AMS Open Math Notes is a repository of freely downloadable mathematical works in progress hosted by the American Mathematical Society as a service to researchers, teachers and students.
These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision.
Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.
Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube.
Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube. No prior experience in group theory is necessary. He’s the author of the recent Fundamentals of Group Theory: An Advanced Approach. [1]
He hopes to eventually also offer lectures on ring theory, fields, vector spaces, and module theory in the near future.
Fundamentals of Group Theory by Steven Roman
References
[1]
S. Roman, Fundamentals of Group Theory: An Advanced Approach, 2012th ed. Birkhäuser, 2011.
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:
https://arxiv.org/abs/1612.09375
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:
…………………………………………………………………..
Dear all,
My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:
https://arxiv.org/abs/1612.09375
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.
References
[1]
T. Leinster, Basic Category Theory, 1st ed. Cambridge University Press, 2014.
Our friend Andrew Eckford has spent some time over the holiday improving his Twitter bot Primes as a Service. He launched it in late Spring of 2016, but has added some new functionality over the holidays. It can be relatively handy if you need a quick answer during a class, taking an exam(?!), to settle a bet at a mathematics tea, while livetweeting a conference, or are hacking into your favorite cryptosystems.
General Instructions
Tweet a positive 9-digit (or smaller) integer at @PrimesAsAService. It will reply via Twitter to tell you if the number prime or not.
Some of the usable commands one can tweet to the bot for answers follow. (Hint: Click on the buttons with the tweet text to auto-generate the relevant Tweet.)
To factor a number into prime factors, tweet: @primesasservice # factor
and replace the # with your desired number
To get the greatest common factor of two numbers, tweet: @primesasservice #1 #2 gcf
and replace #1 and #2 with your desired numbers
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.
The topic for Mike Miller’s UCLA Winter math course isn’t as much a surprise as is often the case. During the summer he had announced he would be doing a two quarter sequence on complex analysis, so this Winter, we’ll be continuing on with our complex analysis studies.
I do know, however, that there were a few who couldn’t make part of the Fall course, but who had some foundation in the subject and wanted to join us for the more advanced portion in the second half. Toward that end, below are the details for the course:
Introduction to Complex Analysis: Part II | MATH X 451.41 – 350370
Course Description
Complex analysis is one of the most beautiful and practical disciplines of mathematics, with applications in engineering, physics, and astronomy, to say nothing of other branches of mathematics. This course, the second in a two-part sequence, builds on last quarter’s development of the differentiation and integration of complex functions to extend the principles to more sophisticated and elegant applications of the theory. Topics to be discussed include conformal mappings, Laurent series and meromorphic functions, Riemann surfaces, Riemann Mapping Theorem, analytical continuation, and Picard’s Theorem. The course should appeal to those whose work involves the application of mathematics to engineering problems, and to those interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.
Winter 2017 Days: Tuesdays Time: 7:00PM to 10:00PM Dates: Jan 10, 2017 to Mar 28, 2017 Contact Hours: 33.00 Location: UCLA, Math Sciences Building Course Fee(s): $453.00 Available for Credit: 3 units Instructors: Michael Miller
No refund after January 24, 2017. Class will not meet on one Tuesday to be announced.
For many who will register, this certainly won’t be their first course with Dr. Miller–yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension
If you missed the first quarter and are interested in the second quarter but want a bit of review or some of the notes, let me know in the comments below.
I look forward to seeing everyone in the Winter quarter!
Michael Miller making a “handwaving argument” during a lecture on Algebraic Number Theory at UCLA on November 15, 2015. I’ve taken over a dozen courses from Mike in areas including Group Theory, Field Theory, Galois Theory, Group Representations, Algebraic Number Theory, Complex Analysis, Measure Theory, Functional Analysis, Calculus on Manifolds, Differential Geometry, Lie Groups and Lie Algebras, Set Theory, Differential Geometry, Algebraic Topology, Number Theory, Integer Partitions, and p-Adic Analysis.
Peter Woit has just made the final draft (dated 10/25/16) of his new textbook Quantum Theory, Groups and Representations: An Introduction freely available for download from his website. It covers quantum theory with a heavy emphasis on groups and representation theory and “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.
I finally have finished a draft version of the book that I’ve been working on for the past four years or so. This version will remain freely available on my website here. The plan is to get professional illustrations done and have the book published by Springer, presumably appearing in print sometime next year. By now it’s too late for any significant changes, but comments, especially corrections and typos, are welcome.
At this point I’m very happy with how the book has turned out, since I think it provides a valuable point of view on the relation between quantum mechanics and mathematics, and contains significant amounts of material not well-explained elsewhere.
📖 On page 24 of 274 of Complex Analysis with Applications by Richard A. Silverman
I enjoyed his treatment of inversion, but it seems like there’s a better way of laying the idea out, particularly for applications. Straightforward coverage of nested intervals and rectangles, limit points, convergent sequences, Cauchy convergence criterion. Given the level, I would have preferred some additional review of basic analysis and topology; he seems to do the bare minimum here.
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.
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
Optimization
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
Planned changes:
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)
Weaverbot
