Thank you for joining. Good morning, Guten Morgen. It is for me a great pleasure to introduce

Nick Trefessen. Most of us know the name, of course. Some of us had the opportunity

to meet the person in the past. Anyhow, he was kind enough to be with us for one week.

He was also visiting the University of Biodiversity, for which we ran a friendly and efficient

cooperation. So Nick was a bachelor student in Harvard. Then he got a PhD in Stanford

University. His topic at the time was pneumatics for wave propagation. Actually, somehow this

is the work I know from him the most. We use very much his seminar papers in the late 80s

on this topic. Then he was a postdoc in MIT, in Cornell Institute, and also in Cornell.

For the last 25 years, he has been a professor in Oxford University. I will not tell the

many honors, the many positions to occupy. I think he was also president of Siam at some

time. When you read his Wikipedia page, you can see a list of the books he has written.

There is something that attracted very much my attention. For instance, it says that in

2013, he proposed a new formula to calculate the body max index of a person. Then I asked

him whether he was cooking an index that would fit better. He said parameter. You can always

modify the definition of the index so that you are perfect. I don't know whether it was

his motivation. This was indeed very successful. You can also have a look at this.

His last book, I wanted to mention, he has done many things in numerical analysis.

Spectra, pseudo-spectra, numerical waves, numerical algebra in general, approximation

theory. He has several books. The last one is something that you might be interested

in reading. It's an applied mathematical topology, Siam 2022. This has been translated to several

languages, including Chinese and Spanish. I think this is an inspiring reading for all those that

either because you are a beginner or you are an amateur mathematician, it's a very inspiring and

insightful reading. He was kind enough to accept our invitation to be here in Erlangen to take

part in this CDSO for MOD, Mathematics of Data Center lectures. Without any further comment,

I'll let you talk. Thank you again for coming. Also, thank you for the 56 colleagues that are

with us online. This online thing, although we all have mixed feelings about this, this allows us to

share and multiply ourselves. That's great. Thank you.

Thank you, Enrique, and thank you, everybody. There's a nice group of people here, and I guess

quite a few of you online. Thank you to everybody for being here. That introduction is very nice.

By the way, I'm not a good enough mathematician to make a formula that makes me perfect. I tried.

I want to talk about a new algorithm for rational approximation, but really, the talk is about

how we might use rational approximations if we have a good algorithm. Something very unexpected

happened to us in 2016, in fact. Eugene Nakatsakasa is my colleague at Oxford, and Olivier Sette was

visiting from Berlin, and we decided to work on rational functions. We tried so many different

things, many different angles, and we didn't quite get the final thing. Finally, it all came together

and became simpler than we imagined and much more effective than we imagined. We tried to

get the final thing, but it was more effective than we imagined. We seem to have stumbled across

an algorithm that, for the first time, makes it easy to compute near best rational approximations.

A rational function means p over q, polynomial divided by polynomial. The idea is we can now,

not provably, but in practice, almost always rapidly compute good approximations to data or

complex approximations. The talk will show you that. The code is called AAA, which stands for

Adaptive Antulas Anderson. The code we use is in CHEBFUN, so I hope you all use CHEBFUN.

I put it on the cover sheet here just to emphasize what a simple thing we are computing.

You have data point z, so that's a vector of points, real or complex, and then you have

function values f of z. You execute this call, and typically in about a tenth of a second,

it comes back with a rational function that matches the data to 13 digits of accuracy.

There's a tolerance, and our default is 10 to the minus 13. That's a rational approximation to f

on the set z. To say a word about the people involved, these have been my main co-authors in

this area. There are a number of other papers related to AAA that I haven't been connected with,

but these are the things I've done. If you're interested in any particular topics,