algebra

Researchers around the world are racing to develop new quantum-based systems for sensing, communication, computing and control that have the promise of outperforming traditional systems. Creating stable, measurable, distinguishable quantum states—which would be the heart of any such system—is a daunting task. Quantum states possess unique properties that can be exploited to develop novel informat…

On simplicial de Rham cohomology?: Johan Louis Dupont: Simplicial de Rham Cohomology and characteristic classes of flat bundles, Topology 15 3 (1976) 233–245 [doi:10.1016/0040-9383(76)90038-0] Johan Louis Dupont: A dual simplicial de Rham complex, in: Algebraic Topology, Rational Homotopy, Lecture Notes in Mathematics 1318 (1988) [doi:10.1007/BFb0077796] On regulators and characteristic classes o…

Raoul Bott (1923–2005) was one of the great 20th century topologists and geometers. Among his famous works, one should mention the Bott periodicity theorem (of importance in K-theory), studies in Morse theory (including the study of Bott–Morse functions), the Borel–Weil–Bott theorem in geometric representation theory, the study of fixed point (localization) formulas (the Atiyah–Bott fixed point t…

On the simplicial de Rham complex and equivariant de Rham cohomology:

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras superalgebra and (synthetic ) supergeometry A super Lie algebra which is a polyvector extension of the super Poincaré Lie algebra (supersymmetry) in for supe…

symmetric monoidal (∞,1)-category of spectra Let and be algebraic theories. The category of -bimodels and their homomorphisms is the category of -models and homomorphisms in . An alternative description is that it is a co--model in . Each such bimodel determines and is determined by a pair of adjoint functors Composition of such adjoint pairs yields a functor The category has a unit object – it w…

A morphism of sites is, unsurprisingly, the appropriate sort of morphism between sites. It is defined exactly so as to induce a geometric morphism between toposes of sheaves (or, more generally, exact completions). Let and be sites. A functor is a morphism of sites if is covering-flat, and preserves covering families, i.e. for every covering of an object , the family is a covering of . If has fin…

The term torsion is used for different concepts in different fields: In algebra, the torsion subgroup of a group is the group of elements of finite order (meaning: elements such that there is such that (with factors in the product)); similarly in ring theory an element of a module over a ring is a torsion element if it is annihilated by a nonzero element of the ring. A module is torsion (resp. to…

I'm studying algebraic geometry and have a question on Kahler differentials. Let $k$ be a commutative ring with unit and $A$ a $k$-algebra. I already have the intuition if we picture $A$ as some ring ...

The Nisnevich topology, also called the completely decomposed topology, is a certain Grothendieck topology on the category of schemes which is finer than the Zariski topology but coarser than the étale topology. It retains many desirable properties from both topologies: The Nisnevich cohomological dimension (and even the homotopy dimension) of a scheme is bounded by its Krull dimension (like Zari…

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.