How to make a 40-minutes presentation about a research paper I've read?
The most common mistake I observe in the presentation of mathematical work is for the presenter to try to cram in too many definitions, equations, derivations, etc., and then rush over them to try to make their talk fit into the allocated time.  Occasionally a misguided student will also decide that it is impressive to show off vast tracts of complicated algebra.  This often makes it difficult/impossible for the audience to follow, even when the audience consists of mathematically-literate professionals (e.g., professors in the faculty).  This error is ubiquitous among students, even at graduate level.

The general problem that occurs here is that students fail to simplify their presentation to a level that allows it to be presented clearly, and they also massively underestimate the amount of time required to explain a single equation/result.  As some general rules for a mathematical presentation, I would suggest the following dos and don'ts:




DO put the problem in context: Put this problem within the context of the broader subject.  Use the literature review in the paper to get an idea of where this work fits in, and what it is adding.  Give a clear statement of what this paper adds, and its limitations.  If it is a small part of working towards a larger theoretical goal, explain the larger goal, and explain how this paper adds to it.  If you found the paper particularly interesting, say why.
DO simplify the problem, then simplify it again, then simplify it again: You should present the problem in your research paper in its most fundamental simplest form.  Minimise the number of variables you are dealing with and minimise notation.  (It is generally best to be consistent with the notation in the paper, but you can still minimise this.)  Present the problem clearly, with a small number of mathematical statements and lots of intuitive explanation.  You do not have to present every equation describing the problem/model used in the paper.  Make professional judgements about which parts need to be shown as mathematical statements, and which parts can be described in general terms without mathematical equations, and which parts are side issues/distractions that you will ignore.
DO obey the crow epistemology - people can remember three things: Do not burden your audience with remembering lots of variables outside the context of their equations.  Pick no more than three important variables in your problem and drum their meaning in with repetition.  Assume your audience will forget the meaning of every other variable.  Definitions should generally be on the same slides as the first statements they appear in, so people don't have to remember them.  I like to give equations with arrows pointing to the variables with their meanings, and I back this up by talking about each of them, and giving reminders.  Some people like to colour-code variables, or use other memory aids to help.
DO allocate enough time - Presenting unfamiliar mathematical statements = 5+ min: If you present a mathematical statement that is not common knowledge to your audience you must stop, and patiently explain the statement, including the meaning of each variable in the statement.  Give an overview of what the statement is saying and what each of the variables is.  You must do this slowly, possibly with repetition to stress key points, and you must give your audience time to "mull" over it, before you proceed.  For every statement like this you have in your presentation, allocate at least five minutes to talk about it; don't rush.
DO allocate enough time - Presenting familiar mathematical statements = 1-2 min: If you present a mathematical statement that is common knowledge to your audience, you will still need to stop, identify the statement, and explain the meaning of each variable in the statement.  If it is a standard statement with variables in standard notation you can probably do this pretty quickly, but you should still allocate at least a full minute to describing it.
DO show graphs (if relevant) and explain them clearly: If the paper presents some of its results graphically, this is usually the simplest tool for presenting the results in a simple manner, that can be absorbed by the audience.  Use your judgment here, but as a general rule, graphs are good.  Again, you must stop, and patiently explain the graph, the axes, the lines, etc.  Explain the intuition of what the audience is seeing.  Tell them why this line goes this way and this other line goes that way.
DO leave time for interjections/questions: Make sure you have left enough time at the end for a reasonable amount of questions.  You may also get some interjections as you go through your presentation, so make a time allowance for that too.





DON'T show a long list of definitions: Under no circumstances should you present a notational slide with a long list of definitions.  If you have a slide like this in your presentation, I am not bothering to read it.  Instead I am watching "Heat" in my head and wishing I could get out of watching such a shitty presentation.
DON'T show a wall of algebra: We get it - you're very clever and you can do algebra.  Unless the derivation of a particular equation is the whole point of the paper you're presenting, we don't want to see that shit - get rid of it.  If you have a slide like this in your presentation, I am not bothering to read it.  Again, I am watching "Heat" in my head and wishing I could get out of watching such a shitty presentation.  Instead of bombarding me with a wall of algebra (which you don't have time to explain), explain the methods and the results (e.g., "To get to the next part, we need to complete the square in this exponential and then use the law of total probability to simplify.  After you do that, you get this next result").
DON'T be a flasher: Whoa!  What was that slide you skipped through in two seconds?  It was your list of references?  If you're not going to take me through it, why flash it up there at all?




I hope some of this is helpful to you.  As a general rule, the quality of student mathematical presentations is so woefully bad that you have a low bar to reach.  If you're the guy people can understand, and you are not rushing them and throwing walls of algebra at them, you will emerge as champion.