Almost 200 years after Évariste Galois's death, there is still one question about Galois groups — the symmetries of the roots of polynomials, that still remains unsolved. This is the Inverse Galois Problem — whether every group is a Galois group of a Galois extension of the rational numbers. In this talk, I will give an overview of the progress that has been made, the approaches that mathematicians are making, and directions for further research.

A note on the requirements for this talk: It is recommended that you be comfortable with groups, fields, field extensions, automorphisms, and basic Galois theory. This requirement can be met by any of the following suggested alternatives:

• having taken PMATH 347, and currently taking PMATH 348;

• having taken PMATH 347, and coming for the prerequisite knowledge presentation (see below);

• having equivalent knowledge to PMATH 347 and PMATH 348;

• otherwise, if you are currently taking or have taken MATH 146, this talk is still accessible, but it is highly recommended that you read An Introduction to Galois Theory by Dan Goodman, and watch this 18-minute lecture by Matthew Salomone (after reading the article) and optionally also come for the prerequisite knowledge presentation

A prerequisite knowledge presentation about Galois theory will be given at 17:00, prior to the beginning of the talk at 17:30. This presentation will last about 20 minutes. If you are unfamiliar with the material, it is recommended that you read some of the linked material above in addition to attending this presentation. Attending this presentation is optional. The reference material used for this presentation is the Galois Theory document.