## Braids, Resolvent Degree and Hilbert’s 13th Problem

February 19 - 21, 2019

## Overview

The purpose of this workshop is to bring focused attention to Hilbert’s 13th problem, and to the broader notion of resolvent degree.  While Abel’s 1824 theorem — that the general degree n polynomial is only solvable in radicals for $$n < 4$$ — is well known, less well known is Bring’s 1786 proof that a general quintic is solvable in algebraic functions of only one variable. Hilbert conjectured that for a general sextic, one needs algebraic functions of two variables, and that for a general degree 7 polynomial, one needs algebraic functions of three variables.  More generally, it is natural to expect that as n → ∞ , so does the minimal number of variables needed to solve the general degree  n polynomial.  In a celebrated theorem, Arnol’d and Kolmogorov proved that, at the level of continuous functions, there is no local obstruction to reducing the number of variables to one.  Thus, a resolution of Hilbert’s problem must lie deeper.  Resolvent degree was introduced by Brauer in order to provide a rigorous statement of these conjectures. While no progress has yet been made on these conjectures, the study of resolvent degree is receiving renewed attention and an influx of ideas from related fields, including:

• the theory of essential dimension
• uniformization of moduli spaces
• braid monodromy