A turning point in my teaching was realizing that I cannot make sense of ideas for students.
📖 Read pages 21-24 of Abstract and Concrete Categories: The Joy of Cats by Jirí Adámek, Horst Herrlich, George E. Strecker
In mathematics, a holyhedron is a type of 3-dimensional geometric body: a polyhedron each of whose faces contains at least one polygon-shaped hole, and whose holes' boundaries share no point with each other or the face's boundary.
The concept was first introduced by John H. Conway; the term "holyhedron" was coined by David W. Wilson in 1997 as a pun involving polyhedra and holes. Conway also offered a prize of 10,000 USD, divided by the number of faces, for finding an example, asking:
Is there a polyhedron in Euclidean three-dimensional space that has only finitely many plane faces, each of which is a closed connected subset of the appropriate plane whose relative interior in that plane is multiply connected?
No actual holyhedron was constructed until 1999, when Jade P. Vinson presented an example of a holyhedron with a total of 78,585,627 faces; another example was subsequently given by Don Hatch, who presented a holyhedron with 492 faces in 2003, worth about 20.33 USD prize money.
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after mathematician Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.
In this episode, Haley interviews Stephen Wolfram at the Ninth International Conference on Complex Systems. Wolfram is the creator of Mathematica, Wolfram|Alpha and the Wolfram Language; the author of A New Kind of Science; and the founder and CEO of Wolfram Research. Wolfram talks with Haley about his professional journey and reflects on almost four decades of history, from his first introduction to the field of complexity science to the 30 year anniversary of Mathematica. He shares his hopes for the evolution of complexity science as a foundational field of study. He also gives advice for complexity researchers, recommending they focus on asking simple, foundational questions.
This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases.
A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.
Mike Miller’s at UCLA beginning in January 2019.
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (sequence A005150 in the OEIS).
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11.
11 is read off as "two 1s" or 21.
21 is read off as "one 2, then one 1" or 1211.
1211 is read off as "one 1, one 2, then two 1s" or 111221.
111221 is read off as "three 1s, two 2s, then one 1" or 312211. The look-and-say sequence was introduced and analyzed by John Conway.
Wikipedia enforces its entries to adopt an NPOV – a neutral point of view . This is appropriate for an encyclopedia.
However, the nLab is not Wikipedia, nor is it an encyclopedia, although it does aspire to provide a useful reference in many areas (among its other purposes). In particular, the nLab has a particular point of view, which we may call the nPOV or the n- categorical point of view .
To some extent the nPOV is just the observation that category theory and higher category theory, hence in particular of homotopy theory, have a plethora of useful applications.
This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases. A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.
Mike Miller has announced in class that he’ll be using Abstract and Concrete Categories: The Joy of Cats as the textbook for hisat UCLA Extension this winter.
Naturally, he’ll be supplementing it heavily with his own notes.
A free .pdf copy of the text is also available online.
Here are the notes from a basic course on category theory. Unlike the Fall 2015 seminar, this tries to be a systematic introduction to the subject. A good followup to this course is my Fall 2018 course. If you discover any errors in the notes please email me, and I'll add them to the list of errors. You can get all 10 weeks of notes in a single file here: here. Their typeset version was based on these handwritten versions:
I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced. The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.
This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra.
Category theory, since its development in the 1940s, has assumed an increasingly center-stage role in formalizing mathematics and providing tools to diverse scientific disciplines, most notably computer science. A category is fundamentally a family of mathematical obejcts (e.g., numbers, vector spaces, groups, topological spaces) along with “mappings” (so-called morphisms) between these objects that, in some defined sense, preserve structure. Taking it one step further, one can consider morphisms (so-called functors) between categories. This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra. Topics to be discussed include: isomorphism; products and coproducts; dual categories; covariant, contravariant, and adjoint functors; abelian and additive categories; and the Yoneda Lemma. The course should appeal to devotees of mathematical reasoning, computer scientists, and those wishing to gain basic insights into a hot area of mathematics.
January 8, 2019 - March 19, 2019
Tuesday 7:00PM - 10:00PM
Instructor: Michael Miller
The new catalog is out today and Mike Miller’s Winter class in Category Theory has been officially announced.
Oddly, it wasn’t listed in the published physical catalog, but it’s available online. I hope that those interested in mathematics will register as well as those who are interested in computer science.
For millennia, Western culture (and most other cultures) declared that men and women were different sorts of humans—and, by the way, men were better. Is that claim not only wrong but straight-up backwards?
Co-hosts Celeste Headlee and John Biewen explore the current state of the nature-nurture gender debate, with help from Lisa Wade of Occidental College and Mel Konner of Emory University.
Music by Alex Weston, and by Evgueni and Sacha Galperine. Music and production help from Joe Augustine at Narrative Music.
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.[a] If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are a universal ordered field.
A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.
I wonder what happens when the reverse process is run on numbers like pi? This could be an interesting thing to take a look at in my current math class.