Proseminar in Algebra and Geometry - Summer 2020

The goal of the proseminar is that you gain experience learning math independently and are also able to present what you learn to the audience. If you are interested in giving any of these lectures please email me. Dates that start with and asterisk are already assigned, choose your topic from the ones that do not have an asterisk.

Each lecture is 60 minutes long with a 5 minute break after 25 minutes. It is mandatory that you meet with me at least one week before you give your lecture. Attendence to all lectures is mandatory. You can give your lecture in English or German.

The calendar for the Summer semester is:

The resultant of two bivariate polynomials

In linear algebra we can use a 2x2 determinant to know if a system of 2 equations in the variables x,y has a common solution. What if we want to know if two polynomials in the variables x,y have a common solution? spolier alert: We use the resultant. Section 5, Chapter 3 from “Ideals varieties and algorithms”. Cox, Little, O’shea.

Fundamental theorem of symmetric polynomials in two variables

A polynomial in two variables x,y is symmetric whenever you interchange the role of x and y you get the polynomials that you start with. e.g. x + y is symmetric but x+2y is not. How do we describe the set of all symmetric polynomials in two variables?

Rigidity of 2-dimensional grids

A framework made of bars and joints is deformable if you allow the angle where two bars meet to change. A framework consisting of one triangle is rigid while one consisisting of a square is not. How can you decide if a framework that is a grid of squares is rigid? Section 2 Graphs and Grids, 2.1-2.4

Euler’s formula for convex polyhedra and classification of platonic solids

Euler’s formula relates the number of vertices, edges and faces of a polyhedra. We can also use it to prove there exist only five platonic solids.

*–Quadratic forms in two variables and diagonalization

What does diagonalization of matrices have to do with polynomials of degree two? Motivation Extra

*– The caterogy of sets and discrete categories (if you like programming)

Category theory is an abstraction of many principles that govern mathematics. It could also be a useful programming approach.

Coloring of graphs, the chromatic polynomial and the four coloring theorem

Are you able to color the vertices of any planar graph with four colors in such a way that no two adjacent vertices have the same color? Four color theorem

Duality and tensor products “Lineare Algebra” Gerd Fischer

For the last part of the seminar the lectures are based off the book by Fischer.

*– 6.1 Dualräume

– 6.2 Dualität und Skalarprodukte

– 6.3 Tensorprodukte

– 6.3 Symmetrisch und äussere Produkt and exterior product

– The session is cancelled this day

– 6.4 Multilineare Algebra