I’ve been told by more than one person over the years that I give decent math talks. These are the presentations that mathematicians give about their research at conferences, seminars, or colloquia.

While I don’t claim to know the perfect formula for a better math talk, I’ve given plenty of talks in my thirty years in the academy (over a hundred, at least) and I’ve seen many more than that. Some of those talks I’ve seen were good, but most were not. The happy news is that your talks can improve with deliberate effort on your part.

Giving talks is a significant part of our profession and a key element in landing your first academic job. My view is that speaking about our work is not an afterthought but should be weighted more highly by mathematicians. One way to excite people about mathematics is to give a great talk. Equally important, one way to turn people off is to give a bad one.

### Clarity

Mathematics is a subject that can inspire fear and loathing, and so mathematicians have our work cut out for us when communicating it. Mathematicians find the subject beautiful. The average person doesn’t. We have a difficult time understanding how people are not fascinated by our latest algorithm, our most recent counterexample, or our proof that spans multiple pages.

Clarity, or the simplicity and efficiency in how we express our thoughts in a lecture, is so important in a mathematics lecture. Let me be blunt: *be clear or go home*. Whether you are speaking to an audience of mathematicians, or a panel for a job interview, or science undergraduates, strive to be as clear as possible when speaking.

The primary function of a lecture is to convey ideas. Use your talk as a platform to express those ideas in a way that is *understandable*. Juggle the topics depending on their importance. Think about ways to reduce notation or jargon to make your points more relatable.

### Technicality

Avoid being overly technical, or risk losing your audience. I’ve seen it all over the years: speakers restating page-long lemmas, a slide with a dozen definitions, or a proof of a theorem that goes over twenty slides. You can guess how many of these talks I remember for their ideas rather than for the less palatable reasons.

Mathematics is, by its nature, a technical subject. But that is not an excuse for a speaker to overload an audience with details. The challenges and opportunities here are to find a balance between the technical and the mundane. Yes, you want to prove your amazing new theorem that will revolutionize your field. But first, think about how you can build up to that without losing the bulk of your audience.

There is a folkloric adage that* each math talk should contain at least one proof*. The missing part to that is to limit the number of proofs. The precise number of proofs to include depends on the audience. In a seminar full of experts, you will likely have more. In a talk for a broad audience, I may only have one or none.

Ask yourself which proofs you can omit or how you can reduce jargon and notation. Do you need the technical parts or can they be softened or eliminated?

### Limiting content

The cliche of *less is more* is highly relevant when giving math talks. Talks can fall apart when the speaker simply put too much material into the talk. The audience becomes bored or lost when talks go on too long.

Stop thinking of your talk as a *race*. You are not in a speed race, you are explaining ideas.

Mathematicians often have much material to cover and have limited time to do so. But realize that you are not going to say everything in your talk. You might not even say fifty percent of what you want. Details, as always, can be found in your papers. When speaking, I’m prepared to skip over material to finish on time.

When crafting your talk, think about limiting the number of slides and limiting the number of lines in any given slide. For the latter, I’ve been told to use no more than nine lines in a talk. I’d suggest even less. A slide with one line can be used for emphasis.

### Parting thoughts

Practice your talk as often as you need to so that you speak *extemporaneously*: neither overly scripted and not fully free-form. No one likes listening to a robotic delivery, and the audience will be annoyed if you ramble.

And remember to have *fun*. The best speakers enjoy giving their talks. If you aren’t enjoying giving your math talk, then don’t expect others to enjoy listening to it.

Anthony Bonato