Essays and presentations

An essay will be due in class on 11/20. You will all be asked to give a thirty minute presentation in class. This will occupy the last four classes of the semester. Both the essay and the presentation will be graded.

Here are suggestions for topics.

• Braid groups


• Möbius transformations: the symmetries of the hyperbolic plane


• Group theory on the Cornell campus. This one might take some imagination! Search for and write about group theory on the Cornell campus (e.g. in the architecture, in the flora, or in any other setting you can find).


• Some aspect of group theory in physics


• Group theory and spectroscopy. This article might be a good starting point.


• Group theory and crystallography. See here for example.


• Cayley graphs. I think the Lovász Conjecture is a particularly fun question in this context.


• Group theory in campenology


• Growth in groups. See Groups, Graphs and Trees by John Meier, for example.


• The lamplighter group. See Groups, Graphs and Trees by John Meier, for example.


• Burnside's problem


• The symmetries of the Convex regular polychoron (the four–dimensional analogs of the Platonic solids)


• The history of group theory: how did the modern concept of a group emerge? The wikipedia page on group theory is a reasonable start.


• The Word Problem for groups. See Groups, Graphs and Trees by John Meier, for example.


• Group theory and Klein's Erlangen Program


• A review (or appreciation) of the book Symmetry by Hermann Weyl

Alternatively, you are welcome to come up with your own topic. Explaining links between group theory and other academic areas of interest to you, would be particularly welcome. Two provisos are that you may not duplicate work you have done elsewhere and there must be some genuine group theory involved.

To gauge the appropriate level and style, imagine you are writing for your classmates. Explain why your topic is interesting and important, and convey the main ideas. Some subjects will allow you to go into the details of the mathematics. Others lend themselves better to a more journalistic approach, as the mathematics may be too technical or too substantial. Whatever style you choose, try to include at least some mathematical detail. Please aim for about 2500 words. Please take care over your presentation and try to make your writing fluent and compelling. I do not insist you type your essays, but I do recommend it. LaTeX and its variants can produce beautifully typeset mathematics, but other software is also viable.

Cornell's Academic Integrity Code states that a Cornell student's submission of work for academic credit indicates that the work is the student's own. All outside assistance should be acknowledged, and the student's academic position truthfully reported at all times. Please include a full bibliography of articles, books, web sites etc. Any text lifted from a source should be presented as a quotation. If you base some exposition on someone else's writing, you should acknowledge the source. You may not collaborate on these essays.

— The aim of your talk should not be to convince me that you have understood a lot of mathematics. Rather, your aim should be to communicate some interesting mathematics to your classmates.
— Choose carefully what material to present in detail and what to give in overview. Where the right balance lies will depend substantially on the subject matter. Time will not permit you to prove many results, so choose judiciously what proofs to omit (and be open about it!). I would like to see at least one proof in every talk, though.
— You might be wise to deliver a practice run-through of your talk to a friend. Or you could pair off with a classmate and practice.
— Please give me (e-mail may be easiest) a brief outline of your talk at least a week before it is due to take place. Just write five to ten lines saying what, roughly, you plan to cover.
— Please attend everyone else's talks. These talks are a continuation of the course: subjects we have learnt about will be developed further, and new important topics will be introduced. I hope you will support and encourage each other, and learn about different ways of giving talks from each other's example.