[siva7]

"harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions."

I'm not sure there is another profession in the world where it's impossible to explain to a layman on what the winners of their most prestigious award have worked on.

[frakt0x90]

I bet she could do it. Maybe not in a sentence, but at least the Kakeya problem is easy to understand, so maybe these other things would be explainable by an expert. I would like to see them try at least!

fwiw, I feel the same way about biology "Lysing action of the (1,2)b-carotene receptive encephalopathy pathway" type shit.

[Jach]

There's been some new videos uploaded with her and the rest of the prize winners (and the winners of other prizes, like the Gauss Medal): https://www.youtube.com/watch?v=mTPyjR3qmTA But after watching them I still don't think they're great for understanding (I'm still mostly confused) but the human-interest aspects of the videos are something.

[Onavo]

> Lysing action of the (1,2)b-carotene receptive encephalopathy pathway

High school biology is enough to get a vague idea of this though.

[nerdsniper]

Right. With a college degree, I understand most of these words. I don’t have a clue about the impact of the research or how it fits into the corpus, but I understand the gist of the article title.

Despite my degree being in engineering, which involved 4 semesters of calculus, and 3 other semesters of math, (7 total) I have absolutely no fucking clue what’s going on in the Fields Medal description above. It’s only barely more intelligible than if it had been written in Chinese (which I truly cannot read at all).

I genuinely don’t understand it at all. Even words that I think I recognize like “harmonic” or “geometric” are useless to me because the actual terms are “harmonic analysis” and “geometric measure theory” and I have 0% clue what those are.

I don’t have to google any of the words in the example biology title.

[Ar-Curunir]

Did your 7 semesters of calculus include any courses that a math major would take?

For example, a single upper-division undergrad level course on probability will expose you to measure theory.

Similarly, harmonic analysis is something that you can take as a reasonably good senior in mathematics.

[j16sdiz]

I think most undergrad level course on probability expose you to measure theory _without_ telling you they are measure theory.

[nerdsniper]

I took multivariate calculus (I to III), differential equations, statistics, discrete math, and linear algebra.

I didn't take those courses you're talking about, but that's my point. I have a top <1% education in math in the nation, maybe top 2-3% of all college grads, and that's not enough to even scratch the surface of the summary.

[yablak]

if you took an engineering course that covered fourier transform, z transform, or laplace transform, then you have seen an introduction to harmonic analysis on the circle and the real number line.

typically when I see mentions of harmonic analysis together with differential geometry I start thinking of fourier transform on more fancy shapes (on the sphere, for example, the replacements for complex exponentials are called spherical harmonics)

my PhD was EE in signal processing and I took some extra math classes at the graduate level, and continue to read some of this for fun and profit.

I still cant understand most of the terminology in the fields medal citations. that probably requires a more thorough couple of years of graduate education in mathematics.

math is by far the furthest of the sciences in terms of depth of human discovery.

[Ar-Curunir]

Every engineering student in the US takes those courses. I did too. That doesn’t make us top 2-3% of college grads.

The stuff you mentioned is first year business for most math undergraduates. You should not expect to know most things in senior level math after taking just first-year math.

It’s like expecting to know computational complexity theory, or distributed systems theory, after taking first year CS.

[BigTTYGothGF]

Harmonic analysis is "just" working with harmonic functions which you have enough background for: f(x,y) is harmonic if f_xx+f_yy=0 (f_xx=2nd partial derivative w/r/t/ x, and so on). Higher dimensions work the same: f_{x1x1} + f_{x2x2} + f_{x3x3}+.... = 0.

[Onavo]

Well, the snobby pure maths folks look down on applied engineering math ;)

[fn-mote]

Try the Quanta magazine articles.

https://www.quantamagazine.org/series/fields-and-abacus-meda...

[goodmythical]

Quanta's video is also excellent: https://www.youtube.com/watch?v=0SNQ4HEOhSI

[mmooss]

That links to The 17-Year-Old Student Who Solved a Major Math Mystery, about Hannah Cairo.

[feyman_r]

This article explains the Kakeya Conjecture in amazing detail and understandability: https://www.quantamagazine.org/hong-wang-wins-2026-fields-me...

[CamperBob2]

Neat article. I still don't grok the Kakeya Conjecture particularly well, but thanks to the article I now understand something I never understood before, which is the idea of fractional dimensionality and what fractals have to do with it.

[gus_massa]

I used to do something similar (at a lower level), and my solution was to lie, just lie. A big part was about https://en.wikipedia.org/wiki/Wavelet_transform so my description was something like:

I work in something related to Image Compression, so when the computer has to download the image from Internet it's smaller and use less data. Anyway, I study the mathematical part, not the programming part.

The idea is that images usually have big plain parts like the sky or the wall of a house, so you use big blobs of "ink" to paint them. For the border you use smaller blobs of "ink". And very close to the border you use smaller and smaller blobs of "ink". In this method, all the blobs of "ink" has the same shape, the only difference is the size. Also, the plain parts are not perfectly plain, so you use some small blobs of "ink" there.

In a typical image, you need very few blobs of "ink" if you pick the shape of the blobs of "ink" correctly. So you can only send the position and size of the blobs of "ink", that is much smaller than sending all the information of the image. The hard part is choosing a shape of the blobs of "ink" to make this conversion automatically and very fast, without asking the computer to do something smart to select the positions.

If the listener has more technical background:

The blobs of "ink" have white "ink" in some parts and black "ink" in other parts. This correspond to positive and negative values and actually all the blobs of "ink" are an orthonormal base so the calculation is only a orthonormal base change, that is super easy and fast. There is no smart selection of the position of the blobs of "ink" positions, just a boring orthonormal base change.

If the listener has even more technical background:

Something something Fourier Transform.

I don't want to count how many lies that description has. Also, all the parts in this description were done by other persons perhaps 10 year before me. I think I only once compressed an image, just for fun, and got a tiny compression because it was a toy method (¿Haar base?).

[Jach]

At least you are aware of the lies. I'm reminded of Feynman's refusal to lie about the nature of magnets (https://www.youtube.com/watch?v=Q1lL-hXO27Q): "You'd soon ask me about the nature of the [rubber] bands." Even in his digression about "why" questions, in giving a very high level reason for why ice is slippery he adds "they say", since I suspect he knew at the time the usual explanation had problems, and only recently do we have better explanations (https://www.quantamagazine.org/why-is-ice-slippery-a-new-hyp...). But at the same time, I've thought, it can be fun to try and come up with nice sounding lies, or analogies, or oversimplifications, especially for some types of people who won't leave you alone about something until they think they've understood something (even if a lie), or who want something repeatable to tell their friends about so they can brag about you. Still I think honesty is a better policy, even if it's not as satisfying, and if departed from at least making sure both sides know there are convenient lies and oversimplifications. As a student I would be (and have been) very put out if the goal is to understand something on a technical level and the teacher starts with lies.

[gus_massa]

From the U(1) gauge invariance symmetry in QED by Feynman's work to magnets sticking to refrigerators, there are like 7 or 10 intermediate steps. Each one add a few approximations and simplifications and perhaps a few experimental constants. I guess a graduate in physics would need half a semester to take a comprehensive course explaining again all the steps.

The main problem is that research is mostly about very specific details. Let's imagine the rounding of metallic nuts for screws, in a society where nobody has seen a nut or a screw. So when people ask, you have to say something like "I make bridges, actually a small part that keep them tight, actually I don't make them but I study how they are more resistant for exaple to corrosion. You are asking what is corrosion? Corrosion changes the color of bridges and they get weak an may fall apart." Notice that I never mention "rounding"! I probably need like half an hour to explain the relation between the rounding of the nut, the alloy, the salt in the sea breeze, the daily heat and cold cycle, and perhaps other things that may be relevant in my totally fictional example.

> Still I think honesty is a better policy, even if it's not as satisfying, and if departed from at least making sure both sides know there are convenient lies and oversimplifications.

Yes, you can call my description an oversimplification. It's hard to keep a balance and make a nice explanation that last less than 30 seconds before people get bored.

[solomonb]

My experience talking with high level mathematicians is that they tend to know their subjects so well and are so excited to share it that they can and will scale their explanation to match their audience.

[_carbyau_]

I mean, they likely have practice!

Dating or maybe a significant other, parents, siblings, any BBQ event IE any social event outside academia where someone asks "So what do you do?"

[Turneyboy]

Not to forget mathematicians from other fields. They might be equally unfamiliar with most of our research topics.

Yes we do have plenty of practice adjusting our explanations to the audience.

[GuB-42]

Besides the usual jargon, what I find interesting is that the tradition of always naming things after their discoverers: Fourier, Falconer, Furstenberg, Kakeya. 4 names in one sentence.

Other fields do it, but it is almost systematic in math, and arguably, it makes things even harder to understand as people names are not descriptive.

[nine_k]

There are no way to name most problems both compactly and descriptively. The inability to attach an acceptably descriptive moniker to a problem leads to calling the problems (and solutions, theorems, etc) after the author.

[CogDisco]

But then there's the opposite in maths where there's an overload on "plain" terminology. "Normal" means a billion different things. Same as "regular" or "simple".

Omitting the human history of a field does not automatically make it easier.

[raverbashing]

Mathematicians might be good with math but their naming skills is atrocious

Same with their ability to summarize and explain things

And their ability to create mathematical objects that are similar, but weird in a funky way, to the actual real objects.