19 June 2011

Free Math Resources You Can Use

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:

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:

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.


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:

Lastly I will mention a blog by Jeff Moser, actually he only has a few math related posts, but his Computing your Skill on Probability and Statistics is a beautiful work of art well worth looking at.

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:

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

GNU Octave

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:


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 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.


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.


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.


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.


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


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.

13 June 2011


Often in our industry the M in RTFM is the code itself.  Yes, there should be more documentation but ultimately if you are a developer the code is all too often either the sole documentation or the most up to date documentation or the only worthwhile documentation.

For me it is no longer just a matter of necessity, admittedly over the years I feel I have developed pretty efficient code grokking skills, but ultimately it has become more of a way of life for me.  If I work with you or for that matter if you publish your code on the internet I will look at it.  If you claim, or are claimed by a third party, to be a good developer and I can look at your code to verify that claim I will do it in a heartbeat, if you have solved a problem that I am trying to solve or you potentially have a better solution or are using a technique that I am unfamiliar with, or are a better coder, I am all over your code.

The best developers are developers those who read code and those who read in general. The great thing about the industry today is there is no shortage of code to look at or general reading material.  As I have previously mentioned I currently work a fair amount with the Spring Framework, looking at the Spring Source code has helped me become a much better programmer and has exposed me to many new concepts.  Also reading the code allowed me to access many useful classes and functionality that are used internally by Spring but are not well documented.

I am frequently amazed (disappointed) by the lack of intellectual curiosity that many developers have for reading code.  I recall on one previous project we had to create some custom Servlet Filters, another developer had created the initial prototype and I refactored it, one thing I changed was I added Spring’s Ant pattern support for the configuration which was needed to exclude some paths, this feature was not well documented at the time and you had to read the code to figure out how to do it.  When the initial developer saw my changes he remarked with surprise that I figured out how to make it work like Spring Security.  I responded that all I had done was dug into the Spring source code to find out how they did it.  Perhaps my most irritating example of this is a project where we were in the process of making some refactoring changes which had a number of deprecated classes and in some cases entire packages of deprecated classes, and this one Cargo Cult Coder rolls in talking about how he was going to fix our architecture, of course he doesn’t bother to look at the existing code base and then he proceeds to add a class, which was also redundant, to a package where every other class was deprecated! Really!?! When a developer, especially a senior level developer works on a project with an existing code base it is their responsibility to learn the existing code base.

If you don’t read code you’re not a real developer!

01 June 2011

Math for Programmers TNG

Steve Yegge wrote a blog entry a few years back called "Math for Programmers", as I previously mentioned I had a few other entries that parallel it, and this is one of those.  I consider this an extension of his list hence the title.

One of the "themes" of my blog is math for programmers, for example see:   "Math You Can Use", "Monoid for the Masses", and "The Combinatorial and Other Math of the Java Collections API".  And here I am going to give some of my opinions and what I feel is a more comprehensive list than is provided in Steve Yegge’s original post, not that it’s not a good post in fact I recommend reading it.  In it he recommends using a technique I like to call Wiki Surfing, which has a nice Hawaiian/Polynesian ring to it, where you surf Wikipedia to learn about Math, I am hoping that this post can in part serve as a giant "root node" to start from.

In my blog "The Math Debate" I talked about my future what and how blogs, this is the "what",  I often see questions on various sites like Reddit from people who want to improve their math skills and asking how and what to learn.  I wanted to talk about what I feel is important. Remember I am neither a mathematician nor do I play one on TV, although I do hope to play one on TV someday, also remember I am still learning about all of these areas so take my advice with a kosher sized grain of salt, and if you see errors please point them out.

In my opinion you can mostly, not worry, for the moment, about things like Calculus and lot of the continuous math, don’t get me wrong you do eventually want to learn or relearn these things but for CS and programming the Discrete Math is more immediately relevant.  I definitely think that if you really want to successfully use math in programming you need to be something of a math generalist, which is really the "breadth first" approach that Steve Yegge mentions.

So the real question, problem if you will, is what to study, this is of course subjective and dependant on your goals. My motivation is multifaceted with two primary motivations, one is pure curiosity coupled with wanting to have a higher understanding of Life, The Universe, and Everything, the other is wanting to do more interesting computer work and to do it better.  Fortunately I mostly do not find these motivations contradictory.  

Math is a huge field and it is intimately intertwined which becomes more evident the deeper you dig, it seems to me that a broad perspective is helpful, of course I am laying out quite a bit here, my recommendation is to try to glean a feel for the various disciplines, also as I mentioned once you start digging sometimes you end up reading about areas that previously seemed unrelated.

If you wish to only focus on a smaller set of areas I would recommend the following for programming: 

Set Theory, Logic, Graph Theory with some Combinatorics, and some basic Abstract Algebra like Groups and Monoids.  Then go on to Linear Algebra, more Combinatorics, Probability and Statistics and some type of intro to Formal Language Theory and Automata Theory. Pretty much stuff you might have taken for a B.S. in CS  How and what you learn is really up to you, here is a more comprehensive list:

Set Theory is in many respects the "arithmetic" of doing advanced and CS related Math it is essential and should be something that you should plan on committing the basics to memory, do you know the following concepts: union, intersection, symmetric difference, set difference, disjoint, cardinality and the power set? Many of the following disciplines rely on Set Theory, learning these basics can save you a lot of time in the sense that you may need to stop and think about them or look them up as you are learning other areas.

Another discipline which could be thought of as an "arithmetic" of Math and CS is Propositional Logic, in fact it really is the "arithmetic" of your computer at the hardware level.  Also Set Theory and Logic have a number of parallels and actually the two are somewhat intertwined.  Here you should ask yourself do you know the truth tables for and, or, nand, nor, and xor off the top of your head? The general relevance of Propositional Logic is enhanced by the added abstraction of Predicate Logic. Also there are Combinitory Logics like Lambda Calculus which will help you better understand functional programming among other things.

The next topic which probably needs no introduction is Graph Theory.  This is another area you should really plan committing the basics to memory also you might want to consider extensive study here if you motivation is advanced programming and software architecture. Also it’s really cool and there’s tons of cool stuff happening in this field both mathematically and in our industry, like Social Network Theory.  Also I think Graph Theory is one of the best maths to study, I find it fascinating and extremely relevant to IT, also it’s what I call a gateway math, once you get hooked on it, it leads to other maths.

Another area which could be considered foundational in an arithmetic sense is Combinatorics, which is science of counting. I admit that this is a math that I love and fits with being a programmer. A good programmer is always looking at things from the perspective of what are all the possible outcomes and contingencies;   Combinatorics gives you a methodology to do that. I have written a programmer intro here.

While we are talking about possible outcomes the natural extension is Probability and Statistics which build in part on Combinatorics and are now pretty much in the forefront of hot "maths" for IT because of areas like Bayesian Statistics, Machine Learning and Data Science.  Also if you delve deep into these then you need to be looking at both differential and integral calculus.

An increasingly important area is Abstract Algebra which includes our old friend the Monoid and Groups, Rings and Fields.  I confess that there are still quite a few things here that I am trying to wrap my head around, especially the intuitively understanding whole Polynomial thing but I feel you can’t go wrong studying this and that it is central to fundamentally understanding what we do. 

One math that I may have implied that you could initially skip is Trigonometry, that really isn’t true, but I think you can get away with just focusing on getting Sine and Cosine under your belt at first and then you have enough to fall back on if you need more and those two will help you with things like Linear Algebra and if you have the inclination: Fourier Transforms.

Linear Algebra is pretty crucial it’s like the Swiss Army Knife of math and to use another weak analogy it’s a little like a Collections API of math.  This is definitely worth studying and is used in a lot of areas including Statistics, Machine Learning and Search Technologies which can involve things likeVector Spaces.

Relations, Functions, Posets, etc., there are some "basic" concepts that you usually get hit with during an introductory CS math course it might be part of a Discrete Math or some type of Introduction to "Advanced" or "Abstract" Math class and it will usually follow pretty close to Set Theory.  These things are pretty essential, and I don’t think I was ever exposed to Posets in my curriculum which I think was one of several deficiencies.  A couple of concepts that you want to know backwards and forwards, pun intended, are Injection, Surjection, and Bijection.  Also it seems like the Ordering/Posets related concept of Infimum aka Greatest Lower Bound and Supremum aka Least Upper Bound come up a lot, at least in the stuff I look at.

Number Theory is actually pretty cool.  Prime Numbers for example fascinate many people including me, and it has some direct but perhaps more specialized applications to CS like Hash Functions and Encryption.

Proofs – This is one area that is important and in which I am severely lacking actually with the exception of a few algebraic combinatorial proofs I’ve been proof free since `93, but it’s definitely on my todo list.   I mention it separately because it can be treated as it as its own discipline even though it’s ubiquitous in Math.

CS Stuff

The following areas I would consider to be very Computer Science related and may generally be somewhat outside of main stream Math.

Algorithms, this is a no brainer, and includes the obvious Big O aka Landau Notation, it’s not only O you know. Also there are Complexity Classes, and really things that overlap with other areas like Graph Algorithms and Computability in general which involve:

Formal Language Theory and Automata Theory, go hand in hand and are needed to really understand Regular Expressions, Parsers and Compilers. Not to mention that the Von Neumann architecture is based on the Turing Machine. I am, however, a little dismayed by the number of developers including CS majors that I have encountered that have minimal or no knowledge of these, and not to be a rude but if you do not have some basic knowledge about these you don’t know jack about computers and software. I mean this in the most positive encouraging way, so get on it!  Do it now! Click the links and get started! Also for Formal Language Theory it helps to understand the String Monoid.

Relational Algebra is the underlying Algebra to how SQL works, yes I know SQL is not cool right now, but learning this underlying math gives you insights that extend beyond SQL.  Actually Relational Algebra relates to Set Theory and Abstract Algebra and some pretty heavy other maths like Category Theory, Topos Theory, and Fibrations, most of these are beyond my understanding at the moment.

Queuing Theory is pretty relevant to things that we do and is an area which I want to improve on so I don’t have a lot to say here, it does of course relate pretty heavily to Probability Theory and things like Markov Chains.

Information Theory (Shannon) and Algorithmic Information Theory (Kolmogorov, et al)1 both relate to a few areas one of course is Probability Theory. Compression and Encryption are related to these in a number of ways.  Put simply, Shannon’s Information Theory relates representing and transmitting information, where as Algorithmic Information Theory is concerned with what is the smallest algorithm to generate strings. This is definitely interesting and relevant stuff that you want to look at.

Taking it up a notch

With Math it can be infinite and there are always more advanced things to learn, now realize that this is a broad list and while you probably want to have a general view of things, you will undoubtedly have areas that you want to focus on.  Actually to do anything with math, unless you are genius, you will have make choices about what you study. 

If you want to go to the next "level" whatever that means to you, you will probably want to look at the some of the above areas in more detail and here’s a list of some areas to ponder as well:

Calculus – At this point you may be thinking, hey wait I had to take this as a freshman in college or like me you took it in high school, how can this be next level stuff, isn’t it basic?  Well yes and no, actually when was the last time you needed to use a Derivative or Integral when programming?  Calculus, specifically Differential Calculus is often cited in arguments against math in programming.  Now don’t get me wrong it’s important and relevant especially as your level of other areas increases and it may be more relevant more quickly, especially Integral Calculus if you go the Statistics/Machine Learning route.

Graph Theory is pretty huge and if you are interested in math in regards to programming work, I don’t think you can go wrong learning more about it, and after you’ve got a lot the "basic" concepts like Coloring, Planarity, Walks, and Cycles ideas, you can look into areas like Spectral Graph Theory and Expander Graphs. I just found out about these two recently and they look pretty interesting.

Other Logic topics like Multi-Valued Logics, extend the ideas of logic creating possibilities that fall outside of traditional two values of true and false.  Fuzzy Logic and Probabilistic Logic add numeric values and probability to the interpretation of logic values. Modal Logics add additional semantics to traditional Logic to describe additional types of attributes like necessity and temporal characteristics. Additionally there is Metalogic and don't forget Gödel's incompleteness theorems.

Order Theory and Lattice Theory take Partially Ordered Sets to a higher level.  These seem to be pretty intertwined, especially lattices, with a bunch of other areas like Algebra, Logic, and Probability. I have to wonder if Lattices should be introduced more formally earlier. 

Model Theory is in some senses a convergence Abstract Algebra and Logic, so if those two maths get you jazzed you probably want to be looking here at some point.

I think an understanding of Topology is going to be critical in doing many math related tasks with computers, in a sense it already is, after all Graphs are topological concepts, in fact Euler’s solution of the "Seven Bridges of Konigsberg" problem is often cited as both the inception of Graph Theory and Topology and Euler went on to create a formula which describes the "planarity" of graphs on Topological surfaces of varying genus, additionally I have seen a number of interesting books and papers involving Topology and Computing, more on those later.

Measure Theory put simply deals with the idea of measuring sets.  It is built on top a structure called a Sigma Algebra which has some similar properties to a Topological Space actually the two can be combined to create a Borel Set, most of this stuff is beyond my current level.  It also relates to probablity theory in that probablity theory can be described in terms of measures especially the relationship of Event Spaces to Measure Spaces.

Analysis - I still have a lot to learn in this area but it seems to be a kind of nexus of continuous math and is a very important area for understanding advanced concepts also I have heard it described as the study of change. One branch of it is Functional Analysis. Analysis ties a lot of pretty heavy areas together including but not limited to Calculus, Measure Theory, Hilbert Spaces and an important, interesting and pretty cool area of Topology called Metric Spaces. This is another area that has a lot of the groundwork for things like Machine Learning.

Algebraic Topology is the combination of concepts of Abstract Algebra and Topology and surprisingly this may be one of the most central maths for Computer Science, one of my goals it to have some decent working knowledge of this field and I recommend that anyone seriously considering attempting to acquire a deeper understanding of Software and Computing consider studying this. Now you may not want to jump right in and there are some related areas that are more immediately relevant like Category Theory, which I have read might be better described as "Abstract Function Theory", it’s sort of the Math of Math, and it is showing up more and more in CS related areas I’m looking at you Monads! Category Theory seems to have a lot of relevance to functional programming which is more intuitive if you think "Abstract Function Theory". This also gets you back to that whole Relational Algebra area with things like Topos Theory, and Fibrations.

Game Theory is perhaps a "Penny Stock" in CS, in that it is probably not on a lot of people’s radar, but it has some very interesting possible applications, especially in terms of algorithms.  It applies heavily to social sciences and considering how prominent Social Networking and the concepts like the Social Graph are becoming it seems like it might be pretty relevant.  It also has applications in general management, perhaps Software Process will benefit from it as well someday.

Complex Numbers, it has always struck me how alien complex numbers seem. I am confused by the fact that they seem to be a way to represent two dimensions, so should we just always use them for that instead of two dimensional vectors?  And you can take it to higher dimensions with Hypercomplex Numbers like Quarternions and Octonions. I know that complex numbers are a key to understanding Hilbert Spaces and:

Fourier Theory has many applications to natural phenomena related to Signals and Signal Processing, it is also used in Image Processing.  This is an area that I still don’t know a lot about but it seems that Euler’s Formula is pretty important here.  

Fractal Geometry: "All of Nature talks to me, if I could just figure what it was trying to tell me." Fractal Geometry and its sister discipline Chaos Theory describe perhaps all natural phenomena from the clustering of galaxies to the pattern of capillaries in your fingers to the Brownian Motion in your hot beverage, these patterns even apply to social phenomena and computer phenomena such as stock market patterns and network traffic and more. I feel that there are a multitude of areas in which we have only scratched the surface on this, and I think that these will become equally commonplace in computing.  There are even constructs called L-Systems, invented/discovered by Aristid Lindenmayer, a botanist, to model plant growth, these structures also generate "pure" fractals like the Koch Curve.  A very compelling thing about L-Systems is their similarity to the structures in the Chomsky Hierarchy.   Benoit Mandelbrot, who sadly, recently passed away, coined the term Fractal and described them as "a set for which the Hausdorff-Besicovitch Dimension strictly exceeds the Topological Dimension." These are both topological concepts so in a sense Fractals are also steeped in Topology.

Now this list might seem a bit crazy and overly ambitious and this is mostly my crazy list but ironically, it partially jibes with, and was partially influenced by a book called Comprehensive Math for Computer Scientists which is published in two volumes. It is a beautifully crafted book but it uses a very modern approach to math so I find it a bit hard to follow at times but still I recommend it, actually one of my goals is to fully understand it.  Here are some top level topics they cover that I did not:

One last and critical point is that this is a pretty broad and probably intimidating list, I know writing it made my brain hurt, if it’s any consolation my knowledge on the above topics ranges from a decent understanding to a vague idea what the topic is about.  If you develop a passion for any or all of these areas you will always feel like there is not enough time to learn what you want to learn, but remember don’t look at it as being intimidating or frustrating, look at it as an opportunity to never be bored with life.

1This is also attributed to Chaitin and Solomonoff