4 least square solution in the OVERdetermined case

b. Systems of linear equations were considered already by Leibniz. Cramer's rule gave the formula for a solution in the $n\times n$ case and Gauss developed a method for obtaining the solutions, which yields the least square solution in the overdetermined underdetermined case. But none of this work yielded a criterion for the existence of a solution. Euler, Laplace, Cauchy, and Jacobi all considered the problem of diagonalization of quadratic forms (the principal axis theorem). However, the work prior to 1850 was incomplete because it required genericity assumptions (in particular, the arguments of Jacobi et al didn't handle singular matrices or forms. Proofs that encompass all linear systems, matrices and bilinear/quadratic forms were devised by Sylvester, Kronecker, Frobenius, Weierstrass, Jordan, and Capelli as part of the program of classifying matrices and bilinear forms up to equivalence. Thus we got the notion of rank of a matrix, minimal polynomial, Jordan normal form, and the theory of elementary divisors that all became cornerstones of linear algebra.

3 Moved a dollar sign to fix apparent LaTeX interpretation bug

Universality for one-parameter families of unimodal continuous self-maps of an interval was experimentally discovered by Feigenbaum and, independently, by Coullet and Tresser in the late 1970s. It states that the ratio between the lengths of intervals in the parameter space between successive period-doubling bifurcations tends to a limiting value $\delta\approx 4.669201$ 4.669201 $that is independent of the family. This could be explained by the existence of a nonlinear renormalization operator$\mathcal{R}$in the space of all maps with a unique fixed point$g$and the property that all but one eigenvalues of its linearization at$g$belong to the open unit disk and the exceptional eigenvalue is$\delta$and corresponds to the period-doubling transformation. Later, computer-assisted proofs of this assertion were given, so while Feigebaum universality had initially appeared mysterious, by the late 1980s it moved into the "99% true" category. 2 -typos; +geometry of numbers, Ars Magna, PDO I would like to preface this long answer by a few philosophical remarks. As noted in the original posting, proofs play multiple roles in mathematics, : for example, they assure that certain results are correct and give insight into the problem. a. It was had been known since the ancient times that it is was impossible to create a perfect (i.e. undistorted) map of the Earth. The first proof was given by Gauss and relies on the notion of intrinsic curvature introduced by Gauss especially for this purpose. Although Gauss's proof of Theorema Egregium was complicated, the tools he used became standard in the differential geometry of surfaces. b. Isoperimetric property of the circle has been known in some form for over two millenia. Part of the motivation for Euler's and Lagrange's work on variational calculus came from the isoperimetric problem. Jakob Steiner devised several different synthetic proofs that contributed technical tools (Steiner symmetrization, the role of convexity) convexity), even though they didn't settle the question (because they relied on the existence of the absolutely minimizing shape)shape. Steiner's assumption led Weierstrass to consider the general question of existence of solutions to variational problems (later taken up by Hilbert, as mentioned below) and to give the first rigorous proof. Further proofs gain gained new insight into the isoperimetric problem and its generalizationgeneralizations: for example, Hurwitz's two proofs using Fourier series exploit exploited abelian symmetries of closed curves; the proof by Santaló using integral geometry established more general Bonnesen inequality; E.Schmidt's 1939 proof works in$n$dimensions. Full solution of related lattice packing problems led to such important techniques as Dirichlet domains and Voronoi cells and the geometry of numbers. a. For more than two and a half centuries since Cardano's Ars Magna, no one could was able to devise a formula expressing the roots of a general quintic equations equation in radicals. The Abel–Ruffini theorem and Galois theory not only proved the impossibility of such a formula and provided an explanation for the success and failure of earlier methods (cf Lagrange resolvents and casus irreducibilis), but, more significantly, put the notion of group on the mathematical map. b. Systems of linear equations were considered already by Leibnizand . Cramer's rule gave the formula for a solution in the$n\times n$case . and Gauss developed a method for obtaining the solutions(in the overdetermined case, which yields the least square solution), but it didn't give solution in the overdetermined case. But none of this work yielded a criterion for the existence of a solution. Euler, Laplace, Cauchy, and Jacobi all considered the problem of diagonalization of quadratic forms (the principal axis theorem). However, the work prior to 1850 was incomplete because it required genericity assumptions (in particular, the arguments of Jacobi et al didn't handle singular matrices or forms. Proofs that encompass all linear systems, matrices and bilinear/quadratic forms were devised by Sylvester, Kronecker, Frobenius, Weierstrass, Jordan, and Capelli as part of the program of classifying matrices and bilinear forms up to equivalence. Thus we got the notion of rank of a matrix, minimal polynomial, Jordan normal form, and the theory of elementary divisors that all became cornerstones of linear algebra. b. Brouwer's proof of the Jordan curve theorem and of its generalization to higher dimensions was a major development in algebraic topology. Although the theorem is intuitively obvious, it is also very delicate, because various plausible sounding related statements are actually wrong, cf as demonstrated by the Lakes of Wada and the Alexander horned sphere. • Analysis The work on existense, uniqueness, and stability of solutions of ordinary differential equations and well-posedness of initial and boundary value problems of for partial differential equations gave rise to tremendous insights into theoretical, numerical, and applied aspects. Instead of imagining a single transition from 99% ("obvious") to 100% ("rigorous") confidence level, it would be more helpful to think of a series of progressive sharpenings of statements that become natural or plausible after the last round of work. c. The Cauchy problem for hyperbolic linear partial differential equations was investigated by a whole constellation of mathematicians, including Cauchy, Kowalevski, Hadamard, Petrovsky, L.Schwartz, LereLeray, Malgrange, Sobolev, Hörmander. The "easy" case of analytic coefficients is addressed by the Cauchy–Kowalevski theorem. The concepts and methods developed in the course of the proof in more general cases(, such as the characteristic variety, well-posed problem, weak solution, Petrovsky lacuna, Sobolev space, hypoelliptic operator) , pseudodifferential operator, span a large part of the theory of partial differential equations. • Universality for one-parameter families of continuous unimodal continuous self-maps of an interval was experimentally discovered by Feigenbaum and, independently, by Coullet and Tresser in the late 1970s. It states that the ratio between the lengths of intervals in the parameter space between successive period-doubling bifurcations tends to a limiting value$\delta\approx 4.669201$that is independent of the family. This could be explained by the existence of a nonlinear renormalization operator$\mathcal{R}$in the space of all maps with a unique fixed point$g$and the property that all but one eigenvalues of its linearization at$g$belong to the open unit disk and the exceptional eigenvalue is$\delta\$ and corresponds to the period-doubling transformation. Later, computer-assisted proofs of this assertion were given, so while Feigebaum universality had initially appeared mysterious, by the late 1980s it moved into the "99% true" category.

The full proof of universality for quadratic-like maps by Lyubich (MR) followed this strategy, but it also required very elaborate ideas and techniques from complex dynamics due to a number of people (Douady–Hubbard, Sullivan, McMullen) and yielded hitherto unknown information about the combinatorics of non-chaotic quadratic maps of the interval and the local structure of the Mandelbrot set.

Agrawal, Kayal, and Saxena proved that PRIMES is in P, i.e. primality testing can be done deterministically in polynomial time. While the result had been widely expected, their work was striking in that at least two respects: it used very elementary tools, such as variations of Fermat's little theorem, and it was carried out by a computer science professor and two undergraduate students. The sociological effect of the proof may have been even greater than its numerous consequences for computational number theory.