mathematical-physics

Peter Cholak, Professor in the Department of Mathematics, has been awarded the Best Paper Award at the 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Sponsored by the European Association for Theoretical Computer Science (EATCS), the award recognizes the…

We are given the following Lagrangian density: $$\mathcal{L}=F_{\mu \nu} A^{\mu} \mathcal{J}^{\nu}$$ where $F_{\mu \nu}$ is the electromagnetic field tensor, $ A^{\mu}$ the 4-vector of the vector potential and $\mathcal{J}^{\nu}$ is the 4-vector of current density. By making use of the Euler-Lagrange equations, determine the differential equations describing the systems' evolution over time. Eule…

I am a math grad student learning mechanics from Landau-Lifshitz's textbook, and I have a question about their derivation of Hamilton's equations from Lagrange's. In short, they use a variational principle with respect to $p$ and $q$ , where they allow arbitrary variations $p\to p+\delta p$ and $q\to q+\delta q$ . However, $\delta q$ determines $\delta p$ , so this is not quite rigorous. Let me s…

Physics is often described as moving between experiment and model. That picture is indispensable but incomplete. In several episodes since the rise of mathematical physics, investigators have also proceeded by fixing a class of possible representations and progressively restricting it through conservation, covariance, locality, isotropy, variational, and constitutive requirements. The result is s…

Subscribe to Anthropic Science Features on AI-assisted discoveries, practical workflows, and field notes across the sciences. We are sharing the first complete computer-checked proof of Fermat’s Last Theorem. Claude worked largely autonomously over 11 days to write the proof in the Lean programming language. Below, we describe how the formalization was done and share some thoughts about what this…

The International System of Units recognises seven base dimensions, but nothing in dimensional analysis fixes that number, and the system’s own history shows it moving. This paper asks what constrains the decision to found a new one. Its principal result is negative. A dimensional formalism can establish that the records of a domain admit an independently re-scalable coordinate; it cannot establi…

In this special live recording from the International Congress of Mathematicians in Philadelphia, July 2026, the central question discussed was what do mathematicians really value about doing mathematics in the first place? The post Live from ICM 2026: What Is Math For in the Age of AI? first appeared on Quanta Magazine

A new mathematical model tracks how plasmids move through competing E. coli populations, revealing how antibiotics, genetic costs, and plasmid loss could shape biotechnology cultures and future probiotic dosing strategies in the gut. The post Math Model Predicts Plasmid Power Struggles appeared first on GEN - Genetic Engineering and Biotechnology News .

That gravity is a product of a time gradient, without spatial distortion. And what’s more — the presence of the variable “time” in practically all formulas of physics probably means that all other “forces” are also derivatives of time. And the speed of light is a speedometer for the speed of time, not an independent […]

Alain Connes (born on April 1, 1947) is a French mathematician, Fields medalist (1982), Crafoord prize winner (2001), Professor at IHÉS, Professor at Collège de France and part-time working as a Professor at Vanderbilt University. His interests include geometry, topology, especially K-theory and index theory, operator algebras, the connections between noncommutative geometry and number theory, an…

What if computational difficulty is not a penalty of execution time, but a law of informational physics written directly into the structure of mathematical languages? I introduce a new framework that replaces the traditional one-dimensional operational "execution clock" with an intrinsic, 4D... Read more

Mathematicians found that a broad class of networks will abruptly shift behavior past a critical point. The post ‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions first appeared on Quanta Magazine

This paper introduces the Minimum Topological Action Principle (MTAP) and defines a scalar field (Phi) that describes the topological distortion of the vacuum. Through this framework, the Golden Ratio (phi) emerges not as an inserted mathematical constant, but as an inevitable physical equilibrium state. Structural stability converges numerically toward a topological quantum number S_nucleus = 35…

This book presents a systematic development of Refined Quadruple Neutrosophic (T, I, N, F) Logic, extending classical neutrosophic logic by distinguishing Neutrality (N) from Pure Indeterminacy (I) and by refining each of the four components—Truth, Indeterminacy, Neutrality, and Falsehood—into semantically labeled subcomponents. The resulting n-valued structure provides a multidimensional represe…

I am studying a physics problem where an object moves from one point to another under frictionless conditions. The trajectory of the object is shaped like a downward parabola, but the exact curvature can vary. All possible paths have the same horizontal displacement (x). Is there a general mathematical or physical method (e.g., variational principles or the principle of least action) that can det…

_Zenodo_. 2026The fourth generator of the Standard Model gauge algebra is the iteration index of the endomorphism's own cascade. The cycle-generator basis {γ₁, γ₂, γ₃} of H₁(Q; ℝ) has rank 3. Under the bilinear form B derived in *The Endomorphic Collapse Traverses the Foundations of Mathematics* (Stewart, 2026e), the Cartan-Killing correspondence on this basis returns su(3) ⊕ su(2). The full Stan…

This paper defends a representational account of mathematics according to which the essence of mathematics lies not in logical closure alone, but in the construction of a high-precision language for expressing, compressing, and extending structure. On this view, mathematical practice is not exhausted by deductive validity. Rather, it consists in the creation of symbol systems that transform prefo…

As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good explanation for canonical transformations. The only one I could find is in the book by DeWitt, but it is explained on page 700 on the second volume and I can't even get past the first volume, it's a bit …

This paper develops the formal mathematical foundations of the ICDP framework — Identity (I), Conscience/Continuity (C), Determinism (D), and Propagation (P) — as a structural law governing biological communication and biological conscience. ICDP is defined as a mathematical tuple equipped with four axioms that constrain the behavior of living systems. Seven theorems are proved. These include: th…

On the theory of computation and introducing the notion of denotational semantics of programming languages by what came to be called domain theory: Dana S. Scott, Outline of a mathematical theory of computation, in: Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems (1970) 169–176. [pdf, pdf] Dana S. Scott, Christopher Strachey, Toward a Mathematical Semanti…