In the beginning of the seminar, you will decide on one among a number of research papers that are being on offer for the seminar, and with this you will get an advisor to guide you through the preparation of the talk. It is important to understand that the paper is an entry point for your preparation. You will not necessarily have to present all material from this paper, and the talk may also use other sources: literature cited in the paper, websites, demos, etc. In this sense, the talk is not so much about the paper itself, but about the topic addressed in the paper.
It is important that you get yourself informed about the current state of research on the topic. Some more recent results may not be covered by the paper that you are reading. For hot topics, it may even happen that new results come out while you are preparing your talk. You are not expected to present such new developments in detail, but you are expected to know about them and be able to answer questions concerning the topic's state of research.The seminar has two goals. On the one hand, you should learn to put together an interesting presentation. This means that you must do some research: get an overview about the topic at hand; what is it all about, and why is it interesting? What is the background, and what have other people already done? Then you should develop an idea of what to present. Your advisor will be of help, but your own initiative is important. For example, you might discover that another paper about the topic is a much better source for material that fits your idea. The second goal is that the audience learns something worthwhile, and you should seriously think about what you want this to be. It can be very painful to endure talks that have not been prepared with this goal in mind, and a bad or even failing grade is a foreseeable consequence.
Usually you have a couple of weeks for preparation during which you are invited to contact your advisor. Here are the main guidelines.
Use electronic slides! It doesn't matter whether it is
Powerpoint, Keynote, Slitex, Beamer, or still other systems. These
days, the only serious alternative to electronic slides are pure
blackboard talks, but giving good blackboard talks is
something for the experts. Well-prepared slides guide you through your
talk almost automatically. It is usually a good idea to change media
and go to the blackboard once or twice during your talk to present a
crucial proof or draw an important figure. This tells the audience
that something special is going on now.
Hint: Please avoid using green colors on white background, as these are either invisible or hurting the eye on most beamers.
Show many pictures! Yes, drawing good pictures takes a lot of time, but it is an absolute must. It is very difficult for the audience to understand even simple definitions without an illustration. You have prepared this for weeks and (hopefully) know it by heart, but the other people hear it for the first time. Pictures add redundancy, so they allow the audience to cross-check whether their understanding of the previous formal definition was correct. Explaining pictures also slows down the presentation and gives people the time necessary to absorb the material.
Use large font, and show one thing at a time! Overfull slides in small font are a nightmare. A slide only helps if it corresponds to what the speaker is telling at the very moment. Slides that stay on for minutes while the speaker is simply droning on makes the audience lose attention. There may be technical slides (that explain an algorithm, say) which are necessarily somewhat denser, but these should be exceptions. Every slide should focus on just one issue.
Be clear and mathematically correct! If you present an algorithm that solves a certain problem, you should make sure that you properly define the problem in the first place. When you use some fact without proving it (which is perfectly ok), you should say whether this fact is (a) easy to see, (b) requires a simple proof, or (c) requires an involved proof. Mathematical definitions and theorems should be stated completely and correctly. If you do not follow this guideline, two things happen: some people just lose attention, since they didn't fully understand the problem, got confused by unclear things that you said, or still try to figure out whether that fact you just used is obvious or not. Other people (the ones that grade you, mainly) will ask you to clarify things. In general, questions during the talk can be very inspiring and lead to interesting discussions, but if it takes five minutes of discussion just to get an unclear definition right, this causes unnecessary delays.
The essential rule here is: keep it simple, but do the simple things well. There is only so much you can say in 45 minutes, so it makes sense to prefer simple over complicated material. Our offered selection of topics supports this approach.
Understand what you present! This may seem self-evident, but let us do a concrete example that makes the point. You mention (and/or write on the slide) that the existence of this and that object follows from a standard result in analysis. Now somebody asks whether you can say what this standard result is. If you don't understand what you present, you now say something like "I don't really know, this is just what they wrote in the paper". The resulting impression on the audience is very bad. If you do understand what you present, you know it/have looked it up/asked your advisor so that you can say "The standard result is that every continuous function attains a minimum over a compact set". As a general rule, you must be able to explain everything that appears on any of your slides. If you can't, don't write it. The fact that the paper does not explain it is no excuse; scientific papers often omit details, and it is part of your preparation to fill them in if necessary.
Understanding what you present is also the key to being more confident and less nervous. It is normal to be nervous, but this is hugely amplified if you don't have full understanding of your presentation.
Prepare for delays! Despite the best preparation, you may run out of time, and this is not a problem per se. After all, the audience is not fully predictable, and your presentation speed isn't, either. But you must prepare for such a case. It is not an option to stop in the middle of an important proof, or to simply go overtime and wait for somebody else to stop you. The best way is to have some material in your talk (preferably close to the end) that you could skip if necessary, without compromising the talk on the whole. For example, this could be an additional result that is not necessary for understanding the big picture that you present.