mathematics

Contents Context -Chern-Weil theory Differential cohomology differential cohomology Ingredients Connections on bundles Higher abelian differential cohomology Higher nonabelian differential cohomology Fiber integration Application to gauge theory Contents Idea A Chern-Simons form is a differential form naturally associated to a differential form with values in a Lie algebra : it is the form trivia…

While our previous papers on the Axiom of Structural Identity (ASI) and Entropic Dispersion established a robust philosophical and meta-mathematical framework for navigating the set-theoretic multiverse, the strict formalization of these concepts necessitates precise model-theoretic boundaries. The conceptual architecture of the Methodological Principle of Operational Integrity (MPOI) fundamental…

I was reading Gelfand&Fomin Calculus of Variations book, in which I found this Lemma 2 If $\alpha(x)$ is continuous in $[a, b]$, and if $$\int_a^b \alpha(x) h'(x)\, dx = 0$$ for every function ...

Let $\mathcal{C}$ be a minimal quasi-category equivalent to the $\infty$-category of spaces. (Such a $\mathcal{C}$ is unique up to isomorphism of simplicial sets.) In particular, equivalent objects of ...

superalgebra and (synthetic ) supergeometry The notion of Jordan superalgebra is the analog in superalgebra/supergeometry of that of Jordan algebra: A Jordan superalgebra is a supercommutative superalgebra with underlying -graded algebra , where: is an ordinary Jordan algebra, is a -bimodule with a “Lie bracket-like” product into , satisfying a super Jordan identity. Simple Jordan superalgebras o…

representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, vector bundle, Atiyah Lie algebroid Eilenberg-Moore category, algebra over an operad, actegory, crossed module The possible actions of well-behaved topological groups (such as compact Lie groups) on topological or smooth n-spheres display various interesti…

The homotopy type of the space of invertible non-negative matrices $\DeclareMathOperator\NNGL{NNGL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Given that the homotopy type of the space of ...

A Malcev operation on a set is a ternary operation, a function which satisfies the identities and . An important motivating example is the operation of a heap, for example the operation on a group defined by . An algebraic theory is a Malcev theory when contains a Malcev operation. An algebraic theory is Malcev iff one of the following equivalent statements is true: in the category of -algebras, …

This paper provides a structural analysis of a topological unification framework based on the universal Hopf fibration and the identification of $\mathbb{CP}^{\infty}$ as a physical base space. We establish explicit criteria for what constitutes a physically admissible theory and show that the analyzed construction fails to meet these conditions. The framework replaces dynamical derivation with t…

Let $u$ solve the heat equation$$ u_t = u_{xx} \quad \text{in } (0,\pi)\times(0,\infty),$$ with Dirichlet boundary conditions $$ u(0,t)=u(\pi,t)=0,$$ and initial data $$ u(x,0)=f(x), $$ where $f \in ...

Galois theory is, famously, typically what is used to prove certain results that are stateable without knowledge of it. For example, the following problems are usually proven in a first course in ...

Koenig's theorem says that, given families of sets $(A_i)_{i \in I}$, $(B_i)_{i \in I}$, if there exists an embedding $\prod_{i \in I} B_i \hookrightarrow \coprod_{i \in I} A_i$, then for some $i \in ...

I was intrigued by a recent article in Quanta magazine on ultrafinitism and how it promped a mathematical existential crisis in Princeton mathematician Edward Nelson. It prompted several questions for ...

Generally in most of the resources discussing deep generative models, we have to maxmize the log likelihood of a distribution using latent variable. $p(x) = \sum_z p(x,z)=\sum_zq(z)\frac{p(x,z)}{q(z)} ...

Let $E/F$ be a finite separable extension of degree $n$. I am trying to prove that the number of intermediate fields $K$ (where $F \subseteq K \subseteq E$) is at most $2^n$. I have already seen a ...

Jeff Giansiracusa is a professor at Swansea University in the Department of Mathematics. He has worked on homotopy theoretic aspects of moduli spaces, operads, topological field theory, and diffeomorphism groups, using topological techniques from algebraic K-theory to study the homotopy theory of moduli spaces arising in algebraic geometry and now includes aspects of topological data analysis, tr…

symmetric monoidal (∞,1)-category of spectra The space of functions on the space of morphisms of a small category (with coefficients in some ring ) naturally inherits a convolution algebra structure from the composition operation on morphisms. This is called its category convolution algebra or just category algebra for short. Often this is considered specifically for groupoids, and hence accordin…

Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology cohomology with constant coefficients / with a local system of coefficients Special notions Variants differential cohomology Extra structure Operations Theorems higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derive…

Mathematics readiness at entry remains critical for success in first-year engineering programs, yet many institutions lack transparent, reusable tools to diagnose risk and design levelling policies. This study develops and evaluates an open Engineering Mathematics Readiness Score (EMRS) using two publicly available datasets: the UCI Student Performance dataset in secondary-school mathematics and …