Instructor: Prof. Askold Khovanskii
Email: askold@math.utoronto.ca

Course Code at the University of Toronto: MAT1103H

Course Dates: January 10th - April 5th, 2024
Mid-Semester Break: February 19th - 23rd, 2024
Lecture Times: Wednesdays & Fridays | 10:30 AM - 12:00 PM (ET)

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 25 students

Format: Hybrid

• In-Person in Room 210, Fields Institute (unless specified otherwise)
• Online via Zoom

Course Description

Many fundamental results of Algebraic Geometry over complex numbers are very visual and can be proved using simple topological arguments. In the course, we will prove topologically the main results of intersection theory of divisors. In particular, it includes Bézout's Theorem and Bernstein-Koushnirenko Theorem which compute the number of solutions of a wide class of systems of polynomial equations.

Besides that, I will briefly discuss algebraic versions of these results over algebraically closed fields. I also will prove topologically the celebrated Sturm Theorem on the number of real roots of a real polynomial and Tarski Theorem which is a very wide generalization of Sturm Theorem. I will provide a detailed presentation of all needed facts from Smooth Topology and Commutative Algebra.

Literature: 

1. J.Milnor. Topology from the Differentiable Viewpoint.
2. Yu. Burda, and A. Khovanskii. Degree of rational mapping, and the theorems of Sturm and Tarski. Journal of Fixed Point Theory and Applications. Vol. 3, No. 1, 2008, 79-93.
3. Topology and Intersection Theory of Divisors. Handout.

Course Assessment

The evaluation method for this course will be as follows: during the course, there will be suggested problems related to the material. Students are expected to submit written solutions. Also, each student will have to make a short (20-30 minutes) oral presentation of a (valuable) part of material chosen themselves.