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Parameter Spaces: Algebraic & Analytic, Past & Present -
Sometimes it’s easy to describe the solutions to a mathematical problem. If we need to find the roots of a polynomial, say f(x) = x^2 − 3x + 2, it’s no problem to describe the solutions: 1 and 2. But what if a mathematical problem has infinitely many solutions? What if we need to classify all circles in the plane, or all rectangles? We’ll see how certain constants, called parameters, can be used to organize solutions to these types of problems, and how they can also reveal important structure underlying our mathematical problems.

We’ll start by exploring examples of parameters used to classify simple objects, like circles and rectangles. Then, we'll see how to define parameters for elliptic curves from both an algebraic and an analytic perspective. We'll finish by summarizing the history of certain parameter spaces, called Shimura varieties, and by discussing a surprising recent development in this area.

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