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Mathematics EE Topic Advice (Combinatorics/Analysis)

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In short, I am an IB student working on a mathematics EE looking for a bit of advice. Importantly, though I make note of technical details in case any reader is familiar with them and would be able to provide more specialized feedback, I require only advice of a more general nature.

The following two paragraphs are a bit of (fairly technical) background on my EE and progress so far. They can be skipped if the reader is lost.

The question I had chosen (last year) was the following: "What mathematical properties of generating functions make them useful in combinatorial fields?" Quickly, generating functions, if the reader of this post is not familiar with their use, are effectively power series around 0 with a sequence of coefficients in the power series that have certain combinatorial meaning. For example, the power series corresponding to the function x/(1-x-x^2) in the open disk of radius (sqrt(5)-1)/2 is the generating function for the number of ordered partitions of an integer into 1s and 2s. (The fibonacci series shifted right by 1) Effectively, generating functions provide a way to leverage powerful analytic tools to solve combinatorial questions, specifically complex analytic.

My general outline is to begin with a brief introduction to the concept of generating functions, an introduction to the concept of an algebra and a quick proof of an injective algebra homomorphism from the algebra of functions complex analytic in some open disk around the origin to the algebra of generating functions which is isomorphic to the algebra of sequences, a connection with either combinatorial classes or binomial posets (I wrote a draft version of binomial posets and the combinatorial meaning of the convolution between, but am considering shifting to combinatorial classes and thereby avoiding defining posets and incidence algebras and also because they give a different and somewhat more interesting explanation) and finally a bit of the analytic side of things with basically the techniques that everyone already knows about using poles to get asymptotics on coefficients.

I have a few concerns. My first issue is that I either have to introduce a lot of complex analysis or content myself with the elementary real analysis taught in further level (which can't be done much with since real analytic functions can be very badly behaved -_-). I would much prefer the former, but I am worried about confusing the reader in a morass of definitions. Since only elementary complex analysis is required, would it be possible to include an appendix or appendices with the relevant background so that the familiar reader may simply skip them? If possible, I might also wish to do this with some other definition heavy material as well. (This was the approach taken in some of the books I read as souces such as Flajolet and Sedgewick's Analytic Combinatorics (what a boss book))

Another concern, perhaps more pertinent, is that I basically have no original research. Most of the ideas discussed in this EE are taken straight from other sources.

The other thing that I was wondering is a possible change in essay topic. Over the summer I did some mathematical research in the field of combinatorics. However, this research seems kind of unmotivated without reference to unpublished work. Is there anything I can do? Given that I've already written quite a bit on the first essay topic, would this even be advisable?

thank you for any help you can give.

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I am also trying to write an extended essay in maths about graph theory, optimization and the traveling salesman problem. This problem was not my original EE topic as I decided to change my topic over the summer holidays. Instead of trying to do a brand new topic and letting my old topic waste away I switched my EE and maths HL IA topics. This has made it easier for me to work on both pieces and not had the setback of starting over from scratch. On the subject of definitions I think that including only what ever is absolutely necessary to your paper at the beginning with more definitions that may enhance understanding as an appendix. The only thing to watch out for when making appendicies is make sure that what ever information is included in the appendicies is not vital for your essay. Your last concern about no original research is one that I have faced as well in my own EE. To make up for that lack I have put other peoples research in practice. I do not fully understand your EE but perhaps that can help you with your problem. Hope that I was able to help and best of luck with your EE.

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