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Solving equations with topology

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If you have a question about this talk, please contact Dexter Chua.

Topology is often useful in showing that equations have solutions without necessarily finding out what the solutions are. The first example of this is the intermediate value theorem: a continuous function f: R → R which takes both positive and negative values must take the value 0; the topological input is that R is connected. The second of these is the fundamental theorem of algebra: a polynomial function p: C → C of positive degree must take the value 0; the topological input is the calculation of the fundamental group of the circle.

I will explain these as well as the less-well known example of solving equations in groups: given a “polynomial” such as w(x)=g_1 x2 g_2 x(-4) g_3 x^3 with g_1, g_2, g_3 in a group G and x a formal symbol, is there a bigger group H ≥ G and a h ϵ H such that w(h)=e ϵ H?

This talk is part of the The Archimedeans (CU Mathematical Society) series.

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