algebra

We propose a three-layered framework for mathematical specialization and research strategy in an era of rapidly advancing artificial intelligence. First, we introduce the "Diffraction Profile," a finite-resource cognitive model that uses a squared-sinc kernel to mathematically describe how deep specialization in a core domain generates non-monotonic, structurally resonant "side lobes" of secondar…

The notion of a normal subobject is the proper generalization of a normal subgroup to other algebraic categories. The notion was found relatively late. In the category of groups, there are two equivalent descriptions of a normal subgroup: as the kernel of a homomorphism of groups and as the equivalence class of the unit of some (necessarily unique) congruence. Given a category admitting finite li…

I would like to put a hat under a symbol. I have the following solution right now: \NewDocumentCommand{\underhat}{m}{\underset{\widehat{}}{#1}} This looks correct, except that tex considers this ...

model category, model -category Definitions Morphisms Universal constructions Refinements Producing new model structures Presentation of -categories Model structures for -groupoids for equivariant -groupoids fine model structure on topological G-spaces coarse model structure on topological G-spaces (Borel model structure) for rational -groupoids for rational equivariant -groupoids for -groups for…

Special and general types Special notions Variants Extra structure Operations Theorems algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on…

The Hodge conjecture asserts that every rational class of type (p,p) on a smooth complex projective variety is a rational combination of classes of algebraic cycles; it is known for p≤ 1 and p≥ n-1 and open in the middle degrees of every dimension n≥ 4. Reductions of the conjecture typically restate it or split it into further open hypotheses, such as the existence of an algebraic representative …

representation, 2-representation, ∞-representation group, ∞-group group algebra, algebraic group, Lie algebra vector space, n-vector space affine space, symplectic vector space action, ∞-action module, equivariant object bimodule, Morita equivalence induced representation, Frobenius reciprocity Hilbert space, Banach space, Fourier transform, functional analysis orbit, coadjoint orbit, Killing for…

Special and general types Special notions Variants Extra structure Operations Theorems Twisted de Rham cohomology is the twisted cohomology-version of de Rham cohomology, a simple example of twisted differential cohomology. For degree-3 twists this is the codomain of the twisted Chern character on twisted K-theory, and in its orbifold cohomology-generalization it is the codomain of the twisted eq…

Hidden correspondences hint at a deeper structure to the mathematical universe. Our columnist unpacks one of those connections and asks mathematicians what they might mean. The post What Is Math’s Mysterious Langlands Program Really About? first appeared on Quanta Magazine

The E6 Root Polytope Posted by John Baez I’ve been thinking about the exceptional Lie algebra E6, as a spinoff of my project on E7, so I want to get a good mental picture of the E6 root polytope. This is 6-dimensional polytope with remarkable symmetry. Let’s climb up to the E6 root polytope starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking …

I'm trying to evaluate a Gaussian integral over Grassmann numbers but not sure if I've made a mistake. What I want to evaluate is \begin{equation} \left(\prod^N_i\int d\theta^*_i d\theta_i\right)\theta_k \theta^*_l \theta_m \theta^*_n \exp\left(-\theta^*_i B_{ij}\theta_j\right), \end{equation} where $\theta$ is a complex Grassmann number, and $B$ is an $N\times N$ invertible matrix. First I expan…

I use the mismath package, which among other math operators, defines \card, but with a definition different from the one I want, which is simply: \DeclareMathOperator{\card}{card} To effect this ...

This paper studies the recursive complex-root relation obtained by repeatedly taking roots of the negative of the preceding value, with particular emphasis on the role of branch selection. We show that different interpretations of the recurrence define fundamentally different mathematical systems. For the principal branch, the dynamics is deterministic and exactly solvable: the modulus converges …

Special and general types Special notions Variants Extra structure Operations Theorems The universal coefficient theorem states how ordinary homology/ordinary cohomology determines homology/cohomology with arbitrary coefficients. For a chain complex (of abelian groups) and a field (the coefficient field), the homology group and the cohomology group are indeed related by dualization: . If the coef…

There are many ways to formulate the group axioms. A standard definition is this: A group is pair $(G,\circ)$ consisting of a set $G$ and a binary operation $\circ$ on $G$ satisfying the following ...

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A […] The post Proof of the rank-trace theorem first appeared on …

Victor Ginzburg (in some 1980s articles spelled Ginsburg) is a professor of mathematics at the University of Chicago. His thesis in Moscow was under Alexandre Kirillov. His main interests are representation theory, especially geometric representation theory, including more recently noncommutative algebraic geometry. Warning: there is another mathematician (global analysis, symplectic geometry), V…

During my second week at the Recurse Center, I've been trying to formalize Dummit and Foote's abstract algebra textbook (appropriately titled "Abstract Algebra") in Rocq. I had quite a hard time with the first proof exercise in the book because the stated proof goal is not true. This was both frustrating and exciting to figure out :) A function f from a set A to a set B (written "f: A -> B") is a…