Solving algebraic equations are among the oldest problems in mathematics. In this talk, we offer a concrete, visual, and historical introduction to resolvent degree (RD), an invariant that aspires to quantify just how hard these problems are. This overview makes contact with the origins of topology, miracles of classical algebraic geometry, Klein’s “hypergalois” program, and centuries-old exploits in reducing coefficients, which dare us to push beyond the solvable/unsolvable dichotomy. Throughout the talk, we will reflect on many human dimensions of mathematical work: the past and future of resolvent problems, the value in grappling with hard questions even if we might not solve them, and what we do and do not know about RD. No background knowledge will be expected or required beyond calculus and linear algebra, and there will be lots of neat animations!
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