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Symmetry in nature and mathematics
Sergei briefly discussed symmetry in one of his lectures. The purpose of this project is to flesh out what he said.
A symmmetry is in non-mathematical terms some sort of transformation that leaves an object unchanged. An example would be
a rotation, which does not change a spherical object. It turns out
that there is a type of mathematical objects, called groups, that
precisely model sets of symmetries.
As part of this project one should explain what a group is and
how it is connected with symmetry. One should also explain where
one finds symmetry in nature and what examples of groups they
give rise to. Examples of symmetric objects can be given, eg flowers, crystals, viruses, and their symmetry groups can be described.
The student who feels very ambitious can make connections
with physics. The fundamental laws of physics possess symmetries
and this has various interesting consequences.
Basic reference material on symmetry and groups can be found
in many places, including online, but one possible starting point is
some sections of “Abstract Algebra: An Introduction” by Hungerford. Johan can help you access the needed sections.
Noncommutative polynomial rings
In the course we have spent considerable time discussing polynomial rings. A generalization of polynomial rings in one variable
was introduced by the Norwegian mathematician Øystein Ore.
Let us imagine that we want to equip the ring of polynomials in
one variable, x, with a new multiplication, where the multiplication
does not have to be commutative. Obviously this can be done in
an infinite number of ways but we want to insist that the degree
of ab is still the sum of the degree of a and b. One can essentially
show that this implies that the relation xr = σ(r)x + δ(r) must
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hold for every number r. Here σ and δ are functions that satisfy
interesting algebraic properties.
The idea with this project is to clarify everything that is unclear
in the description above, explain why one can really define these
non-commutative polynomial rings and explain what the properties
of σ and δ are.
The reference material for this project is foremost an article by
Patrik Nystedt entitled “A combinatorial proof of associativity of
Ore extensions”, published in the journal Discrete Mathematics.
(Ore extensions is the name for the object we are considering.)
This will be complemented by other material in consultation with
the teacher.
Projects in Gröbner basis and related areas
There are several possible topics to choose among if you want write
about Gröbner basis. I list them below. (Observe that the PhD
students will do their projects on this topic. They have special
requirements and should consult Sergei for details.)
Robotics
In the lectures Johan briefly demonstrated how modeling the position of a robot arm leads to a system of polynomial equations.
The purpose of this project is to explore this application more thoroughly, and in particular to explain how Gröbner basis are applied
in robotics.
Chapter 6, §2 and §3, of “Ideals, Varieties and Algorithms” by
Cox, Little and O’Shea contains a discussion of the the application
of Gröbner basis to robotics. The project would consist of reading
that chapter and writing an explanation of it in one’s own words.
The teachers can help you get access to a copy of the text.
A simple Internet search using the right key words will find
plenty of other references. A few interesting examples of them
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might be cited in the report to illustrate that Gröbner bases actually find use in real robotics research.
Algorithms for Gröbner basis
There are several different algorithms for computing Gröbner basis. The purpose of this project is to describe some of them and
their theoretical analysis. Chapter 2 of Cox’, Little’s and O’Shea’s
previously mentioned book contain a discussion. But one would
need to read some research articles to find a good analysis of the
running time.
The report should include at least a description of some important algorithms, a proof of their correctness and a discussion of
their running time. One should also explain why one cannot hope
that algorithms for computing Gröbner basis that are always fast
will be discovered.
Examples of how one applies these algorithms to a set of polynomials should be included.
Diamond lemma
In the course we have considered commuting variables in our polynomials. Thus we can rewrite yx as xy if we want to and express
any polynomial as a linear combination of monomials with the variables in a particular order.
If we drop the requirement of commutativity then xy and yx
will in general be two different objects. The purpose of this project
is to consider a case when the variables do not commute but there
still is some fairly simple relation between xy and yx. For example
one might have xy = yx + 1.
Considering two variables x, y for simplicity, it turns out to be
the case sometimes that one can still write all expression as linear
combinations of monomials xi y j . A famous result known as the
Bergman Diamond Lemma allows one to prove such things and
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the purpose of this project is to understand the Bergman Diamond
lemma and show how it is applied in some examples.
Appendix A in “q-Deformed Heisenberg Algebras” by Hellström
and Silvestrov is a good introduction to this subject and the teachers can help you get your hands on this text.
Singular and Plural
There is a free software package known as Singular that does Gröbner
basis calculations, among other things.
One possible project would be to install Singular, learn how to
use it and explain some of the algorithms it uses. (The last point
means that this project overlaps somewhat with previous project
suggestions.)
If one does this project one should also apply the software to
some real examples of polynomial systems of equations. Some of
the PhD students have encountered systems of polynomial equations in their work which it would be interesting to try to solve
using the computer.
One finds Singular and its manuals at the following URL: http:
//www.singular.uni-kl.de/