This short article is the result of various conversations over the course of the past year or so that arose on the back of two articles/blog pieces that I have previously written about Category Theory (here and here). One of my objectives with such articles, whether they be on aspects of quantum computing or about aspects of maths, is to try and de-mystify as much of the associated jargon as possible, and bring some of the stunning beauty and wonder of the subject to as wide an audience as possible. Whilst it is clearly not possible to become an expert overnight, and it is certainly not my objective to try and provide more than an introduction (hopefully stimulating further research and study), I remain convinced that with a little effort, non-specialists and even self confessed math-phobes can grasp some of the core concepts. In the case of my articles on Category Theory, I felt that even if I could generate one small gasp of excited comprehension where there was previously only confusion, then the articles were worth writing.
I just finished a course on Algebraic Geometry through UCLA Extension, which was geared toward non-traditional math students and professionals, and wish I had known about Smith’s textbook when I’d started. I did spend some time with Cox, Little, and O’Shea’s Ideals, Varieties, and Algorithms which is a pretty good introduction to the area, but written a bit more for computer scientists and engineers in mind rather than the pure mathematician, which might recommend it more toward your audience here as well. It’s certainly more accessible than Hartshorne for the faint-of-heart.
I’ve enjoyed your prior articles on category theory which have spurred me to delve deeper into the area. For others who are interested, I thought I’d also mention that physicist and information theorist John Carlos Baez at UCR has recently started an applied category theory online course which I suspect is a bit more accessible than most of the higher graduate level texts and courses currently out. For more details, I’d suggest starting here: https://johncarlosbaez.wordpress.com/2018/03/26/seven-sketches-in-compositionality/
The Santa Fe Institute's free online course "Introduction to Information Theory" taught by Seth Lloyd via Complexity Explorer.
Many readers often ask me for resources for delving into the basics of information theory. I hadn’t posted it before, but the Santa Fe Institute recently had an online course Introduction to Information Theory through their Complexity Explorer, which has some other excellent offerings. It included videos, fora, and other resources and was taught by the esteemed physicist and professor Seth Lloyd. There are a number of currently active students still learning and posting there.
Introduction to Information Theory
About the Tutorial:
This tutorial introduces fundamental concepts in information theory. Information theory has made considerable impact in complex systems, and has in part co-evolved with complexity science. Research areas ranging from ecology and biology to aerospace and information technology have all seen benefits from the growth of information theory.
In this tutorial, students will follow the development of information theory from bits to modern application in computing and communication. Along the way Seth Lloyd introduces valuable topics in information theory such as mutual information, boolean logic, channel capacity, and the natural relationship between information and entropy.
Lloyd coherently covers a substantial amount of material while limiting discussion of the mathematics involved. When formulas or derivations are considered, Lloyd describes the mathematics such that less advanced math students will find the tutorial accessible. Prerequisites for this tutorial are an understanding of logarithms, and at least a year of high-school algebra.
About the Instructor(s):
Professor Seth Lloyd is a principal investigator in the Research Laboratory of Electronics (RLE) at the Massachusetts Institute of Technology (MIT). He received his A.B. from Harvard College in 1982, the Certificate of Advanced Study in Mathematics (Part III) and an M. Phil. in Philosophy of Science from Cambridge University in 1983 and 1984 under a Marshall Fellowship, and a Ph.D. in Physics in 1988 from Rockefeller University under the supervision of Professor Heinz Pagels.
From 1988 to 1991, Professor Lloyd was a postdoctoral fellow in the High Energy Physics Department at the California Institute of Technology, where he worked with Professor Murray Gell-Mann on applications of information to quantum-mechanical systems. From 1991 to 1994, he was a postdoctoral fellow at Los Alamos National Laboratory, where he worked at the Center for Nonlinear Systems on quantum computation. In 1994, he joined the faculty of the Department of Mechanical Engineering at MIT. Since 1988, Professor Lloyd has also been an adjunct faculty member at the Sante Fe Institute.
Professor Lloyd has performed seminal work in the fields of quantum computation and quantum communications, including proposing the first technologically feasible design for a quantum computer, demonstrating the viability of quantum analog computation, proving quantum analogs of Shannon’s noisy channel theorem, and designing novel methods for quantum error correction and noise reduction.
Professor Lloyd is a member of the American Physical Society and the Amercian Society of Mechanical Engineers.
Yoav Kallus is an Omidyar Fellow at the Santa Fe Institute. His research at the boundary of statistical physics and geometry looks at how and when simple interactions lead to the formation of complex order in materials and when preferred local order leads to system-wide disorder. Yoav holds a B.Sc. in physics from Rice University and a Ph.D. in physics from Cornell University. Before joining the Santa Fe Institute, Yoav was a postdoctoral fellow at the Princeton Center for Theoretical Science in Princeton University.
My response to his post with some thoughts of my own follows:
This is an interesting, but very germane, review. As someone who’s both worked in the entertainment industry and followed the MOOC (massively open online courseware) revolution over the past decade, I very often consider the physical production value of TGCs offerings and have been generally pleased at their steady improvement over time. Not only do they offer some generally excellent content, but they’re entertaining and pleasing to watch. From a multimedia perspective, I’m always amazed at what they offer and that generally the difference between the video versus the audio only versions isn’t as drastic as one might otherwise expect. Though there are times that I think that TGC might include some additional graphics, maps, etc. either in the course itself or in the booklets, I’m impressed that they still function exceptionally well without them.
Within the MOOC revolution, Sue Alcott’s Coursera course Archaeology’s Dirty Little Secrets is still by far the best produced multi-media course I’ve come across. It’s going to take a lot of serious effort for other courses to come up to this level of production however. It’s one of the few courses which I think rivals that of The Teaching Company’s offerings thus far. Unfortunately, the increased competition in the MOOC space is going to eventually encroach on the business model of TGC, and I’m curious to see how that will evolve and how it will benefit students. Will TGC be forced to offer online fora for students to interact with each other the way most MOOCs do? Will MOOCs be forced to drastically increase their production quality to the level of TGC? Will certificates or diplomas be offered for courseware? Will the subsequent models be free (like most MOOCs now), paid like TGC, or some mixture of the two?
One area which neither platform seems to be doing very well at present is offering more advanced coursework. Naturally the primary difficulty is in having enough audience to justify the production effort. The audience for a graduate level topology class is simply far smaller than introductory courses in history or music appreciation, but those types of courses will eventually have to exist to make the enterprises sustainable – in addition to the fact that they add real value to society. Another difficulty is that advanced coursework usually requires some significant work outside of the lecture environment – readings, homework, etc. MOOCs seem to have a slight upper hand here while TGC has generally relied on all of the significant material being offered in a lecture with the suggestion of reading their accompanying booklets and possibly offering supplementary bibliographies. When are we going to start seeing course work at the upper-level undergraduate or graduate level?
The nice part is that with evolving technology and capabilities, there are potentially new pedagogic methods that will allow easier teaching of some material that may not have been possible previously. (For some brief examples, see this post I wrote last week on Latin and the digital humanities.) In particular, I’m sure many of us have been astounded and pleased at how Dr. Greenberg managed the supreme gymnastics of offering of “Understanding the Fundamentals of Music” without delving into traditional music theory and written notation, but will he be able to actually offer that in new and exciting ways to increase our levels of understanding of music and then spawn off another 618 lectures that take us all further and deeper into his exciting world? Perhaps it comes in the form of a multimedia mobile app? We’re all waiting with bated breath, because regardless of how he pulls it off, we know it’s going to be educational, entertaining and truly awe inspiring.
Following my commentary, Scott Ableman, the Chief Marketing Officer for TGC, responded with the following, which I find very interesting: