linear-algebra

The Bianchi identity is a differential equation satisfied by curvature data. It can be thought of as generalizing the equation for a real-valued 1-form to higher degree and nonabelian forms. Generally it applies to the curvature of ∞-Lie algebroid valued differential forms. Let be a smooth manifold. For a differential 1-form, its curvature 2-form is the de Rham differential . The Bianchi identity…

An automorphism of an object in a category is an isomorphism . In other words, an automorphism is an endomorphism that is an isomorphism. Given an object , the automorphisms of form a group under composition, the automorphism group of , which is a submonoid of the endomorphism monoid of : which may be written if the category is understood. Up to equivalence, every group is an automorphism group; …

By algebra of similarity type $\beta=\{0,1,\neg,\wedge,\vee\}$ we mean a set where the operations in $\beta$ are defined (arbitrarily). Analogously, by algebra of similarity type ...

$$\text{Evaluate:}\quad3 + \cfrac{1}{4 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{3 + \cdots}}}}$$ A standard approach to solve this would be assuming it to be a variable ...

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) Given a vector space and an elemen…

Can someone take a look at this and explain to me like a 5 year old what is happening from node to node in this chain. I understand what it does, but I'd like to intuitively get the 'why' as well. The ...

Flux is a compiled, stack-first, general-purpose language with a refreshingly direct philosophy: you own your memory, you write your intent, and the compiler takes you seriously. If you haven't looked at it in a while - or at all - now is a great time to pay attention. Over the past development cycle, Flux has gained several major features that collectively shift it from a capable low-level langu…

symmetric monoidal (∞,1)-category of spectra A mathematical structure is essentially algebraic if its definition involves partially defined operations satisfying equational laws, where the domain of any given operation is a subset where various other operations happen to be equal. An actual algebraic theory is one where all operations are total functions. The most familiar example may be the (str…

I am self-studying Benedict Gross's abstract algebra lectures and am not able to fully follow a point he makes when defining the product of two cosets. The setup is as follows. Let $f: G \to G'$ be a ...

In the 9th century, Muhammad ibn Musa al-Kharizmi helped solidify the concept of algorithms in mathematics and popularized algebra and the use of the zero.

Paolo Salvatore is an Italian mathematician who works in Dipartimento di Matematica Università di Roma “Tor Vergata”. His research interests include configuration spaces of points, string topology, operads, spaces of knots, and cohomology operations. On group completion of configuration spaces of points and aspects of what later came to be called nonabelian Poincaré duality: Generalizing string t…

Context Algebra - algebra, higher algebra - universal algebra - monoid, semigroup, quasigroup - nonassociative algebra - associative unital algebra - commutative algebra - Lie algebra, Jordan algebra - Leibniz algebra, pre-Lie algebra - Poisson algebra, Frobenius algebra - lattice, frame, quantale - Boolean ring, Heyting algebra - commutator, center - monad, comonad - distributive law Group theor…

This entry is about the notion of “crystal” in algebraic geometry. For the notion in solid state physics see at crystal. There are few mutually unrelated notions denoted by “crystal” in mathematics. One can of course talk about mathematical models of physical crystals and their geometry. Another, is an intermediary notion leading to crystal bases of Kashiwara and of Lusztig, thus one associates a…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … An ordered field is real closed if it satisfies the following two properties: Any non-negative element in has a square root in ; Any odd-degree polynomial function with coefficients…

Disclaimer: if your first instinct was to think about string diagrams, that's not was this post is about. Cool guess, though. In algebra there's this idea of representations of some object, especially linear representations. It's so important that even chemists typically have the representation theory of finite groups in their curriculum, because those have a profound influence on the structure o…

In every cohesive (∞,1)-topos there is an intrinsic notion of ∞-Chern-Weil theory that gives rise to a notion of connection on principal ∞-bundles. We describe here details of the realization of this general abstract structure in the cohesive -topos Smooth∞Grpd of smooth ∞-groupoids. For an ∞-Lie group, a connection on a smooth -principal ∞-bundle is a structure that supports the Chern-Weil homom…

homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Discrete homotopy theory (also known as A-homotopy theory) is an area of mathematics concerned w…

Michael Hopkins: The mathematical work of Douglas C. Ravenel, Homology Homotopy Appl. 10 3 (2008) 1-13 [euclid:hha/1251832464] On the Adams-Novikov spectral sequence: On chromatic homotopy theory and introducing Ravenel's spectra and Ravenel's conjectures: On stable homotopy groups of spheres and chromatic homotopy theory: On stable homotopy groups of spheres and chromatic homotopy theory: Doug R…

A couple months ago, Damek Davis and I launched the first mathematical challenge at the SAIR Foundation, aimed at “distilling” the ability to solve 22 million problems in universal algebra into a condensed form. Stage one of that challenge has now been completed, with several effective “cheat sheets” generated to guess the truth or falsity […]

If is a magma, such as a monoid, (which we write multiplicatively) and is an element of , then the element is the square of . Conversely, if , then is a square root of . If is an integral domain, then (in classical mathematics) and are the only square roots of . If has a square root, then we often denote its square roots together as , although there is no meaning of itself. If is a linearly order…