Short answers, pulled from the story.
James A. Garfield was the 20th president of the United States, serving from March 1881 until his death in September of that year. He was a former Civil War general, nine-term congressman from Ohio, and the only sitting member of the House of Representatives ever elected directly to the presidency.
Garfield was chosen as a compromise candidate on the 36th ballot at the 1880 Republican National Convention after delegates deadlocked between Ulysses S. Grant and James G. Blaine. He had attended the convention to nominate Ohio's John Sherman and protested his own selection, but the Ohio delegation overruled his objections and voted for him.
Charles J. Guiteau shot Garfield on the 2nd of July 1881 at a Washington train station. Guiteau was a disappointed office-seeker who believed Garfield's death would end Republican intraparty warfare and benefit the Stalwart faction; one medical expert later described him as possibly a narcissistic schizophrenic, and neuroscientist Kent Kiehl assessed him as a clinical psychopath.
Garfield successfully asserted presidential authority over Senate patronage by forcing the confirmation of William H. Robertson as Collector of the Port of New York, ordered a refinancing of the national debt that saved taxpayers an estimated $10 million, pursued a star-route corruption ring in the Post Office, and appointed several African Americans to prominent federal positions. The Pendleton Civil Service Reform Act he championed was passed by Congress and signed into law by his successor in 1883.
In 1876, Garfield developed a trapezoid proof of the Pythagorean theorem, which was published in the New England Journal of Education. According to the Journal, Garfield arrived at the proof during mathematical discussions with other members of Congress; mathematics historian William Dunham later called it "really a very clever proof."
Garfield was a strong advocate for African American civil rights throughout his career, supporting abolition, black suffrage, and federal funding for universal education to prevent freedmen from becoming a permanent "peasantry." As president he appointed Frederick Douglass, Blanche K. Bruce, John M. Langston, and Robert Elliot to prominent federal positions, and his inaugural address explicitly condemned any legal obstacle placed in the path of a virtuous citizen.