LaTeX Code

After a long time of being very inactive concerning my website, I recently decided to blog more regularly on topics of mathematical physics. For this, it is very important to be able to write formulae and other mathematical expressions which is usually done using \LaTeX. To test if my latex compiler works, I will start with one of the wonderful Stokes’ theorem that combines topology and differential geometry: Given a manifold M with a boundary \partial M, we have

    \[\int_Md\omega=\int_{\partial M}\omega\,\]

where \omega is a (\dim(M)-1)-form.

After 17 Months: PRD Publication

We published a paper based on my Bachelor’s thesis – submitted in July 2011 at Humboldt University. You can read it on arXiv or PRD.

What did we do?
We analysed the efficiency of new analytical expressions for tree-level amplitudes in massless QCD. To give an idea what this means, you have to read the last sentence backwards (almost word for word):

  • QCD
    QCD stands for Quantum Chromo Dynamics and is a quantum field theory to describe the fundamental particles of matter, namely quarks and their interaction through gluons. Protons and Neutrons (in the nucleus of every atom) are built from quarks. Understanding QCD is crucial to understand the behavior of matter on high energies, for example in high energy experiment at the Large Hadron Collider (LHC) at CERN.
  • Massless
    Quarks are not massless (otherwise we would not feel gravity). However, on high energies the mass is negligible compared to the high-energy in their momentum: a bullet shot with a canon will still fall down, but as approximation for a few meters we can assume that it follows almost a straight line. In the same way, we can neglect mass if a particle is accelerated to a very high speed and momentum.
  • Amplitudes
    For many experiments in high-energy physics we want to know how likely it is that accelerated particles scatter and produce new particles. This can be answered by calculating a probability distribution which depends on the absolute value of a complex quantity called scattering amplitude. Roughly speaking, a (scattering) amplitude is the function that allows us to calculate the probability of a scattering event to happen and it depends on the properties of the particles involved.
  • Tree-level
    Calculating scattering amplitudes is difficult: Even with computer there is no way to calculate scattering amplitudes to arbitrary precision. Instead we have to use a perturbation series of very complicated expressions. Feynman found a way to represent these complicated expressions by simple diagrams called Feynman diagrams and the perturbation series
  • Analytical exprressions
    Calculation of such amplitudes is mainly done through a numerical recursion that repeats certain steps a number of times to get a numerical result. In contrast to this, an analytical expression is one formula describing how you get a numerical value by just applying certain operations (like addition, multiplication, squareroots etc.) and is mathematically speaking a function with explicit form. In our case, we have such analytical expressions for the tree-level amplitudes, even though they are very long and complicated.
  • Efficiency
    In order to predict certain outcomes of a scattering experiment one needs to predict probabilities by calculating the amplitudes for different possible processes. Because even on tree-level (leading order) one finds a huge number of possible processes one has to evaluate thousands and millions of amplitudes which makes efficiency very important: the less time one evaluation of an amplitude takes the more realistic is it to make certain predictions. Our whole analysis has the goal to compare the efficiency of just evaluating our analytical expressions with one commonly used numerical procedures (called Berends-Giele) that use a recursion to approximate the value. Furthermore, we compared the precision of the different approaches.

What did we get?
As expected before, our implementation of the analytical formulae (the main work of my bachelor thesis) is faster for the first two orders (MHV and NMHV). The third order (NNMHV) that I implemented, as well, becomes slower than the Berends-Giele approach as the number of involved particles increases. I did not implement higher orders because it was clear that higher orders will be slower than Berends-Giele. Finally, the accuracy of the analytical formulae was a bit higher than using the Berends-Giele scheme.

Into December

The last couple of weeks were very intense, but after I reached maximal stress level last week it is getting more relaxed as the application deadlines for the American universities approach. There were a lot of things going on since I wrote my last post:

  1. Video Production
    We worked on our video production for which we interviewed the director of the IRD, the French institute for research in development. The IRD supports AIMS Sénégal since the beginning and its current director Michel Laurent is also member of AIMS’ executive board. Morgan and I went to Dakar where the annual board meeting took place to use the opportunity to film some interviews with Mr. Laurent and other board members.
  2. GRE Test
    I traveled to Boston where I wrote the two compulsory GRE tests for my applications: The General GRE and the GRE Subject Test for Physics. I don’t have my results yet, but at the moment I’m just relieved that it is over. Especially, as I had the feeling that learning for the GRE used some resources that I would have preferred to spend otherwise (see next projects).
  3. Turning AIMS bilingual
    Back from Boston we had a very interesting discussion among the AIMS tutors how to implement bilingualism at AIMS: up to now many students can speak both, English and French, but those who could not on their arrival did not proceed as much as they could have. A competent language teacher is missing. I suggested for next year that we could advertise the position internationally to find a highly dynamical and committed language teacher and change the student selection to balance the numbers of native speakers in English and French (currently most students have a French background).
  4. African Opportunity Network
    Yannick, Pavel and I started to work on the website of a new project supporting African students pursuing  a career in mathematical sciences. When professor Banyaga from PennState University taught at AIMS Sénégal he gave also a talk on the problems of African applicants at Western Universities: they are not used to the sort of application and especially good recommendation letters are difficult to get as many African professors do not know how to write good reference letters. Yannick, one of our very committed students, asked if there exists a support network to guide talented Africans through Western application procedures. We could not find anything and so we are going to start it: we want to bring professors from all over the world together with African students to guide during their first steps into modern research (Master’s and PhD).

For Christmas I will return to Germany and celebrate New Year in Berlin.

First Post

This is the first post and a first test of my blog. I’m very excited to see it starting. The goal is to post once a months to provide up-to-date information about what I’m just doing.

At the moment, I’m working on my homepage, learn for the GRE (in order to apply for US PhD programs) and last, but not least (in fact, my major occupation) I’m teaching mathematics at the African Institute for Mathematical Sciences in Sénégal.