There are still completely open fundamental questions about one-variable polynomials. One example is Hilbert’s 13th Problem, which concerns formulas for the roots of a polynomial in terms of its coefficients. Work on this problem really goes back hundreds of years; indeed, it inspired a lot of modern mathematics. In this talk I will explain part of the circle of ideas surrounding this problem, including the inspirational viewpoints of Felix Klein and Henri Poincar\'{e}. Along the way we will see some beautiful mathematical objects: algebraic functions, braids, the 27 lines on a smooth cubic surface, and more; all intimately related to each other, all with mysteries still to reveal.

Much of this talk should be understandable to undergraduate math majors.

Frick Fine Arts Building Auditorium