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Topics can include the following

Intro

How conventional vector/matrix notation is inadequate to properly describe algebraic geometry and what works better (tensors) and why.

This simplest possible matrix still contains much of the complexity of larger matrices, but the algebra is simple enough to do explicitly. In particular we can visualize the matrices in a space of four (homogeneous) dimensions and see geometrically the relation between singular matrices, involutions, real and complex eigenvalues, singular value decompositions etc.

Polynomials

Solving quadratic, cubic and quartic polynomials symbolically and numerically. Visualizing their discriminants and seeing a new way to find and understand resultants.

Lines in Space

How to represent 3D lines with tensors and some of the geometric calculations you can do with them.

What possible shapes can an implicit cubic curve take? How to analyze a given set of coefficients to determine what and where singularities happen.

The group structure of the cubic

Given two points on a cubic curve, how do you find the third point collinear with them? What if the two points coincide and you must use the tangent line? The solution to this problem leads to a group structure with the points as elements.

Parametric curves/surfaces

What possible shapes can order 2, 3 and 4 order curves have? How do these relate to the possible shapes of implicit curves? How do you go back and forth?

Pascal’s Theorem

How this theorem is equivalent to finding whether an arbitrary 6 points in the plane lie on a common conic section. A new way to prove this using tensors that has nice generalizations to the next topic:

The 9th point problem

A quadratic curve is uniquely determined by 5 points. A cubic curve is uniquely determined by 9 points… usually. When does this break down and why?

Steiner Surfaces

These are the next simplest type of parametric surface after planes. They form a beautiful zoo of different shapes. I will discuss what geometric singularities they can have and how to detect them algebraically.