Showing posts with label Education. Show all posts
Showing posts with label Education. Show all posts

07 August 2012

Why You Can’t Have a Real Software Engineering Discipline



As we all know the fields of computer science and software engineering are in their infancies.  Many a blog post has been written lamenting the fact that software engineering is not a real engineering discipline, and while I have not written that exact post, I have written about the subject and have deconstructed what others have said about it.  A number of these points do apply to why software projects routinely fail, yet another topic that has received considerable attention.  Now admittedly all engineering disciplines regardless of their maturity and formalization have project failures and creating a real software engineering discipline will not eliminate this problem but one would hope that it would abate it.

I find it odd that at a time when so many people are involved in IT there seems to be little discernible progress in creating a real software engineering disciple.  There are probably millions of people working on creating software.  Also there are many researchers trying to solve this problem from many different angles.  Practitioners have been attempting to solve it with ideas like Agile and Software Craftsmanship, etc.  Additionally there is a long list of failures surrounding the formalization software engineering.  

My biggest complaint is the fact that there really are no good formal standards in general software engineering principles and methodologies lack a formal foundation and even the more vague principles can be easily thwarted and misused, and often, as I have previously complained, it often ends up being based on pure opinion which is usually won through positional authority, perseverance or by those who are just more politically savvy.

The intent of this post is not to attempt to solve or even offer solutions to the problem of creating a real software engineering discipline but to look at what I think are some barriers to creating this discipline and in some of these cases offer thoughts on overcoming those barriers.


Low Entry Requirements


If we are to think software engineering as an engineering discipline, I challenge you to find another engineering discipline that routinely has practitioners with no formal training in the field.  I very much doubt that someone who did not hold a degree in engineering and who built a shed in his back yard is now working as a structural engineer on a construction project, the mere idea seems ludicrous.  Yet I have worked with non technical degree holders who became software developers, one who started by building web pages and now calls himself an "Internet Application Architect", and is probably one of the biggest cargo cult coders I’ve ever had the misfortune to work with.  This egalitarian aspect isn’t all bad as it does let good people into the field as well.   Another non technical degree holder I once worked with became an accomplished developer and went on to get an advanced degree in CS and is now a CS professor.   Although it’s probably the case that for every good non technical degree holder who joins the field it is likely that dozens of mediocre and bad practitioners will also join.   It seems that in other engineering disciplines there are much more stringent educational requirements to ensure proper training.   To be clear here I am not equating having a degree with being competent because I have met many incompetent people with degrees, but still you need something.   I confess I am at a loss for a solution for this one.


The Lack of Differentiation between Science and Engineering


Chemistry is a scientific discipline, chemical engineering is an engineering discipline, and you can make this comparison between other engineering disciplines and the sciences that they employ e.g. electrical, mechanical, and structural map to various areas of physics.  Engineering and scientific disciplines are different types of disciplines often taught in separate schools.  Although each engineering discipline has some scientific overlap and it would probably be possible to move from the appropriate scientific field to the corresponding engineering field in general you probably would not do so without returning to school.   In recent years software engineering curricula have been added to the rosters of many colleges and while this is potentially a step forward, discounting for the moment that software engineering is still not really real engineering, a software engineering degree and a computer science degree in many cases gets you the same job!   

So in other fields there would be a differentiation between a degree that would put you on a track to do scientific research and one that would put you on a track to do engineering work.  As I understand it, if you are fresh out of school with a BS in CS that qualifies you to be a tester at Microsoft1, developers hired right out of school need at least a Masters degree in CS.   This is an extreme case of how this really breaks down in our industry.  In most cases engineering practitioners are the software engineering, MIS, CS or even non technical degrees and the few scientific careers generally go to the advanced degree holders.  I admit this probably fairly normal a BS in chemistry or biology is also more likely to get you a job in IT than a job as a research scientist.  Still compared to more mature engineering fields there seems to be a lack of real differentiation between the degrees that yield a career as a software engineer versus one as a computer scientist.


Bad Management


The entry requirements for managers in software make the low entry requirements for software engineering practitioners look downright rigorous.  I have previously criticized the fact that many organizations find the cheapest people, especially for software management and give them an obligatory certification.  Now I know that engineering management regardless how formal or established the engineering discipline is an area that has problems and failures, but again I very much doubt that you would find a twenty something business or liberal arts major with a freshly minted PMP suffix managing construction or civil works projects.   Yet in my experience this is pretty normal in IT especially in the government sector.  Of course bad software management is executed by older managers as well.

A post by Larry White titled "Engineering Management Is Dying" delves into some these issues.  It is definitely the case that methodologies like Agile have changed how software projects work and the traditional corporate management approach to software really doesn’t work.  One point he makes is that it is not uncommon to see a two hundred person project at Google headed by an engineer.  To me this implies that someone is in some way managing that project.  I think all of this is indicative of the need for software engineering mangers to also be practitioners in engineering or at least very well versed in how software development works.  The situation he talks about at Google is not the case everywhere, in that case the company is its own client, but there are many cases where software is being built for clients and this does create a need for management that deals with the client, perhaps this should be a role that is separated from management and called a client liaison, which is what often happens.  Also I am not sure how things work at Google but I have also seen internal development in organizations where the IT department provides services to an internal client, I have also seen this go horribly awry with contentious unproductive relationships between departments.  In short I feel that just as we need a real software engineering discipline we also need a real incarnation of software engineering management.


The Disconnect between Academia and Industry


Most practitioners do not keep up with, or read academic research, actually many practitioners don’t read at all, but that’s another issue, and most academicians don’t work in the field so they lack the hands on knowledge.  Unfortunately this rift can take on a somewhat disdainful tone as is the case in my exploration of one practitioner’s attempt to define "Real Engineering".   Fortunately some people, I’d like to count myself among them, take a more constructive approach to building bridges across this rift.  Daniel Lemire has an interesting critique the quality of software produced in academia

I think the solution here is for both sides to become more engaged in problems faced on each side. I am optimistic about this one as I feel that these walls are breaking down as the field is growing up and many of the emerging technologies are forcing developers to be more cognizant of and engaged in research topics.  I also have encountered much more research that has ties to the practical concerns of day to day software development, some I have mentioned with more to come.


The Lack of an Effective CS Math Curriculum 


If you read my blog you know it to be a mix of math and software practices. I would describe myself as a software practitioner and a math enthusiast.  My math journey has taken me on some interesting math excursions into areas of math that seem to get little or no mention in CS curricula and I feel that this is a major problem in really applying math to the field of software engineering.  Another problem is in the way that it is taught, my experience was that it was not only taught badly but in a way that made it seem irrelevant to many of the programming courses.  Specifically I would shift the CS math curriculum to include less differential and integral calculus and include more logic, combinatorics, abstract algebra, graph theory, order theory, category theory and probably some topology among others.

Over the last few years I have been progressively learning more math which has been in part motivated by necessity for understanding research papers.  This approach has changed the way I see things, I now see the patterns of math in software. They are there and I believe that they can be exploited to create a real software engineering discipline.

For the math learning problem I do have some ideas many of which are expressed in my blog.  Some of my posts like refactoring if statements with Demorgan’s Laws or the String Monoid, the Object Graph, etc. are about making these mathematical ideas more relevant and accessible to programmers. Other posts like my series on naming and my post on generic programming and entropy and complexity are about my own ideas and ideas in the research literature that I feel may help to create a real software engineering discipline.  These are my attempts at solutions to these problems, so expect more of both of these and more thoughts on the CS math curriculum.


The Software Architect Debacle


This is one that really bothers me.  I feel the software architect role in general has a very negative effect on creating quality software and it dissuades the industry from developing a real engineering discipline.  The Software Architect role tends to be broadly defined, you have enterprise architects, software architects, etc., some architects tend to be involved with hardware and networking, some work on configuration management, some do software design, some do all of these things and more.  I feel this is a problem as each of these is a separate engineering area with different types of problems. By not breaking these down it often leads to a lack of focus on specific problems, also I have seen cases where the architect focuses on their preference and ignores other issues.  The architect role is often divorced from the code.  I have met many architects that were responsible for software but had never even looked at the code.  To me this is a huge failing that architects that are responsible for building software often lack the interest, time, or even aptitude to know the quality or underlying of structure of the software that they are delivering.

To really do justice to these ideas I might need a separate post, my solution would be to break down the architect role into specific engineering roles, which could include:


  • Software Process Engineering - this would be involved with software team and resources planning, some aspects of configuration management such as defining policies, requirements analysis and general project management aspects of software construction including the definition and refinement of the SDLC.  This role is tightly coupled with software engineering management.
  • Software Structural Engineering - this would include requirements comprehension, code infrastructure planning, prototyping work, hands on code structural work including custom frameworks and reusable components, third party library products, and general code quality including reviews and static analysis.
  • Software Quality Engineering - This would include the QA roles, software testing and testing tools, requirements validation and refinement, usability and reliability and general software quality issues.
  • Software Infrastructure Engineering - This would include hardware, networking, database, app server and general service infrastructure, non functional requirements fulfillment and possibly the implementation of configuration management, continuous integration, etc.

This is a rough "sketch" of these possible roles and each of these areas has some overlap implying that all of the people in these roles would work closely together, also some combination of, or all of these roles might be performed by a single individual depending on the organization’s size and structure.  What this approach does is clearly defines roles and responsibilities as opposed to some nebulous software architect role.


The Continuous Turnover of the Work Force


A young CEO once commented that he thinks young programmers are superior.  In general companies prefer younger workers as they often don’t have family commitments so they can work more hours and you can pay them lower salaries, DC area government contractors rely on this to keep their profit margins up.

As an older developer I often find this frustrating.  I confess that I have not moved up the ladder and still mostly work as a developer or what might be called a hands on architect, yes I know but that’s the current term, actually my preferred title would be: Software Structural Engineer.  Working as a contractor I have on several occasions found myself on projects that were dominated by younger developers and unfortunately on more than one occasion I watched as teams would make many mistakes due to a lack of experience, in some cases I was able to help in others I was ignored by the younger developers.  This is not to say that I do not still make mistakes, I’ve just been around long enough to have made a lot of them already.

If you buy into the software craftsmanship thing, I admit to being skeptical of this idea, you might be inclined to draw a parallel to craftsman of the past when older established masters took on apprentices and passed on their wisdom to the next generation.  Given the access to information these days such process is somewhat antiquated and perhaps impractical.  Nevertheless I feel, and the industry supports me here, that older developers who keep their skills up to date have value.  I know several younger developers who have sought me out to learn from me so I think I can safely say I still have some value.

Many older developers have let their skills get rusty and have not kept up on current technologies I have seen this many times.  Also many companies seem to lack the vision to allow for real hands on technical career growth.  I can’t help but feel that this inclination to recycle developers in each new generation is hurting or at least slowing our ability to develop a real software engineering discipline, this potential loss of continuity seems like it leads to some lost wisdom.


The Social Networking Drain and Hollywoodification of Silicon Valley


As I write this Hollywood gears up to deliver a reality TV show that takes place in silicon valley and Mark Zuckerberg now gets the paparazzi treatment.  Another article that I previously mentioned was about how CS enrollment was up because the movie The Social Network had enticed many aspiring Zuckerberg wannabes.  At present there does seem to be a gold rush mentality and it is noticeable and even discussed on sites like Hacker News

As someone who has never worked in the valley I am very much an outsider and may not have a good perspective on these things, but it seems that there are a lot of startup companies focusing on crap. Now I get it.  We live in a shallow consumer society with a rapacious appetite for crap. But are not these supposed to be some of the smartest people and I am not the only one to wonder why so many smart people who should have better taste and higher standards are settling for working on serving up ads and other vapid crap instead of doing more meaningful work like working on real problems that would advance human society or even just work on advancing software engineering. 


1I believe this was the case in the late 90’s and may have changed, this information was conveyed to me by a former Microsoft employee.


18 July 2011

The Certificate Industry

I have been thinking about writing this for a while now and as I write this there is something of an education debate in part spurred on by remarks made by Peter Thiel.  I have no interest in being part of that debate. Actually this is more of a rant that is on the periphery of that discussion.  My complaint is that there seems to be a whole industry that surrounds the idea of certifying people as knowledgeable or learned in certain areas whereas my experience is that many people who carry these certifications then think themselves knowledgeable and or qualified to do certain tasks yet they are often severely lacking in knowledge and ability.  I have met many people who look at education solely as a means to a job and then rest on the laurels of their prior educational accomplishments and see no need to further their education beyond college.  I cannot tell you how many people I have met who seem to have no interest in the very subject that they majored in college, it makes me feel as though the degree was the only goal not the knowledge gained to receive the degree and picking a major was a mandatory option to attain the degree.  It is this type of goal oriented education that seems to be the bane of our educational system which is nicely, albeit nervously, articulated1 by Erica Goldson.

I am lumping college degrees including advanced degrees and corporate, often product related, certifications together, not to mention the mentality by corporations that these,  as Erica Goldson puts it, "pieces of paper that tell us that we are smart enough to do so" seem to be seen as the end all in qualifications. Now obviously not all companies think this way, in fact determining actual skills and knowledge during interviews, especially of programmers, seems to be a subject of much protracted debate on the inter-tubes.  I am not saying that are not people who effectively avail themselves of their educational opportunities and are not passionate about their fields, I just feel these people are not in the majority.

The certificate that I find the most irksome is PMI’s PMP.  In my experience I have dealt with many managers who proudly append this suffix to their names and companies seem to regard this as a concrete qualification for managing a project, however, my experience is that it usually is not. The most egregious example of this in my experience was using twenty-something liberal arts majors with PMP certifications to manage software projects as I have previously mentioned.  I know someone who is pretty involved in the PMI community and she points out that this deficiency is not the methodology itself but due to the fact that the processes are either not applied or improperly applied, so I am in no position to comment on the efficacy of their methodology.  Regardless of that fact it seems like PMI has a pretty lucrative business selling these certifications.

On the other end of the spectrum I have encountered people who hold masters degrees in Computer Science, who seem rather lacking in the fundamentals, I have met relatively recent graduates who strain to tell you the difference between an NDFA and DFA. In one case a young recent graduate saw a document on by desk with the title "Lambda Calculus" and asked me why I was reading about Calculus, when I responded "it’s not Calculus it’s Lambda Calculus", he replied with "what’s that?"  I then said, "You know what a Turing Machine is" thinking I would explain it in terms Turing Equivalence, and he replies "No."  At that point I was dumbfounded and retorted with "How the hell can you have a Masters in Computer Science and not know what a Turing Machine is?"  I then pulled it up on Wikipedia and he claimed to know it, he probably did, it turned out he was pretty bright with a fair amount of potential but kind of an airhead, still it struck me as pretty lame, plus shouldn’t someone with a Masters in CS at least have heard of Lambda Calculus and know about Turing Machines off of the top of their head?  Now this doesn’t only apply to theoretical knowledge, I have worked with some Masters degree holders who were terrible programmers.  Once again this not to say that there aren’t bright knowledgeable Masters Degree holders who are knowledgable and good programmers, but clearly having that piece of paper does not make it so.

Now these are just my experiences but you will find other mention of these types of experiences also related to Microsoft, Sun and other corporate certifications, remember selling certifications are businesses which can be fairly profitable.

I recently came across this article on the New York Times website about how the movie "The Social Network" has renewed interest in Computer Science Curriculums because many starry eyed college students are seeing themselves as possibly being the next Mark Zuckerberg.  This reminds me of a previous coworker who had graduated around the time of the last tech bubble, this guy was actually pretty sharp but he had sporadic interest in the day to day work and would often gripe that he really didn’t like the field, and he was only in it because it was a well paying career.  I admit as someone who is passionate about my career and my field of study I find this mentality extremely frustrating, I feel that these are the exact people that you do not want to hire, now obviously not all of these new enthusiasts will be like that, but as a word of advice, most people working in startups tend to make less money than salaried employees, because even if your company is successful that doesn’t mean that it will have some huge market valuation and if it fails you are out of the street to start anew or find a job, of course if you want try it go for it.  Now there are diverse areas that you can go into in IT but be prepared to find something that you like to do every day, don’t just do it because it pays well or you might get rich, both may happen especially the  former which is a lot less glamorous.  I’m just saying for every Mark Zuckerberg there thousands of us workaday developers who are just out there making a living at our "craft". And you may find that your life in this field ends up being more like the movie "Office Space" than the "Social Network".

Just to be clear here, I really enjoy and admire academia and academic work, but unfortunately I think, perhaps unjustly, that many CS curriculums are broken in terms of practicality and how and what math they teach you. So if you want to learn programming and start a company, you might want to: just do it, but if you are serious about the field and the discipline of software then you might want to go the education route and avail yourself of the possible opportunities that it offers.

Maybe it’s just because I’ve become jaded over the years but I am really skeptical about this Hollywood/Rockstar mentality that seems to now taint our industry and I think that in some ways this is indicative of the real problem with education in our society in that we view education solely as a means to a better job and we do not truly respect knowledge and learning. It’s all about the Benjamins baby. 

1I admire her Courage for speaking out like that.

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.

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:

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

Scilab

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.