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Baker was inspired by Klein while in Göttingen to study algebraic function theory. He wrote Abel's Theorem and the Allied Theory including Theta Functions in 1897 and Multiply Periodic Functions in 1907. He came in contact with the Italian School of geometry and made their work the subject of his 1911 London Mathematical Society presidential address. From 1911 he studied birational geometry publishing a 6 volume work Principles of Geometry from 1922 to 1925.
Elected to the Royal Society in 1898, Baker won the Society's Sylvester Medal in 1910. Baker held the Lowndean chair of Astronomy and Geometry from 1914 until he retired in 1936. He justified the 'astronomy' title in his chair by lecturing on periodic orbits and other astronomical topics.
Baker inspired a younger generation of geometers. He founded the Saturday afternoon seminar or 'tea party' which became the focus of activity in geometry.
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