Transcription
Now, your breakthrough, if I recall correctly, came from reading someone else's thesis in a completely different field. I believe it was quantum chemistry or electronamics or something like that.
>> Yes.
>> Yeah. Okay. And it had to do with the LLAS operator and Green's functions. Tell me about that please.
>> Yes. I mean, in principle um I did my masters at CERN in particle physics. Then I moved on to um cancer research and machine learning because astrophysicists didn't want to take on machine learning at that time. And afterwards I moved to astrophys physics but in principle all I've ever done is analyzing data under mathematical principles. And then a friend of mine, he graduated in quantum chemistry. And so he asked me to um have a look at his thesis, which was more or less the mathematics of how to how to form molecules and all of the all of the the prescriptions, how we can understand which molecules can form based on all of the principles that we know. And if I then had a look at it, I realized gravity and electronamics have a lot of things in common. most most importantly the mathematics >> and I saw the equations and suddenly I mean he also described everything as local positions of electrons and then you had the uh the the how do you call this the the ions in the molecules where you have like the positive the positive um the positive um ions and then you have the electron cloud around this and when I saw all of this I thought this is my gravitational lensing problem. I saw the laplas operator. I saw all of the functions and how to solve it. And suddenly it was pretty obvious this how to solve how to solve my uh my problem to describe all of the degeneracies in this lensing formalism. So how can I wiggle around potential to keep all my observables invariant? Then I saw in this thesis that the mathematics is exactly the same. So I went back to my math book that I still have have on the bookshelf here and I found exactly the theorems in functional analysis to describe my lensing degeneracies. So all I had to do was copy the theorems from mathematics and translate them into the into the the the language of this gravitational lensing problem.
>> So why didn't others see what you saw? I think it was 30 years between when you made an application to an adjacent field. It's not even that far. It's it's adjacent. I mean, it's not directly, but it was 30 years or so. So, what did you see that others missed and what allowed you to see it?
>> I think I mean I'm the most astrophysicist most astrophysicists that I have met, they are more phenomenologists. They look at something and they have an intuition. If I have twice the mass, I have like that many lens power. Or if I have twice the mass, these things should run double fast or something like this. And for me, I cannot say much until I have written down the equations. So for me, it's the mathematical framework that in the end gives me the reasons to inter interpret physical things. So I think that this is why I found this degeneracy because I saw the mathematical formalism and I saw I can one to one I can translate this here and only afterwards I found out that this actually makes sense in physics. So I first solved the equations and I knew this is the solution. This must be right. But then I said okay I have the equations but now they need to get some physical meaning because most people they either live in the world of mathematics and then they have like variables parameters but then these parameters are usually called called called in a certain name and then they say but this is lambda like for instance the cosmological constant it has this lambda name and then people say yeah but this is lambda but this is a mathematical term I want to know what's the physical realism behind all of this what's the physical ical interpretation in this model. And after I had the equations, I then went on to understand what do all of my parameters and variables mean in my equation. Ko Roelli said you should not write down a single thing that you cannot attribute a clear physical meaning. And this is something that I really took to my heart. And I found out that the mathematical formalism completely shows you the degeneracies because it's obvious if I have a parameter that is called the reduced sheer I mean it's a small G this is what's in the equation and then I thought what does it mean and it means that I can only constrain the local shearing power. So what is the local shearing power? But the shearing power is independent of the mass that it takes to shear this. It's just okay. This is the amplitude of the shear and this is the direction. But how much mass physically it takes to create that shear is irrelevant. And so I saw that this is something that makes total sense physically because I do not know the total mass. That was obvious. So I cannot constrain anything that is related to the mass but I can create but I can constrain properties that are more or less something with respect to a certain mass. You always see ratios in these equations and you can ask why yeah well it's always something divided by the mass. So this is the getting rid of the degeneracies that we cannot constrain. So this was the I would say the nice and beautiful part when I realized mathematics works out and then I can learn something about the physics from these equations.
>> Can you tell me who else inspires you or has inspired you? So for instance you mentioned Carlo Relli with the gist of it is that don't write down anything that doesn't have a clear physical motivation something like that. Who else? What else?
Well, my mathematical professor um in my first first few semesters, he was really great. Professor Jagger, Professor Willie Jagger, and he had several honorary doctorates. And in lesson number three, you immediately knew why he was a mathematician doing um calculus, functional analysis, and in this direction. So, more the the um I would say the numerical part of and the practical part of um applied mathematics. And he taught me calculus, functional analysis and also a little bit of finite element theory. And whenever he did something it like proof theorem, whatever he did, he first explained what is it good for. And we had a lecture with a lot of people from biology, chemistry, physics, everybody was sitting there also mathematicians. But he always made sure that we knew the practical applications and what is the problem in the proof where we have to really take care that the reality and the proof still matches.
And this was really inspiring for me and this is I think what went through all of my data analysis that I always remembered that you need to make sense out of this and your mathematics should not I mean should not be somewhere in an abstract space. you you need to to be sure that all the requirements of your proof are actually fulfilled in your physical in your physical problem. This is from the mathematical side and another very inspiring person is George Ellis from Cape Town.
>> Mhm. He is the one who brought forward based on he did his PhD in Cambridge with Dennis Shyama and I think Dennis Shyama was one of the first persons in modern cosmology who tried to ask this inverse problem question and George Ellis and all of the collaborators from the inhomogeneous cosmology community that I very much like and appreciate. they have brought forward that this inverse problem approach should be pursued further and I find this very inspiring and this is half of the camp of the cosmologists who say let's have a look what's in the data and not model too much. You said something that stuck with me. You said that you can place infinitely many black holes. You can stack them in a null set. What are you talking about?
>> Yes. Yes. This is something that completely struck me when when I tried to derive the lensing degeneracies mathematically. Um this all lives in the very abstract notion of a subable space in mathematics and when I was studying solely spaces and all of these what does a function require to be integrated or what does a function require to be differentiated. As a student I thought who the hell needs this? Why on earth should I care? And as a physicist in third, fourth term, you do not encounter these issues because you're not as deep in the in the research that you would actually care. But then when I had this formalism to solve and I suddenly realized it makes a difference if my function is smooth, it's just differentiable twice. Is it even continuous? I mean, what do I know about this lensing potential? I have no clue. So I wanted to have um a function for my lensing potential that is the most agnostic in terms of what do I have to put in here? And suddenly I realized oh it makes suddenly sense to say my function my potential should just be integraable or maybe I want it to be smooth. Let's assume it's smooth. What can I say then? Or then let's just be completely agnostic. What happens if this function is not even stat is not even continuous also and then I discovered suddenly if I say the function should just have the minimum requirements so that it's integraable in my formalism I suddenly end up in these sole spaces >> and in this sole space if a function is integraable you can still say that if you change the contents under the integral meaning your function by a so-called null set, then you wouldn't change the integral. And since all you care for in this formalism is the final result of the integral, you are allowed to change your potential by this null set. And what is a null set? A null set could be in physics. Now you go from mathematics to physics, what does it mean in physics? It could be a countable set of black holes, a countable set of point masses. And if you now think in a physical sense, a potential that is really very nice and smooth, that's something completely different than a potential that can have a lot of point masses everywhere and is actually humpy, bumpy, full of black holes. And this is something that quite struck me back then when I thought suddenly this makes sense because this was this one sentence in um in one of the the books on strong graitational lensing. Oh, and we can also put some black holes in this uh potential. It doesn't matter. And I thought, where's this coming from? I don't know. And for me, this sounded disruptive. I couldn't imagine this. But the moment that I that I worked through the mathematics, it was obvious why I could do this. But this came from a much more I would say fundamental sound mathematical theory that I suddenly understood where is it coming from? But physically, I would say, does this make sense? Could it really be that I have an infinite amount, as long as it's countable, an infinite amount of black holes in my potential?
>> So, I would say mathematically it's clear you can have it, you have this freedom, but physically you could ask, is this reasonable? But the formalism gives it to you. So, your choice. The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new nutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science. They analyze culture. They analyze finance, economics, business, international affairs across every region. I'm particularly liking their new insider feature. It was just launched this month. It gives you, it gives me a front row access to the economist's internal editorial debates where senior editors argue through the news with world leaders and policy makers in twice weekly long format shows. Basically, an extremely high quality podcast. Something else you should know about is that if you go to their app, they not only have daily articles, but they also have long- form podcasts with their editors and writers. This is also available online. Whether it's scientific innovation or shifting global politics, The Economist provides comprehensive coverage beyond headlines. As a toll listener, you get a special discount. Head over to economist.com/toe to subscribe. That's economist.com for your discount.