This course will present four general techniques which appear and are used in many circumstances. They are equally relevant for theoretical and applied mathematicians. We keep the prerequisites low and emphasize instead the main ideas in concrete examples. The course is designed to address both Master students and Researchers.

Class I, October 17/18: "Singularities and Blowups"

Let us call algebraic variety the set of solutions of a couple of polynomials in n variables. At most points, these are smooth, i.e., look like a differential manifold. At other points, the singularities, the local geometry is much more complicated, see https://homepage.univie.ac.at/herwig.hauser/gallery.html for various examples.

The idea and goal is then to present these singular varieties as projections of manifolds, embedded possibly in some higher dimensional space. Such projections are known to exist (say, over C), and called resolutions. The proof is extremely difficult.

In the class, we will present the main technique to achieve this, blowups. There are many equivalent definitions (we will see and compare most of them), and we show how to work with blowups in practice by computing explicit examples.

Class 2, November 13/14: "Symmetry and Invariant Theory"

Symmetry is ubiquituous in mathematics, and can often be used to reduce complicated problems to easier ones (e.g., by reducing the dimension). Underlying is the action of a group on one or several objects, and one wishes to identify objects which define the same orbit. Examples are matrices and their conjugacy class, polyhedra and movements in R^3, polynomials and permutations of the variables, differential equations and analytic coordinate changes, to mention a few.

The key object in this context is the invariant ring, i.e., the ring of polynomials which remain invariant under the group action. Hilbert's famous theorem tells us that, for finite groups, it is a finitely generated algebra. This result, which can be extended to reductive groups, is instrumental for many applications.

To illustrate, Felix Klein's icosahedral group produces the wonderful invariant ring C[x,y,z]/(x^2+y^3+z^5), an example which has stimulated hundreds of mathematicians.

Class 3, December 11/12: Inverse Functions and Henselization

The notorious Jacobian Conjecture, still unproven or falsified, claims that a polynomial map f: C^n -> C^n with constant Jacobian determinant equal to 1 has an inverse polynomial map g : C^n -> C^n, gf = fg = Id. Strikingly enough, to prove it would suffice to show that det = 1 implies that f is injective. This is the Ax-Grothendieck theorem, with a phantastic proof (which we will give) using finite fields.

As the inverse function theorem (for local inverses) does not hold for polynomial maps, one looks for the smallest extension of the polynomial ring (localized at 0) where one has IFT. This is the Henselization, named after Kurt Hensel who introduced the concept in the frame of p-adic analysis through his Hensel Lemma. We will provide a detailed study of this extension and show how it yields to the concept of algebraic power series, a notion which is very popular in combinatorics, counting problems and generating functions.

Class 4, January 16/17: "Control Theory and Integration of Vectorfields"

A mono-cycle is a bicycle with just one wheel. How does one have to control its movement (pedalling and steering) to arrive from one position to another prescribed position. This is the first and simplest example of a control problem. Already the bicycle is much more elaborate, and quickly yields to the problem how one has to control a car to park in a given slot.

The mathematics behind are vectorfields and their integral curves. The first and most basic result in the theory is Frobenius' theorem about the rectification of vector fields. Its use is numerous in control theory, and we will show that it has a very nice and conceptual proof. With this in mind, we will eventually show that (as we already know from experience) one can always park a car (or even a car with remorque) provided the slot is longer than the car.

The course takes place in the Science Center, University of Linz

?? Floor, Room ??

Class 1, Thursday, October 17, 16-18 pm, Friday, October 18, 10-12 am

Class 2, Thursday, November 13, 16-18 pm, Friday, November 14, 10-12 am

Class 3, Thursday, December 11, 16-18 pm, Friday, December 12, 10-12 am

Class 4, Thursday, January 16, 16-18 pm, Friday, January 17, 10-12 am