Simple type theory struggles to express the complex axiomatic networks of algebraic structures found in modern abstract mathematics

Overview Debates Related claims Topics

OpenAI released automated mathematical results claiming solutions or progress regarding Navier-Stokes, the quasi-Riemann hypothesis, and the Unique Games Conjecture

Automated physical law discovery systems can systematically deduce physical laws like Newtonian gravity or Navier-Stokes equations directly from observational data of natural phenomena

Negative numbers achieved widespread acceptance among Western mathematicians after the conceptual acceptance and utility of complex numbers

Kevin Buzzard received a grant to formalize the proof of Fermat's Last Theorem in Lean down to mathematics known in the 1980s

Grigori Perelman proved the Poincare conjecture by publishing his Ricci flow papers on the arXiv preprint repository in 2002 and 2003

In the nineteenth century, Augustin-Louis Cauchy and Joseph Fourier produced conflicting mathematical arguments regarding limits of continuous functions and discontinuous Fourier series

Lean is an interactive theorem prover designed to formally verify software specifications and check the logical validity of mathematical proofs

Virginiya Vassilevska Williams and Josh Alman authored a paper demonstrating an asymptotic algorithm improvement for the three-sum problem

Mathlib is the core mathematical component library used for formalizing mathematics within the Lean interactive theorem proving ecosystem

The ordinary differential equation X dot equals X squared exhibits finite-time blowup due to its quadratic non-linearity

The Boolean satisfiability problem determines whether a given conjunctive normal form formula has an assignment of Boolean variables that evaluates to true

In complexity theory, the class NP consists of decision problems whose positive solutions are verifiable by a deterministic Turing machine within polynomial time

The secretary problem is an optimal stopping problem solvable with an online algorithm achieving a constant competitive ratio by rejecting an initial fraction of candidates

The Riemann hypothesis states that all non-trivial complex zeros of the Riemann zeta function possess a real coordinate part equal to one-half

A quasi-Riemann hypothesis establishes a zero-free vertical strip in the complex plane, proving no zeta zeros exist between a real part of seven-eighths and one

An automated formalization of Fermat's Last Theorem using Lean produced 13 million lines of code

In complexity theory, the class P consists of decision problems solvable by a deterministic Turing machine within polynomial time

Subhash Khot originally formulated the Unique Games Conjecture regarding the hardness of approximating certain label-cover constraint satisfaction problems

Grigori Perelman declined the one-million-dollar Millennium Prize awarded by the Clay Mathematics Institute in 2010 for proving the Poincare conjecture

Grigori Perelman declined the Fields Medal in 2006 for his proof of the Poincare conjecture