mathematical-physics

School starts soon, and I know students are looking to get their textbooks at bargain prices Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on... Read more

superalgebra and (synthetic ) supergeometry Perturbative string theory is defined in terms of certain classes of 2d CFTs. Depending on which class that is, one speaks of different types of string theory. In type II string theory the CFTs in question are -supersymmetric and defined on oriented worldsheets; In heterotic string theory the CFTs in question are -supersymmetric and defined on oriented …

physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples The theory of gravity in dimensions famously has black hole solutions, being the limiting configuration of a point mass gravitational source. In higher dimensional gravity, and in parti…

The nonlinear Schrödinger equation (NLSE) is PDE for a classical field $\psi$: $${\displaystyle i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\kappa |\psi |^{2}\psi }.\tag1$$ The Hamilton–...

Hungarian mathematician Paul Erdős’ ‘planar unit distance conjecture’ has stood since 1946 – but an AI disproof has made mathematicians pay attention.

This problem has been addressed before under https://www.physicsforums.com/threads/trouble-understanding-coordinates-for-the-lagrangian.1006528/ I also copied the following problem statement with Landau's very sketchy solution from this old post, because I don't have the English edition of... Read more

His revolutionary idea? Before “computer science” was even a field, Church invented the lambda calculus (λ-calculus)—an elegant, abstract system for expressing computation through pure mathematical functions. In 1936, he used it to prove that no universal algorithm could ever decide the truth of all mathematical statements, solving Hilbert’s famous Entscheidungsproblem in the negative. This becam…

This article proves two no-go results against the conventionality of geometry. I then argue that any remaining conventionality arises from scientific incompleteness. I illustrate by introducing a new kind of conventionality arising in the presence of higher spatial dimensions, where the incompleteness is resolved by introducing new physical theories like Kaluza–Klein theory. Thus, conventional ch…

superalgebra and (synthetic ) supergeometry A supermanifold is a space locally modeled on Cartesian spaces and superpoints. There are different approaches to the definition and theory of supermanifolds in the literature. The definition is popular. The definition has been argued to have advantages, see also the references at super ∞-groupoid. See at geometry of physics – supergeometry the section …

Introduction: The Timeless Nature of Algorithms When you crack open a standard Data Structures and Algorithms textbook, it’s easy to assume the contents are products of the digital age. But here’s the kicker: many of these algorithms were devised centuries before computers existed . Take the Euclidean algorithm for finding the greatest common divisor—it dates back to 300 BCE . Or the sieve of Era…

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…

I have started to read about Geometric Langlands from Frenkel's book https://arxiv.org/abs/hep-th/0512172. As said in Theorem $3$ of the book, the Langlands correspondence is stated as follows ...

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts physics, mathematical physics, philosophy of physics theory (physics), model (p…

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical) quantum mechanical system, quantum probability interacting field quantization physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples In physics, a scalar field…

physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples The spinning string is a variant of the string in direct analogy (but one worldvolume dimension higher) to how the spinning particle is a variant of the particle. (Which means that “spi…

A random matrix is a matrix-valued random variable. Random matrix theory studies mainly the behaviour of eigenvalues and various functions of random matrices; as such it has large importance in physics. Review: Leonid Petrov, Random Matrices, lecture notes 2019 (pdf slides, pdf, webpage) Madan Lal Mehta, Random matrices, 3rd ed. Pure and Applied Math. (Amsterdam) 142, Elsevier/Academic Press 2004…

I came to know that there was a Monstrous moonshine seminar 2020-2021 that took place in IAS organized by Akshay Venkatesh and Jacob Lurie. However, I did not find the webpage or any resources for ...

I ran an experiment today to see whether Claude could generate Lean code to prove a calculation at the bottom of this post, six lines of calculus. I started with this prompt This page contains a mathematical proof that a Fourier coefficient, a_n, is given in terms of a Bessel function. The LaTeX source for […] The post Formally proving a calculation with Claude and Lean first appeared on John D. …

New mathematical research suggests dark energy may not be needed to explain the accelerating expansion of the universe, challenging the foundations of the standard cosmological model. Mathematicians are questioning whether dark energy is actually responsible for the universe’s accelerating expansion. In a new study published in Proceedings of the Royal Society A, researchers at the [...]

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…