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Bialgebras in topology -
A central structure in mathematics is that of an algebra: a vector space equipped with a multiplication satisfying certain properties. For example, the set of functions on a geometric object (such as continuous functions on a topological space, smooth functions on a manifold, the coordinate ring of an algebraic variety) often has the structure of an associative and commutative algebra. A less familiar notion, but perhaps more fundamental, is the dual concept of a coalgebra: a vector space equipped with a “comultiplication" satisfying certain properties. A comultiplication is an operation that takes an element of a vector space and “splits” it by producing an element in the tensor product of the vector space with itself. Coalgebras are often found when one linearizes some geometric object and they are related to diagonal maps. In different areas of mathematics, such as algebraic topology and representation theory, we often find that the structure of some object is organized by a bialgebra: a vector space equipped with algebra and coalgebra structure such that the multiplication and comultiplication satisfy certain compatibilities. These compatibilities take different forms. In this talk, I will discuss different flavors of bialgebras and relationships between them. The examples will include Hopf bialgebras, Frobenius algebras, infinitesimal bialgebras, and Lie bialgebras. Then I will explain how these types of bialgebras appear in topology and geometry. I will end by stating some questions and possible projects accessible to students of all levels.
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