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In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations on Lie groups continue to hold when the (convergent) power series locally representing the group law was replaced by a suitable formal analogue.  It was not long before this formalism found far-reaching uses in algebraic number theory and algebraic topology.

Unfortunately, few students see more than two or three explicit (i.e. closed form) group laws before stumbling into the deep end of abstract nonsense.  In this article, we’ll see in a rigorous sense why this must be the case, providing along the way a complete classification of polynomial and rational formal group laws (over any reduced ring).

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In its early years, the rigorous study of games (game theory) looked only upon games of so-called perfect information, in which each player knows the moves carried out by all players.  Such games laid the foundation for early economic models of perfect competition, wherein consumers have full knowledge of both market conditions and each others’ consumer tendencies.

Of course, perfect competition — taken literally — cannot exist, and the failure of this model and others prompted economic theorists to study a more-general class of games, games of limited information (also known as games of imperfect information).

In this post, we’ll look at one-player games of limited information (sometimes classified as puzzles, not games) through a topological lens, and create for each game a poset of topologies under which topologically indistinguishable points correspond to outcomes that are indiscernible in a limited-information context.  Expanding this dictionary, we’ll describe a topology on the outcome space under which the “safe” or “warranted” extension of one’s limited information relates to the continuity of certain maps.

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