So recently I've been attending seminars for the graduate students (and VAP's) at my local university, and after yesterday's seminar, the professor asked if I would like to give a seminar next quarter; he noted that the seminar could be on any topic of my choice as long as it was relevant and interesting. The seminar groups are entitled "Fractal Research Group" and "Mathematical Physics and Dynamical Systems". These titles are fairly loose, albeit the topics still seem to relate to the group in some way.
The last few lecture titles:
Framework for Single Image Super-resolution Methods and Digital Photograph Expansion for Natural Images
Applications of Quantum Complexity Theory to Parallel Computation and Public Key Cryptography
Interior and Trace Embedding Results for Variable Exponent Sobolev and Maz'ya Spaces
on "Bad" Domains
Pressure and Zeta Functions for Graph Directed Markov Systems
So naturally these are high level topics (many professors and assistant professors also attend the seminars). My question is as follows:
Does anyone have any suggestions or ideas for a topic of discussion that I can present on?
Best Answer
An idea which absorbed me the last couple of years and stirred true explorative enthusiasm and joy was the concept of fractional iteration of functions, here of exponentiation (aka tetration). It has a very wide range of aspects, and fits with the subject of complex dynamics. I'm a complete amateur but it has been worth for me to become confronted with the difficulties (and partial solutions) up to undergraduate level and learn from historical articles in archived mathematical journals which can even be found online. In the "tetration-forum" there are even graduate mathematicians involved and have discussed/discuss things on higher levels (where I usually could then no more follow) and thus I think this subject has also some flair as a subject for a whole semester. Students can explore different approaches, try their own intuitive ideas and learn from detecting the subtile reasons for possible or even likely failures.
My own approach, for instance, was an accidental (re)discovery of the concept of Carleman-matrices for the composition/selfcomposition of functions - and for some of those matrices fractional powers can be defined to determine formal powerseries for fractional iteration; difficile aspects of convergence radii occur etc...
For some examples of my own exploring you might take a look at my site at the subsection for tetration and a very nice eyeopener is possibly this comparision of 4 more-or-less naive and one seriously discussed methods, the latter namely of Hellmuth Kneser in the 40ies of the previous century, which I'd just updated today.
If this proposal sounds as a possibly reasonable base for such a course to you, we could continue the discussion about a more precise workout of list of topics by email (or via the tetration-forum)