erickson has some Algorithms Course Materials

Steven Skiena author of the The Algorithm Design Manual offers some pretty comprehensive course notes for his cse541 LOGIC for COMPUTER SCIENCE not to mention the opportunity to learn how to bet on Jai-alai in the Cayman Islands.

Gregory Chaitin's Algorithmic Information Theory.

### Computer Science

Foundations of Computer Science by Jeffrey Ullman and Al Aho.

The Haskell Road to Logic, Math and Programming by Kees Doets and Jan van Eijck

### Discrete Math

Discrete Mathematics with Algorithms by M. O. Albertson and J. P. Hutchinson.

### Analysis

Analysis WebNotes is a self-contained course in Mathematical Analysis for undergraduates or beginning graduate students.

Introduction to Analysis Lecture Notes by Vitali Liskevich.

Applied Analysis by John Hunter and Bruno Nachtergaele.

REAL ANALYSIS by Gabriel Nagy.

### Probability Theory

Introduction to Probability Theory by Ali Ghodsi.

Introduction to Probability by Charles M. Grinstead.

The first three chapters of Probability Theory: The Logic of Science by E. T. Jaynes. Can be found here.

Think Stats: Probability and Statistics for Programmers by Allen B. Downey.

LECTURE NOTES MEASURE THEORY and PROBABILITY by Rodrigo Bañuelos.

Principles of Uncertainty by by Chapman and Hall.

Information Theory, Inference, and Learning Algorithms by David MacKay.

### Machine Learning/Date Mining

Machine Learning Module ML(M) by M. A .Girolami.

Alexander J. Smola's and and S.V.N. Vishwanathan's draft of Introduction to Machine Learning.

The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Second Edition) by Trevor Hastie, Robert Tibshirani and Jerome Friedman.

Introduction to Information Retrieval by Christopher D. Manning, Prabhakar Raghavan and Hinrich Schütze.

Mining of Massive Datasets by Jeffrey Ullman.

### Fourier Theory

Lecture Notes for EE 261 The Fourier Transform and its Applications pdf By Brad Osgood.

### Abstract Algebra

Abstract Algebra by Thomas W. Judson.

James Milne has a number of sets of extensive notes on Algebraic topics like goup theory here.

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS by John A. Beachy.

Elements of Abstract and Linear Algebra Edwin H. Connell.

Abstract Algebra by Elbert A. Walker.

A series of chapters on groups by Christopher Cooper.

### Linear Algebra

Linear Algebra by Robert A. Beezer.

A course on Linear Algebra with book chapters.

Really cool interactive tutorial on Singular Value Decomposition by Todd Will.

### Model Theory

Fundamentals of Model Theory pdf by William Weiss and Cherie D'Mello.

### Set Theory

A book on Set Theory pdf by William Weiss.

### Graph Theory

Reinhard Diestel makes his excellent and comprehensive book Graph Theory available, pdf here.

### Logic

You can find Introduction to Mathematical Logic by J. Adler, J. Schmid, Model Theory, Universal Algebra and Order by J. Adler, J. Schmid, M. Sprenger and other goodies here.

Introduction to Logic by Michal Walicki.

Logic for Computer Science: Foundations of Automatic Theorem Proving by Jean Gallier.

The Novel Research Institute has a number of free academic books including: Logic and Metalogic:Logic, Metalogic, Fuzzy and Quantum Logics and Algebraic Topology, Category Theory and Higher Dimensional Algebra-Results and Applications to Quantum Physics

### Category Theory

Some course notes on Category Theory by Tom Leinster.

Basic Category Theory pdf by Jaap van Oosten.

Abstract and Concrete Categories The Joy of Cats by Jiri Adámek, Horst Herrlich, George E. Strecker.

A gentle introduction to category theory --- the calculational approach pdf by Maarten M. Fokkinga.

Steve Easterbrook's An introduction to Category Theory for Software Engineers.

### Algebraic Topology/Topos Theory

Eugenia Cheng of Catsters fame has a course in Algebraic Topology with some substantial notes.

The above links of Eugenia Cheng refer to Algebraic Topology by Allen Hatcher.

An informal introduction to topos theory pdf by Tom Leinster.

### Topology

A free, protected, password available by request, e-book on topology: Topology without Tears by Sidney A. Morris.

Chapters for a topology course by Anatole Katok can be found here.

### Computational Topology

Jeff Erickson has some nice notes on Computational Topology, pdf's can be found on the schedule page.

Afra Zomorodian has some nice resources on Computational Topology including a nice introductory paper.

### Spectral Graph Theory

Fan Chung Graham has a lot interesting stuff, some pretty advanced, relating to graph theory including social graph theory and spectral graph theory.

Dan Spielman has some course notes on Spectral Graph Theory.

### Expander Graphs

Avi Wigderson's Expander Graphs and their Applications.

### Fractal Geometry

The Algorithmic Beauty of Plants pdf by Przemyslaw Prusinkiewicz and Aristid Lindenmayer is available on the Algorithmic Botany site.

### Game Theory

Thomas S. Ferguson's course at UCLA on Game Theory also Game Theory for Statisticians.

A course in game theory by Martin J. Osborne and Ariel Rubinstein, requires registration.

### Algebraic/Enumerative Combinatorics

MIT Open Courseware in Algebraic Combinatorics

An uncompleted book and notes on Enumerative Combinatorics by the Late Kenneth P. Bogart also here.

Lionel Levine's notes on Algebraic Combinatorics

Richard P. Stanley new edition of Enumerative Combinatorics Volume one.

A Course in Universal Algebra by Stanley N. Burris and H.P. Sankappanavar

### Misc

Pat Hanrahan's CS448B: Visualization.

Sean Luke's "Essentials of Metaheuristics".

### It's all a click away

The links in this entry, especially the academic links are susceptible to link rot, people move from institution to institution or leave academia for jobs in the private sector. I will endeavor to revisit this entry and try to keep these up to date and perhaps even add to them, however, if you encounter this page and have any interest in any or all of these resources I recommend downloading them now so that you have them.

Using the resources of this blog you should be able to get your hands on a huge amount of free resources on a wide range of topics. This can be helpful if you are on a budget or just want to try before you buy an expensive book on a topic. I hope you avail yourself of some of these, there's lots of great stuff and if you know of some that I do not please add them in the comments.