Short answers, pulled from the story.
Joseph Fourier is best known for initiating the investigation of Fourier series, which developed into Fourier analysis and harmonic analysis, and for their applications to heat transfer and vibrations. His 1822 treatise Theorie analytique de la chaleur introduced the heat equation, now taught to every student of mathematical physics. The Fourier transform and Fourier's law of conduction are also named in his honour.
Joseph Fourier was born on the 21st of March 1768 in Auxerre, Burgundy, France. He died on the 16th of May 1830 in Paris, following a fall down a flight of stairs on the 4th of May 1830 that aggravated a pre-existing heart condition.
Fourier accompanied Napoleon Bonaparte to Egypt in 1798 as a scientific adviser and was appointed secretary of the Institut d'Egypte, which Napoleon founded at Cairo. Cut off from France by the British fleet, Fourier organized the workshops supplying the French army with munitions, and contributed mathematical papers to the Cairo Institute.
Fourier is generally credited with the first scientific proposal of what is now called the greenhouse effect. In articles published in 1824 and 1827, he considered whether the Earth's atmosphere might act as an insulator, drawing on an experiment by Horace Benedict de Saussure using glass panes to trap heat. Fourier never used the term 'greenhouse effect' himself.
The 1822 treatise contained one mathematical and two physical contributions. Mathematically, Fourier argued that any function of a variable can be expanded in a series of sines, a breakthrough that laid the foundation for the Fourier transform. Physically, he advanced the concept of dimensional homogeneity in equations and proposed the partial differential equation for conductive heat diffusion now known as the heat equation.
Fourier was buried in the Pere Lachaise Cemetery in Paris. His tomb is decorated with an Egyptian motif reflecting his role as secretary of the Cairo Institute. His name is one of 72 names inscribed on the Eiffel Tower, and Joseph Fourier University in Grenoble was named after him.