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Abstract:
Given four spheres of radius one in three-dimensional space, how
many lines can be simultaneously tangent to all four? The answer is easy
to state, but understanding where it comes from requires some interesting
mathematics, including Bezout's Theorem and the projective plane. This
lecture will explain how these tools apply to the four sphere problem and
put this problem into a larger context by introducing other counting
problems that arise from algebraic equations (this is "enumerative
algebraic geometry"). I will give numerous examples, including some that
arise in string theory in mathematical physics.
For most of the lecture, knowledge of first semester calculus will be
sufficient. In some places, I will use dot product and cross product from
third semester calculus.
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