of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by . Status unclear Uniform convergence. In his Cours d'Analyse of 1821, Cauchy "proved" that if a sum of continuous functions converges pointwise, then its limit is also continuous. However, Abel observed three years later that this is not the case. For the conclusion to hold, "pointwise convergence" must be replaced with "uniform convergence". It is not entirely clear that Cauchy's original result was wrong, because his definition of pointwise convergence was a little vague and may have been stronger than the one currently in use, and there are ways to interpret his result so that it is correct. There are many counterexamples using the standard definition of pointwise convergence. For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuous function such as a step function. Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed out that his proof was incomplete. As of 2016 the problem is still open. Italian school of algebraic geometry. Most gaps in proofs are caused either by a subtle technical oversight, or before the 20th century by a lack of precise definitions. A major exception to this is the Italian school of algebraic geometry in the first half of the 20th century, where lower standards of rigor gradually became acceptable. The result was that there are many papers in this area where the proofs are incomplete, or the theorems are not stated precisely. This list contains a few representative examples, where the result was not just incompletely proved but also hopelessly wrong. In 1933 George David Birkhoff and Waldemar Joseph Trjitzinsky published a very general theorem on the asymptotics of sequences satisfying linear recurrences. The theorem was popularized by Jet Wimp and Doron Zeilberger in 1985. However, while the result is probably true, as of now (2021) Birkhoff and Trjitzinsky's proof is not generally accepted by experts, and the theorem is (acceptedly) proved only in special cases. Jacobian conjecture. Keller asked this as a question in 1939, and in the next few years there were several published incomplete proofs, including 3 by B. Segre, but Vitushkin found gaps in many of them. The Jacobian conjecture is (as of 2016) an open problem, and more incomplete proofs are regularly announced. discuss the