The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. It is a generalization of problems including factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes it especially important in the theory of quantum computing because Shor’s algorithms for factoring and finding discrete logarithms in quantum computing are instances of the hidden subgroup problem for finite abelian groups, while the other problems correspond to finite groups that are not abelian.

Generalization of many hard problems in theoretical CS.

What I term the “monad tutorial fallacy,” then, consists in failing to recognize the critical role that struggling through fundamental details plays in the building of intuition. This, I suspect, is also one of the things that separates good teachers from poor ones. If you ever find yourself frustrated and astounded that someone else does not grasp a concept as easily and intuitively as you do, even after you clearly explain your intuition to them (“look, it’s really quite simple,” you say…) then you are suffering from the monad tutorial fallacy.

We establish new, yet intimate relationships between the 2-adic integers /sub 2/Z from arithmetics and digital circuits, both finite and infinite, from electronics.

VERY interesting paper! Sadly didn’t answer all my questions about 2-adic circuits, but it did answer quite a lot.

see: https://www.di.ens.fr/~jv/HomePage/pdf/cirnum.pdf

see: https://scispace.com/pdf/on-circuits-and-numbers-3du2olohli.pdf

via: https://en.wikipedia.org/wiki/Two%27s_complement#Two’s_complement_and_2-adic_numbers

Very nice set of slides on a phenomenon I’ve been trying to put a name to. The discussion of discrete dynamical systems over the integers can be found in “arithmetic dynamics.”

see: https://en.wikipedia.org/wiki/Arithmetic_dynamics

see: https://en.wikipedia.org/wiki/Height_function

The p-adic numbers are bizarre alternative number systems that are extremely useful in number theory. They arise by changing our notion of what it means for a number to be large. As a real number, 1 billion is huge. But as a 10-adic number, it is tiny!

Fascinating.

This post reviews CNNs from a representation theoretic perspective. It shows how the weight sharing of CNN layers is derived from translation equivariance.

Quite the powerful correspondence.

This blog post gives an introduction to equivariant neural networks. It explains what they are, why they are relevant and how they are constructed.

The road to fixing AI. Also a really good explanation of symmetry.

I’ve always wanted to understand Sporadic Groups better, so I designed some physical twisty puzzles (like the Rubik’s cube) to explore them and get a feel for how they work. I couldn’t be happier with the results.

So dope. Man, I really need to learn GAP.

mathematical knowledge is defined by its computation-granting abilities. Knowledge exists if and only if you can compute answers to questions exhaustively in the domain of that knowledge.

It could be said that no human yet has a “full understanding” of integration, or that Liouville’s theorem shows the impossibility of solving this problem completely. And so to add to “can you compute” you must allow a response, “impossible given the constraints.”

Once a field that doesn’t have strong computational roots becomes computational (see, topology becoming algebraic topology), you get mathematicians calling algebra the “devil” and complaining that a subject is no longer beautiful.

why are some algebraic structures more useful than others in software applications?

A “strong” structure admits constructive characterization theorems, and efficient algorithms to convert objects into various canonical forms.

Very nice post. Explains the immense power of linear algebra and graph theory.

What holds the protons and neutrons together in the nucleus?   Early work on this question combined Heisenberg’s idea that the proton and neutron are two states of the same particle, and Yukawa’s idea is that the nuclear force is carried by a new kind of particle.   The results were revolutionary, and laid some of the groundwork for the Standard Model.

Fascinating application of functional analysis to integral calculus. Very curious if this technique can be further generalized.

Edit: On that note, a commenter (@jimskea224) gives insight:

A simplified case of the Risch algorithm for Liouvillian functions.

see: https://en.wikipedia.org/wiki/Risch_algorithm

For our 6.5610 (Applied Cryptography) final project, we present an backdoorable DRBG based on Dual EC DRBG that, unlike Dual EC DRBG, is a true DRBG under suitable assumptions. Our algorithm, which we call the Twisted Dual EC DRBG, involves iteratively multiplying points on an elliptic curve or its quadratic twist.

The introduction contains background on how the NSA and NIST initially worked to backdoor a DRBG; LOL.

In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, and has seen many uses in the field of universal algebra since then.

!!

This summer at the Topos Institute, under the supervision of Dr. Sophie Libkind, I studied the composition of attractors. The project itself started earlier with my advisor, Dr. William Kalies, who asked me the following question: how do attractor lattices behave when we combine dynamical systems? In this post, I explain how attractor lattices in decoupled product systems can be characterized algebraically in terms of the lattices of their component systems.

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.

Very nice philosophy paper by one of the progenitors of category theory on structure. The idea to show a correspondence between Bourbaki and category theory seems like a nice grad school project.

In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.

via: https://doi.org/10.1093/philmat/4.2.174

We recall the definition of the fundamental group develop in the previous lecture then prove that it is indeed a group. Finally, we show that the fundamental group of the circle is isomorphic to Z, the integers.

We give a quick review of group theory then discuss homotopy of paths building up to the definition of the fundamental group.