n-Category Café

The link references at the bottom have some more detail on the structure. For example, the N-Category Cafe talks about how the dragon curve shows up at the edges [0]. There are other gross level symmetries due to the symmetries of the Littlewood like polynomials involved. For example, if $p(x)$ is a Littlewood-like polynomial, then so is $p(1/x)$ etc. I think the "holes" that show up on the unit disc because of the factors of the degrees of certain polynomials. If I'm not mistaken, the holes follow a Farey sequence [1]. I wrote a little blurb about it but it's still incomplete and haphazard. One thing I'm still curious about is how quickly the holes diminish in size. [0] https://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathe… [1] https://en.wikipedia.org/wiki/Farey_sequence [2] https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.html

One of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_pro… (Category Theory, of all things, being the best tool for formalizing dualities and all sorts of analogies between seemingly different objects of study.)

One of my favorite magic tricks is one I've learnt. You have 31 cards prepared in an order determined by a linear shift register. The audience is allowed to cut the deck as often as they want (thus maintaining cyclic order) after which they draw 5 cards. Simply by asking which audience members hold red cards you can compute which cards they're holding. I've also performed this trick by combining two decks with different patterns on the back, allowing you to 'divine' their cards without ever asking a question. See here for details: https://golem.ph.utexas.edu/category/2015/01/mathematics_and_magic_the…

I feel like the author should just write call Set FiniteSet, and then a lot of confusion would go away! https://golem.ph.utexas.edu/category/2021/06/large_sets_1.html also, I highly recommend this series for anyone interested in sets outside of set theory.

This is a pleasant but intentionally very elementary summary of the relevant concepts. If you're interested in much more on this, one place to start is MO (https://mathoverflow.net/questions/4841/what-precisely-is-categorifica… or the n-Category Café (https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.… or Baez's TWF 121 (http://math.ucr.edu/home/baez/week121.html). Warning: all are much math-ier than Riehl's very nice slides, although the first one has enough content that you may find an answer that you like. If your math itch still isn't scratched, there's Baez and Dolan's paper "Categorification" (https://arxiv.org/abs/math/9802029). I don't know a programmer-focussed view on categorification; can anyone recommend one?

Probability and probabilistic reasoning can be described by this framework. The theory is not yet as well developed as the classical theories. A discussion is here: https://golem.ph.utexas.edu/category/2007/02/category_theoretic_probab…

The linked course pages contains a good synopsis, but Robert Harper's own blog post "What’s the big deal with HoTT?" [0] gives a wider view about its motivation and history. Another two pages with worthwhile information on the HoTT book are [1] and [2]. [0] http://existentialtype.wordpress.com/2013/06/22/whats-the-big-deal-wit… [1] http://golem.ph.utexas.edu/category/2013/06/the_hott_book.html [2] http://math.andrej.com/2013/06/20/the-hott-book/

There aren't big population-level differences, aside from basic cosmetic differences due to genetic drift [0]. Humans act as one single species, with one single pedigree [2], lacking any selfish genes which speciate us further. Genes within a single species' gene pool follow a Poisson distribution because they are randomly permuted [1][3][4], assuming the pool is large enough. This means that often, some genes will be popular for no good reason; that is, these popular genes are cosmetic. When, for example, most people in the world follow one of several popular skin-coloration patterns, that does not therefore imply any sort of fitness function is selecting for skin colors, because the null hypothesis that they are randomly permuted has so little problem explaining how they could arise without any effect on fitness. These questions aren't dangerous because they somehow subvert existing institutions, but because they're a direct introduction to ethnofascism via eugenics. We should roundly not just reject them based on empirical observations and data, but also based on theory and modelling; this is because we can then use contraposition to flip around various deductions. For example, people often ask whether "race" affects "intelligence" and point to survey data which shows correlations; using contraposition, and understanding that "race" is a social construction, we can refute the idea that "intelligence" is innate and suggest instead that it is tied to socioeconomic status…

So, here's the thing I'd like somebody to explain to me. Telegram encryption spec is published, and the client is open source. This means you can verify that the server is following the spec by creating a clean room client implementation. You could even create a green field server implementation if you wanted. Telegram regularly has contests to break its encryption with a reward of 300K USD https://telegram.org/blog/cryptocontest If it's not secure, then surely people would be cashing in on that sweet money. So, why is it that we constantly see articles talking about how insecure Telegram encryption is, but nobody is showing a proof of concept attack or collecting the prize? Unless somebody puts their money where their mouth is and shows an actual exploit with code, it seems like pure FUD to me. On top of that, it appears that attacks on Telegram often come from people associated with Signal in some way. Signal is endorsed by NSA who have a history of promoting weak encryption that they have found backdoors into. I hope everybody still remembers this debacle https://golem.ph.utexas.edu/category/2014/10/new_evidence_of_nsa_weake…

The progress in n-category theory is largely about that. Instead of just studying objects, operations and equations, like "a group is a set g, together with multiplication + certain axioms" one studies objects, relations between objects, relations between relations etc. To study this up to level two, you visualize objects as regions in 2d space, relations between objects as lines between them and relations between relations as points where lines meet. Equations are then equalities between such pictures. If you look at those equalities as a dynamic process, which starts at time t=0 with one picture and time t=1 with the other, you get a picture in three dimensional space, which you can visualize as a movie (to be mathematically precise after the choice of a morse function you get a movie). One of the easiest examples would be the zig-zag identity. See for example https://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_comp… and scroll down. One person who has completely gone overboard with that is Jacob Lurie, he has practically single handedly written down a foundational work of higher algebra and algebraic topology, which essentially precisely corresponds to what you are asking. How does mathematics look like if we weren't constrained to 2d paper to write down equations. Of course to actually write down those things you have to somehow project the equations to several essentially 1d sections or invoke very abstract methods to reason about them. Of…

Huh? Nature Publishing Group, where Rowan Manning works, makes its profit margins from extorting scientific libraries rather than superior code. http://golem.ph.utexas.edu/category/2010/06/boycotting_the_nature_publ… Boycotting the Nature Publishing Group The Nature Publishing Group publishes not only the prestigious journal Nature, but also many others. When this company bought Scientific American, it raised the institutional subscription price seven-fold. Now they are insisting on quadrupling the fees for 67 journals to which the University of California subscribes. Right now, we pay them an average of $4,465 per year for each journal we subscribe to. After the increase, this would soar to $17,479 per year. In response, the University of California is considering a system-wide boycott of the Nature Publishing Group — for example, cancelling subscriptions to all their journals. Rowan Manning works for a gang of monopolistic rent-seekers who currently have the scientific community over a barrel. His salary comes from a closed-access model which is soon to be obsolete. Thus "cutting-edge" technology is actually highlighted as a fail! https://github.com/rowanmanning/joblint/blob/master/lib/rule/expectati…

I have already explained in the post above why optimization is very a much a part of machine learning. Now, you say that that classification and optimization are two different things and that is true. But really, it can be more fruitful to look at supervised learning as a special case of unsupervised learning [1]. It is best to seek the most general framework from which to understand things as it leads to a deeper understanding, broader applicability of concepts and easier cross fertilization across fields. For example, understanding the spectral theorem makes SVD (hence PCA) and the DFT class of algorithms much clearer. Understand the notion of Lp-Norms, convexity, adjoints, loss functions and regularization and a whole bunch of seemingly different algorithms collapse into facets of the same thing. Hook it up to automatic differentiation then some optimization algorithms and you can write anything from Neural networks, SVMs, regularized logistic regression to Non negative tensor factorization in a few lines. You stop making arbitrary divisions between classification or optimization. Much the same kind of collapse can be done for the dual [2] notion of probabilistic algorithms by thinking in terms of graphs, simplices, parametrizations, families and conjugacy. The best thing from all this is you stop thinking of which algorithm should I use and start thinking of what do I want to do? What is the best mathematical model for this? What would really be great would be a…

Article was from 2008, Wired did a follow up story a year later... http://www.wired.com/dangerroom/2009/10/impossible-drive-designers-dre… ---------------- Last year, professor Yang Juan of the College of Astronautics at Northwestern Polytechnical University (NPU) in Xi’an was happy to confirm that they were building an Emdrive which would be tested by the end of the year. But following the publication of this news in Danger Room, the situation changed. I was informed that the publicity was very unwelcome, especially any suggestion that there might be a military application. (Yang had previous published a study on the use of plasma as a weapon against low-orbiting satellites. [.pdf]) No further information has been forthcoming, and no Chinese papers have been published on the Emdrive, though Yang has recently published work on (unrelated) microwave plasma thrusters (.pdf). ---------------- Nextbigfuture did an article stating the tech was sold... http://nextbigfuture.com/2011/03/controversial-emdrive-claims-to-have.… A few responses from physicists... http://scienceblogs.com/sunclipse/2008/09/the_emdrive_story_or_how_to_… http://golem.ph.utexas.edu/category/2006/10/new_scientist_reacts.html

Personally I’ve found this article a much more approachable overview to the question in the title: https://golem.ph.utexas.edu/category/2019/09/the_riemann_hypothesis_pa…

Yeah, see this thread --- I assume these guys haven't heard of the other meaning neither https://golem.ph.utexas.edu/category/2013/06/torsors_and_enriched_cate… Consider in particular that use of ‘distance’ >I think you can look at adjoint profunctors from the unit category and show that they consist of giving a consistent ‘distance’ to every object, which in a torsor will be represented.

Related sentiment to the above post: https://golem.ph.utexas.edu/category/2009/09/the_mathematical_vocation… .

It's nice that pure mathematics is funded, but it should not be funded by the army. https://golem.ph.utexas.edu/category/2020/12/the_lie_of_its_just_math.… My pet theory: the MIC buys goodwill and scares academics to criticize the MIC.

This looks like a very good book on Category Theory with a lot of illustrations. Anyone can compare between this book and The Joy of Abstraction by Eugenia Cheng for learning Category Theory beginner? https://golem.ph.utexas.edu/category/2023/02/the_joy_of_abstraction.ht…

It can give semantics for it, such as how cartesian closed category gives semantics to simply typed lambda calculus https://golem.ph.utexas.edu/category/2006/08/cartesian_closed_categori…

Also John Baez's posts https://golem.ph.utexas.edu/category/2022/12/dividing_a_square_into_si… https://golem.ph.utexas.edu/category/2022/12/guillotine_partitions_and…

Either in person, on Math Overflow (https://mathoverflow.net/), or specialist sites like (maybe) the n-category Cafe (https://golem.ph.utexas.edu/category/).

Nah, here's a recentish overview: https://golem.ph.utexas.edu/category/2020/01/profunctor_optics_the_cat…

> The reason for that is that it is not nice enough a logic, full stop. So why would it be nice enough for computer scientists? Type theory has many attractive properties over traditional foundations like set theory. See, for example: https://golem.ph.utexas.edu/category/2013/01/from_set_theory_to_type_t…

It's not very precise talking about whether a monad commutes -- it's not clear what "commutes" should really mean (certainly not F . G = G . F since that's way too restrictive). I started working it out by hand, but then figured nLab had it somewhere, and indeed: https://golem.ph.utexas.edu/category/2017/02/distributive_laws.html F, G, and F.G are monads if and only if there is a "distributive law," which is a natural transformation G.F -> F.G satisfying some properties. It's like something that satisfies half the braiding properties, where braidings are already a weaker version of commutativity.

The book in the course [1], specially ch. 4, I think, provides some analytical category-theoretical tools to model feasibility/performance in that kind of situation. In [2] the same technique (enriched profunctors) applies for critical path analysis in project management PERT diagrams. Looks to me (having just skimmed the Uber post) a very similar dependency structure. [1] https://math.mit.edu/~dspivak/teaching/sp18/ [2] https://golem.ph.utexas.edu/category/2013/03/project_planning_parallel…

If it were abstract algebra, the title would be ‘Basic Algebra’. https://golem.ph.utexas.edu/category/2013/12/the_long_grind_of_writing…

Note that the actual message appears to be in the comments below the article: https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arit…

There are financial incentives to solve scheduling issues in the UK. Trains must meet a schedule within time limits. Failure to maintain reliability at a regulatory 90.9% can mean the loss of the license to operate. The UK interconnects with European networks via Eurostar, and the major hubs feed it. The tube runs trains every minute or two so connecting between these hubs, for example Waterloo to Kings Cross isn’t usually an issue. But arriving late at a hub can be very disruptive. Middle managers, in europe generally, focused on targets don’t need to understand the math, but are in a competitive market that can require mathematicians to model and solve scheduling issues. https://golem.ph.utexas.edu/category/2018/05/tropical_mathematics_and_… https://www.independent.co.uk/news/uk/home-news/late-network-rail-brea…

There are multiple representations of pi as stated by the other commentor. One way of deriving it is given in the link, very elegant using the simple series formula. (Euler came up with it) [1]: https://golem.ph.utexas.edu/category/2020/08/eulers_continued_fraction…

https://golem.ph.utexas.edu/category/2018/02/mlab.html it might as well be from a random text gwnerator