# Tag Info

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There have been recent developments in dependent type theory which relate type systems to homotopy types. This is now a relatively small field, but there is a lot of exciting work being done right now, and potentially a lot of low hanging fruit, most notably in porting results from algebraic topology and homological algebra and formalizing the notion of ...

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Algebraic geometry is used heavily in algebraic complexity theory and in particular in geometric complexity theory. Representation theory is also crucial for the latter, but it's even more useful when combined with algebraic geometry and homological algebra.

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Algebraic geometry Noether's Normalization Lemma (NNL) for explicit varieties is currently only known to be in $\mathsf{EXPSPACE}$ (like general NNL), but is conjectured to be in $\mathsf{P}$ (and is in $\mathsf{P}$ assuming that PIT can be black-box derandomized). Update 4/18/18: It was recently shown that for the variety $\overline{\mathsf{VP}}$ it is in $... 15 Your knowledge of field theory would be useful in cryptography, while category theory is heavily used in the research on programming languages and typing systems, both of which are closely related to the foundations of mathematics. 12 [Computational complexity and category theory] seem like such natural pairs. Given the prominence of computational complexity as a research field, if they were such natural bedfellows, maybe somebody would have brought out the connection already? Wild speculation. Let me entertain the reader with thoughts about why a categorical rendering of ... 12 There is a more general question on mathoverflow, and I asked a similar question on CS.SE. jbapple provided a good answer. To quote: "Necklaces, Convolutions, and X+Y", By Bremner et al. shows a$O\left(\frac{n^2(\lg \lg n)^3}{\lg^2 n}\right)$algorithm for this problem on the real RAM and a$O(n \sqrt{n})$algorithm in the nonuniform linear ... 12 The order of permutation groups can be computed in polynomial-time. In fact, I believe even in$\mathsf{NC}$and also nearly linear Las Vegas time. See, e.g., the book by Seress. For reference, subgroups of$S_n$(and algorithms related thereto) are typically called "permutation groups" rather than merely "subgroups (of$S_n$)". So you can google "... 12 For NP, this seems hard to construct. In particular, if you can also sample (nearly) uniform elements from your group - which is true for many natural ways of constructing groups - then if an NP-complete language has a poly-time group action with few orbits, PH collapses. For, with this additional assumption about sampleability, the standard$\mathsf{coAM}$... 11 Field theory and algrebraic geometry would be useful in topics related to error correcting codes, both in the classical setting as well as in studying locally decodable codes and list decoding. I believe this goes back to work on the Reed-Solomon and Reed-Muller codes, which was then generalized to algebraic geometric codes. See for example, this book ... 11 It seems there is a paper answering this exact question, and even in the more general case of$\omega$-regular languages, but I cannot find an open-access version. If somebody finds a link without paywall it would be great. I requested the full-text on ResearchGate. Title: Which Finite Monoids are Syntactic Monoids of Rational omega-Languages. Authors: ... 11 In a more elementary way than Denis's answer, the following is extracted from Pippenger's "Theories of Computability", p.87, and immediate to check. Definition: Let$M$be a monoid, and$Y \subseteq M$. Define the congruence relation$\equiv_Y$over$M$by$x \equiv_Y y$iff$\big[\forall w, z \in M$,$wxz \in Y \Leftrightarrow wyz \in Y\big]$. Definition:... 11 The terminology rigid seems to be relatively new compared to the term disjunctive used in the late 70's (and probably before, I didn't check for earlier references). A subset$P$of a monoid$M$is disjunctive if and only if the syntactic congruence of$P$in$M$is the equality relation. Thus a monoid is the syntactic monoid of a language if and only if it ... 11 From a programming language theory perspective, as opposed to the computability perspective other answers and comments have offered, C++ templates combined with concepts correspond to bounded polymorphism or constrained genericity. Concepts themselves correspond to the constraints or bounds placed on a type. A template is type-level function, parameterised ... 10 As a complement to Joshua Grochow's answer: Computing the order of a permutation group given generators is in P by Schreier–Sims algorithm, see also p. 8-9 of these lectures notes by Luks. Just as membership in permutation groups, the problem was believed to be P-complete by many researchers, but it was finally shown to be in NC by Babai, Luks & Seress. ... 10 To my knowledge (which is definitely incomplete), there has been relatively little work on this, presumably because it requires assimilating two relatively intricate bodies of knowledge. However, little does not mean nonexistent. Thierry Coquand and his collaborators have written quite a few papers on the connections between commutative algebra and ... 9 There is a proof in the book of Kuich & Salomaa, Semirings, Automata, Languages and another one in the paper of Panholzer, "Gröbner Bases and the Defining Polynomial of a Context-free Grammar Generating Function", J. of Automata, Languages and Combinatorics 10 (2005), 79–97. I wish there were a simple and clear proof of the result. 8 The reason that sub exponential algorithms for the discrete log in$\mathbb{Z}/p\mathbb{Z}$do not carry over to groups based on elliptic curves is because$\mathbb{Z}/p\mathbb{Z}$has extra structure that doesn't seem to be present in elliptic curve groups. In particular,$\mathbb{Z}/p\mathbb{Z}$has both addition and multiplication, whereas the elliptic ... 8 Let$\mathbb K$be a field of characteristic$0$or at least$d(d-1)+1$, and$p\in\mathbb K[x_1,\dotsc,x_n]$be a polynomial of total degree at most$d$. If$d$is fixed and$n$is growing, one has the following complexity bounds for the reduction of the factorization of$p$to the factorization of a degree-$d$univariate polynomial: (The notation$\tilde{\...

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This question is solved for deterministic automata and for unambiguous automata in the book [1] [1] J. Berstel, D. Perrin, C, Reutenauer, Codes and automata, Vol. 129 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2009. In the case of deterministic automata, the characterization is given in Proposition 3.2.5. Recall that a ...

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Algebra (and algebraic geometry) has had a pretty big role to play in cryptography, with elliptic curve groups, (number-theoretic) lattices, and of course $\mathbb{Z}_p$ being the basis for nearly all modern cryptographic work.

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For the question of when does $p=0 \Rightarrow q=0$, randomness can indeed help, as follows. First, factor $p$ (uses randomness when $p$ is given as a straight-line program; doesn't need randomness if $p$ is given as a coefficient vector). Let $\tilde{p}$ denote the square-free version of $p$ - that is, if $p=p_1^{k_1} p_2^{k_2} \dotsb p_\ell^{k_\ell}$ then $... 7 Short answer The general answer is that computability is easy for locally compact spaces, doable for separable spaces, and hard for non-separable spaces. The short anwer for a particular example, namely$\ell^2$is: in order to get computability on$\ell^2$working, represent an element$x$of$\ell^2$by a program which accepts$n$and gives (the list of ... 7 There are some problems in computational learning theory, machine learning and computer vision that can be solved using commutative algebra and algebraic geometry. For instance, convergence of the Belief Propagation algorithm, a message passing algorithm for Bayesian inference, can be formulated in terms of characterizing the affine variety of system of ... 7 Such algebras are called functionally complete. Also, what you call terms are actually called polynomials. In standard terminology, term operations have a more restricted definition that allows variables and the basic operations$f_i$, but not constants from$E$. Algebras that satisfy the stronger condition that every operation is represented by this kind of ... 7 Yes, the initial algebra is by definition one of the members of the class for which it is initial. You may however be interested in the category-theoretic concept of a limit. Given a diagram (a collection of objects and morphisms between them), we can ask for an object which "best approximates" the diagram, but needs not be an element of the diagram. There ... 7 You might be interested in bounded synchronization delay expressions. See [1] for details on these expressions. To sum up, they are equivalent to star-free expressions, but instead of using complement, they restrict the use of the Kleene star to certain languages: the prefix codes with bounded synchronization delay. This way, you can have your ... 6 Grobner bases are used for the fastest list decoding algorithms for Reed-Solomon codes: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.320.1170&rep=rep1&type=pdf 6 Have you thought about looking at computer algebra? Axiom is a computer algebra system where the type system is modelled after Category Theory (or Universal Algebra, depending on your view). There are two further derivatives of Axiom FriCAS and OpenAxiom. If you're interested in Category Theory, then the type system may be one thing to look at. In Axiom, ... 6 There is no relationship. They both use the word "free", but with different meanings of the word "free". It's just an accidental collision, which will happen when you have a language like English with a fixed number of words and the number of concepts we want to talk about exceeds the number of words in the language. A free group is, roughly, a group that ... 6 Pavlovic et al. view Turing machines over a binary alphabet as coalgebras for the functor$\lambda X. \, 2 \times \mathcal{P}_{\mathrm{fin}}(X \times 2 \times \{\lhd,\rhd\})^2$. The symbols$\lhd$and$\rhd\$ represent thereby the tape moves. Bart Jacobs has presented in "Coalgebraic walks, in quantum and Turing computation" an approach by using a monad. He ...

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