Every two years, the Australian Mathematical Society (AustMS), with support from the Australian Mathematical Sciences Institute (AMSI), organises a lecture series.
It's called the Mahler Lecture series, where a prominent mathematician tours Australian universities, giving lectures for audiences at a variety of levels.
This year's guest lecturer is Dr Holly Krieger from the University of Cambridge, where she is the Corfield lecturer in mathematics as well as a Fellow at Murray Edwards College. Before that, she was at MIT.
Dr Krieger is very active in her outreach activities and promoting mathematics to female students and young researchers. In addition, her mathematics videos with Numberphile have millions of views on YouTube.
In this episode, Dr Krieger talks about her research, her outreach, and her Australian maths tour.
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Thank you to Matthew Mack for this transcription
Every two years the Australian Mathematical Society (AustMS) with support from the Australian Mathematical Sciences Institute (AMSI) organises a lecture series.
It's called the Mahler lecture series where a prominent mathematician tours Australian universities giving lectures for audiences at a variety of levels. This year's guest lecturer is Dr. Holly Krieger from the University of Cambridge, where she's the Corfield lecturer in mathematics as well as a Fellow at Murray Edwards College. Before that she was at MIT. Dr. Krieger is very active in her outreach activities and promoting mathematics to female students and young researchers. In addition, her mathematics videos with Numberphile have millions of views on YouTube. Dr. Krieger recently spoke in Melbourne as part of the lecture series and that's where our Sarah Belet from AceMS at Monash University caught up with her for this episode of The Random Sample.
Welcome to The Random Sample, a podcast by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers. I'm Emily Duane. In this show we share stories about mathematics, statistics and the people involved. You can learn more about what the Centre does by visiting our website acems.org.au. That's acems.org.au. Here is the show.
Sarah Belet: So today we're talking to Dr. Holly Krieger who's visiting Australia to give the Mahler lecture tour. Holly, welcome!
Dr. Holly Krieger: Thank you!
Sarah Belet: What's it like being in Australia? Is this your first time coming to Australia?
Dr. Holly Krieger: This is my first time in Australia and I think it's sort of an unusual experience to actually go around and see so many different cities. Because I'm visiting I guess 10 different universities in the course of the month, and so I'm getting to see quite a few different places.
Sarah Belet: lovely I think early we were talking about how you're experiencing like all this smoke from the bush fires which is kind of a... it's not a common experience in Australia but you've seemed to come at an interesting time.
Dr. Holly Krieger: Yeah I mean this is sort of the Irish curse about interesting times.
[Laughter]
Sarah Belet: Holly do you mind telling me more about your career progression and what eventually took you to take up a post at Cambridge University?
Dr. Holly Krieger: Yeah definitely so my position right now at Cambridge is that I am a lecturer which is in the US system it would be the analogue of an assistant professor for example. The mathematical career progression I had is one that is relatively typical for academia, which is that after undergraduate I went on to study for a master's; I did a year at Michigan State and then I moved to University of Illinois at Chicago, and continued on there to a PhD. After I finished my PhD I went to MIT in Cambridge, Massachusetts for my postdoctoral position for three years and then I took up this permanent job at at Cambridge - the other Cambridge, the original Cambridge. So yeah so that's pretty typical I think once people decide that academic careers offer them that kind of progression.
Sarah Belet: Excellent. Could you tell us a little bit more about this lecture series that you've come to Australia to give?
Dr. Holly Krieger: Yeah so this is a fantastic lecture series it's given every other year. It's in honor and in memory of Kurt Mahler who was originally not actually Australian but did quite a bit of his significant work in Australia at ANU. He was a fantastic mathematician with a particular focus and number theory so the study of sort of the basic whole numbers, the building blocks of all of the mathematics that we do and really made a lasting sort of very broad contribution. The lectureship was established to bring a mathematician over to Australia and give this series of lectures to the various Australian universities and bring pieces of their mathematics which generally the person has chosen so that their research interests have some overlap or some connection to Mahler's work, to bring their mathematics to to Australia.
Sarah Belet: So how does your research overlap with Mahler's work?
Dr. Holly Krieger: Oh that's a good question! So I mentioned that Mahler was particularly focused on number theoretic questions and in particular the study of what's called Diophantine approximation and transcendence theory. This is the study of the types of numbers that we work with. So we have the counting numbers, you're in school and you learn like oh there's not just two oranges or two apples or two slices of pizza there's a thing (concept) called 'two'; and I can multiply things by two and I can add two to things. Then you learn "Okay I want to multiply" that means then I want to be able to divide so that I can undo what I just did when I multiply things. Well then you need fractions, so rational numbers as mathematicians would usually refer to them.
What Diophantine approximation is is the question of how well real numbers (sort of you know the more complicated numbers like \(\pi, \sqrt{2}\) etc.) can be approximated by these rational numbers. So that piece of Mahler's work which is not everything he did by any means, but one of his focus is, that sort of work has connections to my own because I study analogous questions inspired by Diophantine approximation, Diophantine geometry, in what's called complex dynamics which is the study of long-term behavior of chaotic systems. Wow, [Laughter] that was quite a mouthful!
Sarah Belet: So when you give this lecture series are you talking about your research in particular or are you talking about topics that are sort of slightly related to your research, or is it like a variety of different topics and lectures that you're giving?
Dr. Holly Krieger: so it's a little bit of both so the university talks that I'm doing for the department they do tend to be quite closely connected to my research or at least sort of inspired by some some particular piece of work that I've done. One of the public lectures was focused on the Mandelbrot set which is this sort of famous picture but it's also an objective serious mathematical study in complex dynamics. The other is on the mathematics of life and that's a little bit further removed from my own research - there's connections with chaos theory and dynamics but my research, I have to say, doesn't have quite so much application to the real world in general.
Sarah Belet: That's fair enough so what brought you into doing research in the field that you're looking at?
Dr. Holly Krieger: Oh that's a good question. So I had initially I had always been interested in these kinds of Diophantine questions. So I mentioned Diophantine approximation, there's a related field called the Diophantine geometry which is basically about finding solutions to equations involving whole numbers. I mean you would think we would know everything that there is to know about this kind of thing but we don't. e.g. like Fermat's Last Theorem is a very famous, relatively recently solved question in this field. As you can imagine it's such a basic foundational question that there's a lot of sort of different means of attacking these kinds of questions. I had initially been working in a field called automorphic forms that studies these questions in one way. It turned out not to be for me. [Chuckles] You know you learn something for a year or two and you get to know some of the people in the field and you decide like "Mmm maybe this doesn't feel like a good fit", and so then I moved into dynamics more or less I think what felt to me like an accident but what might have been orchestrated by some of the faculty in my graduate school. [Laughter]
Sarah Belet: [Indecipherable] ...to get a hold on to it.
Dr. Holly Krieger: I don't know if that was the case, but in any case it worked out, so lovely!
Sarah Belet: So you mentioned Fermat's Last Theorem which is of course a very famous theorem that's deceptively simple and back in about 1600s, I don't know my history. Someone thought that they had a proof for it they said "I can't be bothered writing it down but I've got a proof for it whatever like it's true" but it took us until what 2,000? I say us, but really I'm so great.
Dr. Holly Krieger: The completion was in the 90s actually but it was Fermat himself who had written like "Oh I have an elegant proof but there's no space for it in the margin" and it's sort of now a running mathematical joke.
Sarah Belet: Yeah.
Dr. Holly Krieger: Um...but... what was the question (again)?
Sarah Belet: My question was: Is there a question in your field that may or may not be deceptively simple to ask but is sort of like this overarching problem that you're trying to work on; you're trying to solve in your field or perhaps many people in your field are trying to solve?
Dr. Holly Krieger: So in my particular field which is this sort of interaction between number theory and complex dynamics, I would say one of the big open question so it's a relatively new field so there's nothing like Fermat where it's like we've been thinking about it for hundreds of years. I think one of the questions that was one of the initial interesting questions is what's known as the dynamical uniform boundedness conjecture which sounds quite fancy but I can explain in a very straightforward way.
Sarah Belet: Let's do it!
Dr. Holly Krieger: All right so a dynamical system of the type that I study is just a function. I take in some number and I output some other number and the types of functions that I'm particularly interested in (these are the ones related to this Mandelbrot set) are the ones which taken a number says add and output something of the form \(z^2 + c\) where \(c\) is some constant.
Sarah Belet: What's \(z\), sorry?
Dr. Holly Krieger: So \(z\) is my input so to describe the function, what I need to do is I need to choose a constant \(c\) and then the function that I'm going to be interested in applying and then applying again and applying again is the function which takes in a number and squares it and then adds \(c\), so \(z\) is just representing that number. Some of the things that can happen: if I do this process sometimes when I apply the function over and over and over again I get back where I started yeah so like if I think about \( z^2 - 1 \) that map takes in the number zero for example \( 0^2-1 = -1, (-1)^2 - 1 = 1-1 = 0 \) which (goes) back to 0, so after two applications of this function I return to back where I started.
So here's the question: I can do that, I can cook up an example like that which returns back where I started say after two applications and I can do it after three applications. We know we can't do it after four iterations of the function,(at least not with rational numbers. We know we can't do it after five applications with rational numbers, to get back where we started. We think we can't do it with six (iterations) and we don't know otherwise. We don't know if there's some function like that and some input rational number that has the property that when I apply the function say 27 times, I get back where I started. It's like completely bizarre somehow, you wouldn't expect it to be able to happen, but we can't prove it yet. So that's probably one of the simpler to explain (problems) but still sort of like unhappily unsolved.
Sarah Belet: Yeah, that's an interesting problem. So do you know...presumably someone might have some idea about why it's that threshold of 3 or 4?
Dr. Holly Krieger: Yeah so it's exactly actually the same thing as Fermat's. So Fermat's is about solving equations of the form like \( x^n+y^n = r^n \) and if \( n \) is small like say 1 or 2 then those equations are actually pretty easy to see. Like I can always find integers which sum up to another integer and it's also not hard to find squares which sum up to another square (these are Pythagorean triples that we see with right triangles). But there's a there's a change as soon as the exponent gets large enough between being able to have infinitely many solutions and then only having finitely many solutions. So that change which which is measured by something called the genus of the the curve described by the equation. That change happens right at that threshold both for Fairmont and also for this dynamical question.
Sarah Belet: Cool so going back to the question about this problem you have about finding your way back home: Are you fairly certain that you indeed can't find a function like that beyond this threshold?
Dr. Holly Krieger: It would be very surprising, I wouldn't say I'm certain. There's sort of very infamous examples of people making conjectures based on extremely solid evidence. There's an old conjecture that was disproved called Mertens conjecture and it's true for the first - I don't know - \( 10^30 \) whole numbers or something like that; the exception is huge. So I would be a little hesitant to be like "I'm certain there's no (such equation)" but I would be very surprised if it's not true that that 3 is sort of as high as you can go.
Sarah Belet: Yeah well I'll keep my ear out to see if anything comes out of that.
Lovely! So I read that you also did some work on YouTube is that right? Could you tell me about how you came to do stuff like that?
Dr. Holly Krieger: Yeah, so I work with a channel called Numberphile and this is run by Brady Haran who's actually originally from Australia. He has done Numberphile for quite a few years and has a lot of regular hosts, mathematician hosts. I was visiting a math Institute in Berkeley, California called MSRI (and MSRI is one of the sponsors of Numberphile) and so I met Brady there and was asked if I wanted to do one of these videos and I was sort of like "Oh god no like a camera and my face sounds like a horrible idea", but we did one and of course Brady is fantastic to work with: he made me very comfortable like it was a good process for the filming of Numberphile and then it turned out to be incredibly rewarding. One of the downsides of doing really sort of abstract pure mathematics is, you know, I don't go home at the end of the day like a doctor who saves someone's life. There's not there's not like a helping the world type life satisfaction and I really think Numberphile reaches a huge audience and it resonates with a lot of people that they have this sort of chance to interact with experts in this way and the level of interest was so high that I found it really satisfying so I kept doing it.
Sarah Belet: That's amazing. Do you pay attention to the sort of audience at these videos reach? Like do you know do you have like hardcore fans that come back every time or do you reach like a more general audience or maybe like particularly enthusiastic and interested high schoolers or something like that?
Dr. Holly Krieger: So I think it's a combination, I think it's the channel itself which has like the hardcore base and attracts really the wide audience but I get emails and nice messages and thanks from people and it's really from quite a range. One of the more satisfying things for me I think is to hear from like young women who are interested to see that like you know this is a woman interested in maths and she's sort of like able to talk, happy to have a conversation and breaking a little bit of some of the stereotypes that I think people have about adult mathematicians and so that's really nice.
I remember one of my favorite things is pretty early on and when I was doing Numberphile so the videos are filmed and the write the mathematical writing is done on these like sheets of brown paper and they used to be auctioned off as part of this sort of like "have a piece of Numberphile in your house" and the person who bought my first one sent me a picture and he had put it up in his shortly to-be-born daughter's nursery with the hopes that she would sort of be inspired and you know what that kind of thing I mean it it means something like even to an icy heart, so that was pretty sweet.
Sarah Belet: That's so lovely so you've done like YouTube you're doing this lecture series do you do many sort of radio things back in England are you on the TV much do you try and put yourself out there, or is it more like people just come to you?
Dr. Holly Krieger: It's mostly the latter not for a lack of interest per se on my part, but more about time. My primary time is spent being a professor doing research and teaching and having students and having a community of mathematicians that I talk to. I would say like I do have a preference: I like outreach that is either very directed e.g. I'm really involved with the undergraduate women in Cambridge; I'm a fellow at a college that has all female undergraduates and so I get to know all of them really well, there's not that many of them. Or like very broad audience like the Numberphile channel stuff, because I find it it's very easy.
I think especially right now when there's so much interest in women in STEM and that kind of thing, it's very easy to get approached for many many things and not have a good idea of what's actually effective in terms of "Is somebody getting interesting information that they didn't have before from me? Are they actually learning something that they wouldn't have otherwise learned?" It feels odd to talk about myself in this way, but like to be inspired in a way that they wouldn't have otherwise been inspired and so I really try and put some thought into like what do I feel like is is evidence-based like going to potentially make an impact for someone type of outreach work and that's really what I'm what I'm interested in.
Sarah Belet: Yeah that's a really interesting point because I've been doing a lot of outreach stuff for a while and I've sort of eventually come to this realization that I say yes to a lot of things because I'm like "Yeah this is a good cause!" Then it's like "Some of it is really good and some of I'm not quite sure" like I said yes to it I agreed to help out with it but at the end of the day I'm not quite sure if it's really like helping the message or if it's just helping a university with their KPIs (key performance indicators) or something. But it's nice to hear that other people are also thinking about these sorts of things.
Dr. Holly Krieger: Yeah I mean it's tough right, because as you say it's not like it's ever a bad call, and so you have to make some decisions and make some filters and say no occasionally but I think it's really important. One of the things I'll really emphasize: I think we fall short a little bit on evaluating the quality of our outreach work and it's to our detriment, like I think if a little bit more thought was put into really trying to detect what works and what builds like a vibrant mathematical community and that kind of thing, we'd be a bit better off with our time management if nothing else.
Sarah Belet: I agree!
In the news this week in Australia there's been talk about the OECD (Organisation for Economic Cooperation and Development) rankings and Australia in absolute terms has plummeted quite a bit in mathematics, so basically the message is that the standard of mathematics education, the standard of mathematics that our high school students are coming out of school with is declining over the years. Obviously there's going to be some structural issues to do with the Australian education system that's probably causing that, I'm not going to ask you to solve for us or anything like that. I was just wondering this sort of outreach stuff you're talking about, do you think stuff like that could help alleviate this or do you think it doesn't play as much of a part, we should just focus on fixing our school system basically?
Dr. Holly Krieger: That's a tough question I mean I think that at heart there is an issue of incentivization...
Sarah Belet: Mmm-hmm right.
Dr. Holly Krieger: ...and making massive priority for educators. First of all, for people who want to become educators, making maths competency and math fluency interesting to them and worthwhile for them - I think that part of outreach does actually have a chance of influencing this kind of thing but I would be surprised if it's the major (cause) [Indecipherable]. I think that really it has to be a bit more hard work than just outreach.
Sarah Belet: Because at the end of the day I guess she can watch heaps of YouTube videos and you can read lots of popular science books but the end of the day if you can't solve a quadratic equation...[Laughter]
Dr. Krieger: Sorry I mean this is actually disputed a little bit I think amongst people who do maths outreach, but I have a very strong opinion on it which is that I think that in order to express creativity, technical proficiency has to come first. Right I think that I could pick up a saxophone today and I might have beautiful music in my head but it's definitely not going to come out of the other end to the saxophone right and so while I think it's very important to present mathematics to people in a way where it can be appreciated without the technical proficiency, in order to actually do it as a profession or as an educator or whatever, you've really got to get your hands dirty. I mean it's the same way we wouldn't ask somebody to be an art teacher without them drawing (for) hours and hours, (having) hours and hours of training trying to do that kind of that technical learning. I think that what obstructs that from happening in math is something that outreach can affect a little bit which is this notion of "Well I didn't do well in math when I was six years old and therefore I am doomed to forever do terribly at mathematics and somehow it's like in my blood that I'm not good" and that kind of thing, like people telling their stories about how they got interested and especially this sort of like less-obvious-path-type stories where they didn't know that they were good at maths or enjoyed maths until later in life can be quite useful.
Sarah Belet: Yeah definitely, and I think it's a really interesting point especially about the saxophone because if you look at say artists that kind of broke the mold a little bit - if you go back at the earlier work they have very good technical proficiency and it's because of that solid background, I think they were able to go their own way and be creative and extend upon that.
Dr. Holly Krieger: That is certainly my opinion and yeah it's not it's not a universal opinion but I really do believe that's the most successful approach towards communicating any sort of beauty or science or humanities, anything like that really - to develop the expertise and put in the time and the hard work is (important) for most people after that kind of thing.
Sarah Belet: Yeah I think that's a popular opinion in this room...
Dr. Krieger: ...from our perspective anyway!
Sarah Belet: All right, Holly, to end with I reckon one more question: say we have a young student in high school, maybe fifteen or sixteen (years old), and (they) came to you and said exactly what you were saying before: "I recall when I was younger I found a math test so I'm dreamed I don't have the talent for it there's no hope for me." What would you say to them?
Dr. Holly Krieger: Probably I wouldn't say that much! What I've tried is to get them to do things right. The main thing I think it's like a lot of skills where you're discouraged which is like, (to get) small-step successes right. First of all get them thinking about mathematics in maybe a slightly unfamiliar way meaning like some more advanced mathematics like not this sort of trigonometry or whatever that they're unhappy with or algebra or whatever, get them something that they find interesting that they connect to because I think that there's a lot of mental block just around like "Oh god there's cosine" and then start building their confidence with small successes. I guess this is almost the same as my answer the previous question, like doing (maths) is really the way around this kind of mass fear which is you you do something that you can actually get sort of your teeth in and you have someone there to to direct you towards something that you might be interested in and once you have those small successes then you have the confidence to to fail. I mean that's the main thing is like maths fear is so often about being afraid of getting it wrong, and even professional mathematicians (like I meet them every day) who are terrified of somehow saying something that is untrue or making the wrong guess or whatever and it's so detrimental to your work. I mean, it's such a such a disadvantage and so happily being wrong is the way that I would push any student like that.
Sarah Belet: "Happily being wrong", that's a great sentiment. Well thank you very much for taking the time to come speak to us today.
Dr. Holly Krieger: Thanks so much it was my pleasure.
Sarah Belet: ...and I hope you enjoy the rest of your Australia tour!
Dr. Holly Krieger: Thank you.
Sarah Belet: Thank you.
At the release of this podcast Dr. Krieger has two more stops in Australia on Friday December 13th, she's speaking at the University of Adelaide. On Thursday December 19th she will be speaking at the University of Western Australia. We've also included a link to her videos from Numberphile on YouTube for you to check out as well.
Thank you for listening to The Random Sample, a podcast brought to you by the Australian Research Council Center of Excellence for Mathematical and Statistical Frontiers. If you like what you heard please rate us on iTunes because we think stories about maths are worth sharing, and we want others to discover that as well.
You can learn more about what we do at acems.org.au. You can find us on social media at "ACE Maths Stats".