mathematical-physics

There is a $1 million bounty for the first person providing a solution to the Navier-Stokes existence and smoothness problem.

OpenAI announced today that it had solved the problem, one of seven “Millennium Problems” seen as among the most important in higher mathematics.

The company’s announcement is the most dramatic sign yet that artificial intelligence is fundamentally transforming the field of higher mathematics.

Prior to the announcement, rumors were circulating online among mathematicians over the proof’s origins—OpenAI has denied all allegations

Nature, Published online: 07 September 2026; doi:10.1038/d41586-026-02822-9 Claude produced a 13-million-line, computer-checked proof of the famed conjecture — a major milestone in mathematics.

Suppose we have a configuration space manifold $M$ . The Lagrangian $\mathcal{L}(q, \dot{q})$ is a function on the tangent bundle $TM$ . From $\mathcal{L}$ we can define the action functional $S$ which accepts paths on $M$ and returns the action for that path. A law of physics says that the extremal paths of the action functional are physical trajectories. With some calculation we find that the e…

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch what I tried until now and why it is not working: Analogous to the mentioned publications I introduced $$b_n:= \frac{1}{2}Exp{(\frac{q_n-q_{n+1}}{2})}\\ a_n:= -\frac{p_n}{2} $$ where it follows dire…

Could math help explain why some relationships survive life’s inevitable stresses while others slowly fall apart? Researchers have spent decades developing mathematical models that treat romantic feelings as changing systems, revealing patterns in attraction, emotional reactions, conflict, and recovery.

We develop moment and prime-trace formulations of the Riemann hypothesis and prove two classes of spectral exclusions. The analytic results use explicitly stated entire-function, moment, and operator hypotheses; a separate TCGS--SEQUENTION application formulates the source-to-readout realization problem. Writing $F(w)=\xi(\tfrac12+i\sqrt w)/\xi(\tfrac12)$ by its entire power series, we establish …

The problem I'm supposed to solve is finding $Q$ , such that $(p,q)\rightarrow(P,Q)$ is a canonical transformation. In this case $\mathcal{H}=\frac{p^{2}+q^{2}}{2}$ and the new hamiltonian $\mathcal{K}$ is $\mathcal{K}=P$ . This means $\dot{q}=p$ and $\dot{p}=-q$ Since $\mathcal{H}$ and $\mathcal{K}$ are time independent $\mathcal{H}=\mathcal{K}$ and $P=\frac{p^{2}+q^{2}}{2}$ . Now I use a genera…

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 .