TASK DEFINITION: In this task you will be given a text passage from a trivia quiz. You should choose the category this question belongs to from these categories: History, Science, Social Science, Fine Arts, Literature. You can only choose one category, so if in doubt, please choose the closest match.
PROBLEM: Loewner's inequality relates the area and systole of Riemannian metrics on one of these spaces. Arnold's cat map is a chaotic mapping from one of these spaces to itself. This name is given to connected, compact, abelian Lie subgroups, and any compact Lie group has a maximal one. Dupin cyclides are geometric inversions of these objects, and two Villarceau circles can be created by cutting one of these objects diagonally. N-dimensional versions of these objects, which have genus one, are homeomorphic to the Cartesian product of N circles. Other than the Klein bottle, these are the only closed, compact surfaces with an Euler characteristic of zero. They can be created by revolving a circle around an axis coplanar to the circle's diameter. For 10 points, name these mathematical objects that often look like doughnuts.

SOLUTION: Science

PROBLEM: This man achieved full power within his party at the Zunyi Conference. That conference occurred while this man and other members of his party were traveling from Jiangxi (jee-AHNG-shee) to Yanan as part of the Long March. This leader's third wife was part of an influential group that was named the Gang of Four. This ruler was responsible for his country taking part in the (*) Great Leap Forward and the Cultural Revolution. For 10 points, name this political rival of Chiang Kai-shek who was the first leader of Communist China.

SOLUTION: History

PROBLEM: Sodium phenylburate is often used to treat suffers of disorders of this pathway. Defects in transporters found in this pathway can lead to HHH syndrome and Lysinuric Protein Intolerance. N-acetylglutamate is an allosteric activator of a key enzyme in this pathway. Aspartate and Citrullyl-AMP are used in one step of this reaction to form arginosuccinate. Like a connected pathway, fumarate is a byproduct of this reaction. In the mitochondria, citrulline is formed using a transcarbamoylase. Two amino acids that are part of this pathway are arginine and ornithine, and this cycle is sometimes named for the later. This cycle allows the body to deal with overconsumption of protein. For 10 points, name this cycle which generates a namesake compound allows the body to remove excess nitrogen.

SOLUTION:
Science