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 and graduate students as well as interested researchers.

Class I, October 16/17: "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 6/7: Commutative Algebra: Free
Resolutions, Hilbert Functions, and Flatness

The dimension of an algebraic variety defined by polynomials
f_1,...f_k is strongly related to the module of linear relations
sum r_i f_i = 0 between f_1,...,f_k. This gives rise
to the concept of free resolutions of polynomial ideals
and the associated homology groups. In particular,
one constructs the Hilbert function and Hilbert
polynomial of the variety, an important invariant
which has also geometric significance
(Euler characteristic, genus, ...).

In a similar vein, flatness is a mean to characterize
the continuous variation of a family of varieties.
While being algebraically defined, it has strong
geometric implications, which we will discuss in detail.

Class 3, December 11/12: "Symmetry & 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 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 central result in the theory
is Frobenius' theorem about the rectification of vector fields. Its use is numerous, 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 trailer) provided the slot is longer than the car.

The course takes place in the Science Center, Johann-Kepler-University Linz

Class 1, Thursday, October 16, 16-18 pm, ESH 3, Friday, October 17, 10-12 am, MZ 005A
"Singularities and Blowups"

Class 2, [NEW DATE AND TOPIC] Thursday, November 6, 16-18 pm, S3 055, Friday, November 7, 10-12 am, HT 177F
"Commutative Algebra: Free Resolutions, Hilbert Functions, and Flatness"

Class 3, Thursday, December 11, 16-18 pm, K 012D, Friday, December 12, 10-12 am, S3 048
"Symmetry and Invariant Theory"

Class 4, Thursday, January 15, 16-18 pm, K 012D, Friday, January 16, 10-12 am, T 406
"Control Theory and Vector Fields"