The abstract concept of symmetry idealizes many different phenomena which are common in everyday life. No matter where we look, whether in art, nature, science, or simply at the next person who passes by, we see something symmetric. In this talk we will illustrate various manifestations of symmetry, and explain how symmetry is described in mathematical terms. We will then see how symmetry is exploited in different fields of mathematics, from algebra to numerical analysis to dynamical systems. Curves, for example, are manifolds.
Surfaces are also manifolds.
An algebraic variety is an object defined by the zeros of one or more polynomials. When considering these two classes of objects an obvious question arises. Nash's theorem investigates this question. In fact, Nash's theorem displays a compact differentiable manifold as some part of an algebraic variety. We will talk about manifolds, varieties, the proof of Nash's theorem, and, possibly, applications of these ideas.