Andrew Jordan reviews Peter Woit's Quantum Theory, Groups and Representations and finds much to admire.
Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky’s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.
This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.
My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:
It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.
Huh? What’s a ‘universal property’?
In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.
A universal property amounts to a precise description of what an object does.
Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:
through representable functors (which are how you actually hand someone a universal property),
through limits (which are ways of building a new object out of a bunch of old ones),
through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).
If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.
Here’s what he wrote, over on the category theory mailing list:
My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:
It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.
There are lots of good introductions to category theory out there. The particular features of this one are:
• It’s short.
• It doesn’t assume much.
• It sticks to the basics.
Instagram filter used: Clarendon
Photo taken at: UCLA Bookstore
I just saw Emily Riehl‘s new book Category Theory in Context on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one wouldn’t purchase the physical version.
I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.
“contains significant amounts of material not well-explained elsewhere.”He expects to finish up the diagrams and publish it next year some time, potentially through Springer.
Advanced Data Analysis from an Elementary Point of View
by Cosma Rohilla Shalizi
This is a draft textbook on data analysis methods, intended for a one-semester course for advance undergraduate students who have already taken classes in probability, mathematical statistics, and linear regression. It began as the lecture notes for 36-402 at Carnegie Mellon University.
By making this draft generally available, I am not promising to provide any assistance or even clarification whatsoever. Comments are, however, welcome.
The book is under contract to Cambridge University Press; it should be turned over to the press before the end of 2015. A copy of the next-to-final version will remain freely accessible here permanently.
Table of contents:
I. Regression and Its Generalizations
- Regression Basics
- The Truth about Linear Regression
- Model Evaluation
- Smoothing in Regression
- The Bootstrap
- Weighting and Variance
- Additive Models
- Testing Regression Specifications
- Logistic Regression
- Generalized Linear Models and Generalized Additive Models
- Classification and Regression Trees
II. Distributions and Latent Structure
- Density Estimation
- Relative Distributions and Smooth Tests of Goodness-of-Fit
- Principal Components Analysis
- Factor Models
- Nonlinear Dimensionality Reduction
- Mixture Models
- Graphical Models
III. Dependent Data
- Time Series
- Spatial and Network Data
- Simulation-Based Inference
IV. Causal Inference
- Graphical Causal Models
- Identifying Causal Effects
- Causal Inference from Experiments
- Estimating Causal Effects
- Discovering Causal StructureAppendices
- Data-Analysis Problem Sets
- Reminders from Linear Algebra
- Big O and Little o Notation
- Taylor Expansions
- Multivariate Distributions
- Algebra with Expectations and Variances
- Propagation of Error, and Standard Errors for Derived Quantities
- chi-squared and the Likelihood Ratio Test
- Proof of the Gauss-Markov Theorem
- Rudimentary Graph Theory
- Information Theory
- Hypothesis Testing
- Writing R Functions
- Random Variable Generation
- Unified treatment of information-theoretic topics (relative entropy / Kullback-Leibler divergence, entropy, mutual information and independence, hypothesis-testing interpretations) in an appendix, with references from chapters on density estimation, on EM, and on independence testing
- More detailed treatment of calibration and calibration-checking (part II)
- Missing data and imputation (part II)
- Move d-separation material from “causal models” chapter to graphical models chapter as no specifically causal content (parts II and IV)?
- Expand treatment of partial identification for causal inference, including partial identification of effects by looking at all data-compatible DAGs (part IV)
- Figure out how to cut at least 50 pages
- Make sure notation is consistent throughout: insist that vectors are always matrices, or use more geometric notation?
- Move simulation to an appendix
- Move variance/weights chapter to right before logistic regression
- Move some appendices online (i.e., after references)?
(Text last updated 30 March 2016; this page last updated 6 November 2015)
If you’re not already doing so, you should follow Barabási on Twitter.
— Laszlo Barabasi (@barabasi) August 3, 2016
Download a pre-publication version of the book which will be published by Cambridge University Press. The book arises from notes of courses taught at the second year graduate level at the University of Minnesota and is suitable to accompany study at that level.
Complex systems are usually difficult to design and control. There are several particular methods for coping with complexity, but there is no general approach to build complex systems. In this book I propose a methodology to aid engineers in the design and control of complex systems. This is based on the description of systems as self-organizing. Starting from the agent metaphor, the methodology proposes a conceptual framework and a series of steps to follow to find proper mechanisms that will promote elements to find solutions by actively interacting among themselves.
This is the draft version of a textbook, which aims to introduce the quantum information science viewpoints on condensed matter physics to graduate students in physics (or interested researchers). We keep the writing in a self-consistent way, requiring minimum background in quantum information science. Basic knowledge in undergraduate quantum physics and condensed matter physics is assumed. We start slowly from the basic ideas in quantum information theory, but wish to eventually bring the readers to the frontiers of research in condensed matter physics, including topological phases of matter, tensor networks, and symmetry-protected topological phases.
I’m sure, as always, that there are a few who are interested, but who couldn’t make the Fall lectures. Never fear, there’s a group of us that can help you get up to speed to keep pace with us during the second quarter. Just drop us a note and we’ll see what we can do.
Algebraic Number Theory: The Sequel
In no field of mathematics is there such an irresistible fascination as in the theory of numbers. This course, the second in a two-quarter sequence, is an introductory, yet rigorous, survey of algebraic number theory, which evolved historically through attempts to prove Fermat’s Last Theorem. Beginning with a quick review of the previous quarter’s work, the course continues discussions on the structure of algebraic number fields, focusing particular attention on primes, units, and roots of unity in quadratic, cubic, and cyclotomic fields. Topics to be discussed include: norms and traces; the ideal class group; Minkowski’s Translate, Convex Body, and Linear Forms theorems; and Dirichlet’s Unit Theorem.
UCLA: 5137 Math Sciences
January 5 – March 15
11 meetings total
We’ll be using Introductory Algebraic Number Theory by Saban Alaca and Kenneth S. Williams (Cambridge University Press, 2003, ISBN: 978-0521183048).
Like a kid anxiously awaiting Christmas morning, I spent some time refreshing UCLA Extension’s web page over the weekend in hopes of seeing the announcement of Mike Miller’s Fall math course with no results.
I checked again a half hour ago and their site was down!
My salivating hit a fever pitch!
Refreshing, refreshing, refreshing… and now it’s live again with:
Mike Miller is teaching Algebraic Number Theory in the Fall!
Register quickly before it fills up. And let the pool for the guesses about which textbook he’ll recommend begin!
Algebraic Number Theory
MATH X 450.8 | 3.00 units
In no field of mathematics is there such an irresistible fascination as in the theory of numbers. This course, the first in a two-quarter sequence, is an introductory, yet rigorous, survey of algebraic number theory, which evolved historically through attempts to prove Fermat’s Last Theorem. Beginning with a quick review of primality and unique factorization for ordinary integers, the course extends these notions to more exotic domains: quadratic, cubic, cyclotomic, and general number fields. This development is then applied to the representation of integers as sums of squares and, more generally, to classic Diophantine equations. Topics to be discussed include: Euclidean, principal ideal, and Noetherian domains; integral bases; binary quadratic forms; algebraic field extensions; and several remarkable theorems/conjectures of Ramanujan.
UCLA: 5137 Math Sciences
September 22 – December 8
11 meetings total
(no mtg 11/17)
See you all in just a few weeks!
A Note For the Reticent
Exercise Your Brain
As many may know or have already heard, Dr. Mike Miller, a retired mathematician from RAND and long-time math professor at UCLA, has been offering incredibly inexpensive upper level undergraduate and graduate level math courses for 30+ years through UCLA Extension.
Whether you’re a professional mathematician, engineer, physicist, physician, or simply a hobbyist interested in mathematics you’ll be sure to get something interesting out of this course, not to mention the camaraderie of 20-30 other “regulars” with widely varying backgrounds (actors to surgeons and evolutionary theorists to engineers) who’ve been taking almost everything Mike has offered over the years. Once most new students have taken one class, they’re incredibly prone to want to take them all (and yes, he’s THAT good — we’re sure you’ll be addicted too.)
Even if it’s been years since you last took calculus or linear algebra, Mike (and usually the rest of the class) will help you get quickly back up to speed to delve into what is often a very deep subject. Though there are a handful who will want to learn the subject for specific applications, naturally, it’s simply a beautiful and elegant subject for those who just want to meander their way through higher mathematics for the fun of it (this will probably comprise the largest majority of the class by the way.)
Whether you’ve been away from serious math for decades or use it every day or even if you’ve never gone past calculus, this is bound to be the most entertaining thing you can do with your Tuesday nights in the fall. If you’re not sure what you’re getting into (or are scared a bit by the course description), I highly encourage to come and join us for at least the first class before you pass up on the opportunity – there’s no need to preregister or prepay if you’re unsure. I’ll mention that the greater majority of new students to Mike’s classes join the ever-growing group of regulars who take almost everything he teaches subsequently.
For the reticent, I’ll mention that one of the first courses I took from Mike was Algebraic Topology which generally requires a few semesters of Abstract Algebra and a semester of Topology as prerequisites. I’d taken neither of these prerequisites, but due to Mike’s excellent lecture style and desire to make everything comprehensible to the broadest number of students, I was able to do exceedingly well in the course. Also keep in mind that you can register to take the class for a grade, pass/fail, or even no grade at all to suit your needs/lifestyle.
Textbook: Introductory Algebraic Number Theory
Update (8/19/15) Per my email conversation with Dr. Miller, despite that neither the Extension website nor the bookstore have a book listed for the class yet, he’s going to be recommending Introductory Algebraic Number Theory by Saban Alaca and Kenneth S. Williams (Cambridge University Press, 2003, ISBN: 978-0521183048).
Introductory / General Readership / Popular Science Books
These books are written on a generally non-technical level and give a broad overview of their topics with occasional forays into interesting or intriguing subtopics. They include little, if any, mathematical equations or conceptualization. Typically, any high school student should be able to read, follow, and understand the broad concepts behind these books. Though often non-technical, these texts can give some useful insight into the topics at hand, even for the most advanced researchers.
Possibly one of the best places to start, this text gives a great overview of most of the major areas of study related to these fields.
One of the best books on the concept of entropy out there. It can be read even by middle school students with no exposure to algebra and does a fantastic job of laying out the conceptualization of how entropy underlies large areas of the broader subject. Even those with Ph.D.’s in statistical thermodynamics can gain something useful from this lovely volume.
A relatively recent popular science volume covering various conceptualizations of what information is and how it’s been dealt with in science and engineering. Though it has its flaws, its certainly a good introduction to the beginner, particularly with regard to history.
One of the most influential pieces of writing known to man, this classical text is the basis from which major strides in biology have been made as a result. A must read for everyone on the planet.
The four books above have a significant amount of overlap. Though one could read all of them, I recommend that those pressed for time choose Ben-Naim first. As I write this I’ll note that Ben-Naim’s book is scheduled for release on May 30, 2015, but he’s been kind enough to allow me to read an advance copy while it was in process; it gets my highest recommendation in its class. Loewenstein covers a bit more than Avery who also has a more basic presentation. Most who continue with the subject will later come across Yockey’s Information Theory and Molecular Biology which is similar to his text here but written at a slightly higher level of sophistication. Those who finish at this level of sophistication might want to try Yockey third instead.
In the coming weeks/months, I’ll try to continue putting recommended books on the remainder of the rest of the spectrum, the balance of which follows in outline form below. As always, I welcome suggestions and recommendations based on others’ experiences as well. If you’d like to suggest additional resources in any of the sections below, please do so via our suggestion box. For those interested in additional resources, please take a look at the ITBio Resources page which includes information about related research groups; references and journal articles; academic, research institutes, societies, groups, and organizations; and conferences, workshops, and symposia.
Lower Level Undergraduate
These books are written at a level that can be grasped and understood by most with a freshmen or sophomore university level. Coursework in math, science, and engineering will usually presume knowledge of calculus, basic probability theory, introductory physics, chemistry, and basic biology.
Upper Level Undergraduate
These books are written at a level that can be grasped and understood by those at a junior or senor university level. Coursework in math, science, and engineering may presume knowledge of probability theory, differential equations, linear algebra, complex analysis, abstract algebra, signal processing, organic chemistry, molecular biology, evolutionary theory, thermodynamics, advanced physics, and basic information theory.
These books are written at a level that can be grasped and understood by most working at the level of a master’s level at most universities. Coursework presumes all the previously mentioned classes, though may require a higher level of sub-specialization in one or more areas of mathematics, physics, biology, or engineering practice. Because of the depth and breadth of disciplines covered here, many may feel the need to delve into areas outside of their particular specialization.
Dearth of (Great) Textbooks on The Entertainment Business
In having previously taught several classes on the business of the entertainment industry, I was never quite able to pick out even a mediocre textbook for such a class. There are a handful that will give one an overview of the nuts and bolts and one or two that will provide some generally useful numbers (see the syllabi from those classes), but none comes close to providing the philosophy of how the business works in a short period of time.
A Short Term Solution
To remedy this problem, I was always a fan of producer and ex-agent Gavin Polone, who had a series of articles in New York Magazine/Vulture. I’ve recently gone through and linked to all of the forty-four articles, in chronological order, he produced in that series from 9/21/11 to 5/7/14.
I’ve aggregated the series via Readlists.com, so one can click on each of the articles individually. Better yet, for students and teachers alike, one can click on the “export” link and very easily download them all in most ebook formats (including Kindle, iPad, etc.) for your reading/studying convenience.
My hope is that for others, they may create an excellent starter textbook on how the entertainment business works and, more importantly: how successful people in the business think. For those who need more, Gavin is also an occasional contributor to the Hollywood Reporter. (And, as a note for those not trained in the classics and prone to modern-day stereotypes, I’ll make the caveat that I use the title “Machiavelli” above with the utmost reverence and honor.)
I’m still slowly, but surely making progress on my own all-encompassing textbook, but, until then, I hope others find this series of articles as interesting and useful as I have.
Gavin Polone is an agent turned manager turned producer. His production company, Pariah, has brought you such movies and TV shows as Panic Room, Zombieland, Gilmore Girls, and Curb Your Enthusiasm. Follow him on Twitter @gavinpolone.
I’ll also give him a shout out for being a mathematician with a fledgling blog: Rick’s Ramblings.