### Math ClubZero Sets of Polynomials: Analytic vs Geometric RegularityMatthew Badger (UConn)

Wednesday, February 27, 2019
5:45pm – 6:35pm

Storrs Campus
Monteith 321

The zero set of a polynomial is the set of points where the polynomial has the value zero. Polynomials like $$p(x,y) = x^4 + y^4 - y^2$$ or $$q(x,y,z)=xyz$$ may seem like simple algebraic expressions, but they generate a lot of interesting geometry by considering their zero sets. It can be hard to predict what the zero set of a polynomial looks like, and despite being studied for many years, there are many open problems.

In this talk, I will describe two different ways that we could say the zero set of a polynomial looks "regular" at a point. One possibility comes from calculus (a non-zero derivative) and another possibility comes from metric geometry ("flatness"). I'll give a lot of pictures and examples to illustrate the difference between the analytic and the geometric definition of regularity, and also pose some questions for research that an interested student could pursue.

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