Science and math

Modular form

Jovan Stosic January 30, 2020

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that f (z) be holomorphic in the upper half-plane. Instead, modular functions are meromorphic.

Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups.

Source: Modular form – Wikipedia

Goro Shimura – Wikipedia

Jovan Stosic January 30, 2020

Gorō Shimura (志村 五郎, Shimura Gorō, 23 February 1930 – 3 May 2019) was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat’s Last Theorem.

Source: Goro Shimura – Wikipedia

Modularity theorem

Jovan Stosic January 30, 2020

The modularity theorem (formerly called the Taniyama–Shimura conjecture) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat’s last theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles’ techniques to prove the full modularity theorem in 2001.

Source: Modularity theorem – Wikipedia

Andrew Wiles

Jovan Stosic January 30, 2020

Sir Andrew John Wiles KBE FRS (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat’s Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society. He was appointed Knight Commander of the Order of the British Empire in 2000, and in 2018 was appointed as the first Regius Professor of Mathematics at Oxford. Wiles is also a 1997 MacArthur Fellow.

Source: Andrew Wiles – Wikipedia

SymPy

Jovan Stosic October 26, 2019

SymPy is a Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live or SymPy Gamma. SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies.[2][3][4] This ease of access combined with a simple and extensible code base in a well known language make SymPy a computer algebra system with a relatively low barrier to entry.

SymPy includes features ranging from basic symbolic arithmetic to calculus, algebra, discrete mathematics and quantum physics. It is capable of formatting the result of the computations as LaTeX code.[2][3]

SymPy is free software and is licensed under New BSD License. The lead developers are Ondřej Čertík and Aaron Meurer.

Source: SymPy – Wikipedia

Q.E.D. – Wikipedia

Jovan Stosic October 13, 2019

Q.E.D. or QED (sometimes italicized) is an initialism of the Latin phrase “quod erat demonstrandum”, literally meaning “what was to be shown”.[1] Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument to indicate that the proof or argument is complete, therefore used with the meaning “thus it has been demonstrated”.

Source: Q.E.D. – Wikipedia