Instructor: Prof. Askold Khovanskii
Email: askold@math.toronto.edu
Registration Deadline: September 24th, 2021
Lecture Times: Wednesdays and Fridays | 10:30 am - 12:00 pm 
Course Dates: September 10th - December 8th, 2021
Mid-Semester Break: November 8th - 12th, 2021
Grading: One final presentation or written report

Course Overview

Topological Galois Theory
Solvability and nonsolvability of equations in finite terms

Numerous unsuccessful attempts to solve a series of algebraic and differential equations “in explicit form” led mathematicians to belief that explicit solutions for these equations simple do not exist. Galois theory explains why algebraic equation is usually not solvable by means of radicals It belongs to algebra. It is very understandable and has a lot of other applications. Liouville theory explains why integral of elementary function usually is not an elementary function. Liouville theory is very elegant and understandable.

About 50 years ago, I started to develop a topological version of Galois theory for functions in one complex variable. According to it, there are topological restrictions on the way the Riemann surface of a function representable by radicals or by quadratures covers the complex plane. If the function does not satisfy these restrictions, then it is not representable by radicals or by quadratures. Beside its geometric clarity the topological results on nonsolvability are stronger than the algebraic results. 

In this course, I plan to present Liouville Theory and Galois Theory in detail and to discuss topological results on nonsolvability by radical and by quadratures. I expect some basic knowledge in complex analysis and in elementary algebra.

References

1) Askold Khovanskii. Topological Galois Theory.
2) A few original papers will be suggested as additional material.

See Detailed Course Information