Giving a Talk

How to give a talk

Here are a few wisdoms from our very own First Pedro on how to give a good talk in PedroSoc (and elsewhere!) The IT guy found the original message a little bit too aggressive so decided to edit it a little bit (and, as the WCR guy, added his view as a postgraduate). If you want the original version you could check out the section below.

Rule 1 - What should you aim for?

A PedroSoc talk should be a hands-on experience for undergraduate mathematicians to explore a certain mathematical topic beyond what’s covered in standard undergraduate lectures.

Rule 2 - Take care of undergraduates!

PedroSoc is, unfortunately, not the best place for you to give a research talk.

This does not mean you cannot present your original research here, though bear in mind that many of the audiences here are undergraduates.

Where is the line? It should become clear as you read through our advice below, but if you are impatient, you can skip to rule 8.

Rule 3 - No broad overview!

A broad overview of a certain branch of mathematics is equally undesirable.

Instead, we prefer having a talk that walks through the proof of a certain theorem, or exemplifies a certain commonly-used technique. Again, we are here to help our audience to interact with the maths you are presenting. You should always aim to prove or “semi”-prove any statements in your talk.

Rule 4 - Technical details?

This does not mean one must give all technical details of the results used in your presentation.

Rather you can (and is often better to) rely on the intuition of the listeners.

Example. One can explain with pictures that the fundamental group of the circle is (isomorphic to) Z (the set of integers), pointing out it needs more technical proof.

Rule 5 - Hard or unproven results.

Hard or unproven results can be used, as long as they are stated clearly, intuitively explained and exemplified.

Example. one might use Weierstrass Factorisation Theorem to prove something regarding the Zeta function, but one should state the theorem clearly and give examples that illustrate why the result is true (e.g. with trigonometric functions).

Rule 6 - Take away!

You should expect a majority of the audience will be able to interact with the topic you are presenting after listening to the talk, i.e. knowing why the theorem is (or should be) true, and/or how to apply the theorem.

Ask yourself: what will the listeners take away from that, and why should they care?

A BAD TALK IS ONE IN WHICH THE STUDENT DOES NOT LEARN HOW TO DO OR PROVE THINGS (EVEN IF HE LEARNS THAT SOME THINGS ARE TRUE).

Rule 7 - Timing

Plan your talk for the amount of weeks you are given. We have booked the lecture theatre for two hours so please take your time to disseminate the results.

If you need two (or more) weeks, do ask for them, but do not rush to fit two weeks worth of content in one.

Make sure to leave time to address questions/misunderstandings from the audience, and even second guesses on your own material (we all have been there)! It is natural to cover much less content than we think we will :’)

Rule 8 - Original Research

Try not to make your original research the main focus of your talk.

Yes, we welcome talks on original research. It is, however, more preferable to have a talk on a simpler result/technique that is accessible by undergraduates, and note at the end that your current work is one of the many applications of the result/technique.

We aim to have at most two per term talks that focus on original research, since we want to make sure everyone has a space to share their voice (and who does not love classics like the Dirichlet's Prime Number Theorem?). Therefore, don't be too frustrated if the committee (or the Pedro) asks you to defer your talk to next term for this reason. In fact, it is preferred to give a preliminary talk this term to prepare the audiences for your actual research talk next term!

Rule 9 - Have Fun!

Most importantly: have fun and let others have fun, after all, it is maths!

It is OK to make mistakes on the board and to hesitate to do things! Don't worry too much! Talk about something you enjoy and others might enjoy. This is not the defence of a thesis, but rather a friendly gathering of people who are willing to support each other and have fun with that.

Examples

This is a good time to give a few examples. (Note: what makes a Bad talk is not the topic one chooses, but rather a bad (overly or not enough technical) approach to that topic.)

Good Talk	Bad Talk
Proving the Dirichlet's Prime Number Theorem.	A broad overview of analytic number theory.
Prove Stoke’s theorem for differential forms.	A survey on alternating tensors that leads to nowhere.
Go through the “interesting” (cohomology) part of the proof of Jordan's Curve theorem (and explain how it is used to study 2D dynamical systems, i.e. proving Poincaré-Benedixon theorem).	Go through all (especially the “boring" analysis-topology) technical details of the Jordan's Curve Theorem.
Explain how to solve Riemann-Hilbert (RH) problem and give examples of how other problems can be used to solve an RH problem.	Recent researches on how to simulate equilibrium measure (an application of RH problem)
Motivate Ito’s formula and demonstrate how stochastic processes give rise to certain Partial Differential Equations (e.g. Black Scholes, Schrödinger equation).	Go through all details of construction of Brownian motion and/or stochastic integrals.
Prove some algebraic graph theory results assuming the spectral theorem.	Prove the spectral theorem itself, since this is already done in regular courses… … or focus on your research on graph partitioning and rush through the algebraic graph theory results. (It’s ok to add your research as a side comment, but not as a focus).
How to use Lovász Local Lemma (LLL) in application for combinatorial proofs? (without actually proving LLL itself).	Stating a stronger version of LLL without any concrete proof or application of the results.
Prove the Fundamental Theorem of Algebra (using that the fundamental group of the circle is isomorphic to Z, Liouville's Theorem and Galois Theory)	Introduce Galois Theory, Complex Analysis and fundamental groups all at once without having time to explain the Fundamental Theorem of Algebra and its proofs, which was the goal.

Original Version

The is the original version of The First Pedro's message sent during his conversation with the IT/WCR guy. If there are disputes between this version and the above edited version, then the Pedro will resolve it. (All times are GMT+1)

[10:16 pm, 19/09/2024] Lack Pedro: Hi. Just my "original" version as I wish it to be presented to save you some work:

[10:16 pm, 19/09/2024] Lack Pedro: Guidelines for a PedroSoc Talk:

This is not a presentation on "new results in the area X", in other words, your main aim is not to present interesting results to people, but actually to prove or "semi-prove" them.
This does not mean one must prove careful all the results used. Rather one can and should rely in the intuition of the listeners. For example, one can explain with pictures that the fundamental group of the circle is Z, pointing out it needs a more technical proof.
Hard or unproven results can be used, as long as they are stated clearly, intuitively explained and exemplified. For example, one might use Weierstrass Factorisation Theorem to prove something regarding the Zeta function, but one should state the theorem clearly and give examples with trigonometric functions.
The listeners has to improve as a mathematician by listening to the talk. One might structure the talk by proving a certain theorem or exemplifying a certain technique. That is the allowances in 2 and 3 are made with the goal of proving a certain given theorem. A BAD TALK IS ONE IN WHICH THE STUDENTS DOES NOT LEARN HOW TO DO OR PROVE THINGS (EVEN IF HE LEARN THAT SOME THINGS ARE TRUE).

For example, a good talk can teach how to use Lovász Local Lemma in application for combinatorial proofs without actually proving the theorem. A bad talk states stronger versions of this lemma only hinting at proving them without any concrete proof or application of the results. (As an easier example) one might give a goog talk by assuming the spectral theorem for finite dimensional self-adjoint operators to prove certain results using graphs adjacency matrices, without actually proving the result. A bad talk would be to prove this result, since this is already done in regular courses.

Finally, a good talk would be to prove the Fundamental Theorem of Algebra (using that the fundamental group of the circle is Z and using Liouville's Theorem and Using Galois Theory). A bad talk would be to try to introduce Galois Theory, Complex Analysis and Fundamental Group theory all at once in the same lecture without having time to explain the Fundamental Theorem of Algebra and its proofs, which was the goal.
Remeber to always ask yourself: what will the listeners take away from that?
Plan your talk for the amount of weeks you are given, including misunderstandings by the audience, questions and even second guesses on your own material (we all have been there). If you need two weeks, do ask for them, but do not rush to fit 2 weeks worth if content in one. It is natural to cover much less content than we think we will, so do plan for that!
Most importantly: have fun and let others have fun, after all, it is maths! It is OK to make mistakes in the board and to hesitate to do things! Don't worry too much! Talk about something you enjoy and others might enjoy. This is not the defense of a thesis, but rather a friendly gathering of people who are willing to support each other and have fun with that.
(additional topic prompted by the advisory comitee) If you wish, you are more than invited to share your own original research. However, we aim to restrict this to, at most, twice a term, since we want to make sure everyone has a space to share their voice (and who does not love classics like Dirichlet's Theorem?). Therefore, don't be too frustrated if you have to wait until next term, instead you can give a talk this term in preparation for that! Moreover, you can also give a talk on your topic of research and state (at the end of your presentation) what your current problem of research is. When presenting your own research, remember that you still have to follow the tips above (this is not a postgraduate talk with people familiar to that area).

Best wishes,

the First Pedro (Pedro Lack)

[11:20 pm, 19/09/2024] Lack Pedro: Don´t forget my version (it might even be a link to a pdf document, it doesn't need to be all the text in one page).

[11:23 pm, 19/09/2024] Lack Pedro: (The PDF Document of his wisdom.)

[11:23 pm, 19/09/2024] Lack Pedro: If you prefer, here it is!