Why Mathematicians Must Speak Up with Bryna Kra

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What if I told you that the number seven moves?

And what if I told you that the future of mathematics doesn't just depend on brilliant

theorems, but whether mathematicians are willing to step out of silence.

Today we're joined by Dr. Bryna Kraw, a professor of mathematics at Northwestern University,

and former president of the American Mathematical Society.

Before we start today's episode, we just have to know that Bryna's opinions are her own

and do not reflect any organizations she's associated with.

So on that note, I'm Autumn Phenath.

And I'm Noah Transferkisa, and this is Breaking Math.

Bryna, can you tell us what kind of math you do?

I work in dynamical systems and dergodic theory.

So what this is is it's branch of dynamics where I study abstract properties of systems

and how they evolve in time.

So for me, the system doesn't really matter.

What's a system?

It's just a bunch of objects and a rule with how they evolve over time.

And what I try to do is make general predictions as to what kind of behavior has to show up in

the long term.

It doesn't one could apply it to concrete things like planets rotating around the sun.

That's a very concrete dynamical system because we have a rule for how the planets move.

But what I do is one is that all systems share these general properties that I tried to

study them from a very abstract point of view.

But it's all about making long term predictions and behaviors of these systems.

Now when result your brother famous for is actually number theory and help me understand

this because numbers to me feel very static like seven is seven eight is eight.

So how are you using dynamical systems things that are about evolution in time to prove something

about numbers that I thought don't change?

So numbers there's a hidden dynamical system in numbers and this goes back to first

number for making the connection.

So hello, first thing is an amazing mathematician who basically found a way to view the positive

integers like one, two, three, the counting integers as a dynamical system.

And what he did is he said, look at them as adding one.

So you take all of the integers or you take a subset of the integers.

If you're looking at the events, for example, and what you do is you move them all over

by one by adding one that would move all of the events over and give you the odd numbers.

And then if you did it again, add one again, you'd come back and you'd be at the even numbers.

So you take every this whole set of the even numbers, shift them all over by one, turn

them into the odd numbers and shift them again over by one, turn them back into the even numbers.

That's a super simple set to look at and the dynamical system you get from it is just

two points, even and then odd and then even and then odd.

But what he did is find a way to take a subset of the integers and turn it into a very specific

dynamical system that we can then study.

And so I've spent a lot of my work studying the types of systems that come out of that

turning numbers looks seemingly static into something completely moving in motion and

iteration.

They're mostly not as simple as just going from events to odds.

But when they get more complicated is when it gets really interesting.

Now you've written some beautiful expository work.

How do you approach explaining this to a general audience, especially things that are very

abstract in your work?

What are some of your applications for non specialists?

So I am upfront with non specialists that my work by itself does not directly have applications.

But that's how mouth research is, right?

And so I think every mathematician has to be able to explain where the applications might

be in the future or where we've seen similar things from past mathematics is now being

applied.

And to me, it's an obligation for the health of our field, for the future of our field,

we have to be able to communicate what we do in a better manner.

So to me, it's a priority.

So where the types of problems I work in, dynamics, well, we live in a dynamical system.

Their dynamical systems all around us and we take them and we try to make them find simpler

ways to study them because the dynamical system we live in is a little complicated.

So when we try to turn them into what we call symbolic things of like using just finitely

many symbols to encode what's going on, this is an extremely important way to study dynamical

systems.

Concrete applications for related things?

Well, I've worked on problems that Erdos posed.

Erdos was a famous, itinerant mathematician who among many others think, well, first of

all, he had huge numbers of co-authors.

But he was perhaps most famous for posing questions.

Now, the questions are as important as the answers sometimes, right?

And Erdos was one who threw out questions all over the place and many of those questions

have turned into actual, the research that people did on them, applications that we're

not even, we don't even think about right now how we use them.

For example, you know, you use your cell phone in a crowded room and there might be a thousand

other people in a small space using their cell phones and well, why does it work?

This is related to sphere packing, which relates to questions that Erdos posed, you know,

we think about cannonballs piling them up.

You know how they're supposed to be packed to be most efficient.

Well, Erdos posed lots and lots of questions about how is the efficient packings in some

of these have led to huge innovations and breakthroughs.

And one of the applications happens to be that you can now use your cell phone in a crowded

room and everybody else can use it at the same time.

So a lot of parts of math are very, very old, even ancient, like algebra goes back thousands

of years in geometry.

But to my understanding, dynamical systems is relatively recent.

And curious what drew you to the field in the first place and what excites you about

at the most?

Well, when I started, I was not, when I started in grad school, there was, I didn't even

know what dynamical systems was.

In fact, I was absolutely convinced I was going to be a number of theorists.

I wanted to study number theory and I went to a series of lectures actually by Hillel

Firstenberg, who I had already mentioned and these, it was a summer that he was spending

at Stanford where I was a grad student.

And to me, the lecture was really most beautiful thing I'd ever heard, the whole theory of exactly

this idea of taking a static object and turning it into a dynamical system and turning it into

something where we have tools that are about motion and about recurrence that go back quite

a long ways.

I mean, this goes back to Pancarré.

Well, okay.

He's not an ancient Greek, but it's not 100, it's late 1800s is when Pancarré began

approving things, he was studying stability of this solar system and he proved various

statements about recurrence.

Things come back to themselves in certain ways in a very precise sense.

And Firstenberg was explaining how you use that to study numbers.

Well, I don't know.

To me, this connection between two seemingly unrelated fields is part of the beauty of math.

Like, that's, I find that an amazing thing.

And it's beautiful that you had this interest in number theory coming out of college, put

it aside when it's a dynamical system and then came back to it to prove some very powerful

results in number theory.

Yes.

That's how math is, right?

It's, it's all one interconnected beast and you need to, you know, like one of the problems

I think is we're all siloed.

And the more we interconnect and see the different parts, the better off math researches

and the more we advance it.

Now I'm curious, do you have any quirky story or anecdote about your research that you

would like to share?

Ah, quirky stories.

Well, there are many quirky stories in math like how collaboration start?

Usually quirky.

I've had one never knows because this thing about connecting with people who think about

it differently.

This is an amazing part of math.

One of my papers started well in a bar and it was at a big conference and there was

a shuffleboard table, you know, one of these old things that you play shuffleboard in a

bar.

And there were so many of us from the conference there that we decided to pair up and

play two on two and met some person who I'd never met before.

We decided to pair up with people we hadn't met.

We were sitting out waiting our turn to play shuffleboard and he asked me, what do you

think about?

What kind of math do you think about?

And I told him something, well, I'm trying to solve this problem.

He said, that sounds fascinating.

I actually think about a related problem.

And we ended up writing a paper a few years later, started as a shuffleboard.

So this is not a joke of a math, a pair of mathematicians walk into a bar, you know, it

comes to collaboration.

It could be.

It could be.

Fantastic.

All of my collaborations, not all, many of my collaborations have been started by serendipity

of some way.

One of my collaborators, long-term collaborators is a French mathematician Bernard Host.

And I was a postdoc in France when we started working.

But the reason we started working together was, well, pretty by action.

I was there for my first year of my postdoc and I decided it was a one-year postdoc

at an institute there.

And I had just arrived September and I realized I wanted to stay for a second year already

rather than returning to the U.S. to a position that I had.

And I found this grant that I could apply for, except it needed a sponsor.

So I didn't know anyone yet.

And I went to some seminar and met somebody and the only person who I knew of him.

And he said, well, I'd be happy to be your sponsor, but this is something official.

It needs like a stamp from the university and all of this.

And he said, and we're not at my university and I'm not going to be there before the deadline

because it was a little last minute, three days before the deadline.

And he said, but here, let me introduce you to this person, Bernard Host.

And maybe he'll write you the sponsor letter.

So Bernard was friendly and he said, sure, I'll write you the sponsor letter.

You have to tell me what you want to work on.

So we had, we ended up spending the whole afternoon talking about research.

I guess he wrote a decent enough letter because I got the grant.

And then we started working together on that topic and we've worked together.

We worked together for over 20 years.

So it's like my identity that he was there at that seminar and it was his institution.

The lesson I'm getting from this is there's a lot of chaos theory involved

in the collaborations of dynamical theorists.

For how?

So moving on.

Last year, he served as the president of the American mathematical society.

And a question I want to ask is just what was that experience like?

But let me step back by saying, I've been in mathematics for many years, autumn as well.

I really, and I'm being honest, have no idea what the president of the AMS does.

And that's because mathematicians are very independent, autonomous.

You know, we just kind of do our own thing.

So I'm curious, you know, you can go in any direction you want.

But just the idea of trying to lead a group that doesn't necessarily like being led

is one thing that I'm very interested in.

But again, take it where you wish.

Yeah.

So, okay.

There are many misconceptions about professional societies, I think.

And the AMS is no exception.

Many people view it as its monolithic entity,

where it only serves a narrow portion of the population of mathematicians.

And one of the amazing things is the AMS is incredibly broad.

And there are so many different parts.

I had been involved at the AMS for over 20 years before I became president.

And yet I learned many new things while I was president about it.

So it's an incredible thing that brings together people from different groups.

So what does the president do?

You said, you asked me.

Sort of everything.

And yet, by themselves, they can do nothing at the same time.

So when I became president, I had an idea of something that I really wanted to happen.

The idea was that people who are primarily under graduate institutions,

many of them are doing amazing research.

And they have almost no resources to support it.

Getting something from the NSF is impossible.

Well, for everybody's difficult and for people who's teaching loads are higher.

It's almost possible.

And there's not really other sources.

And so I wanted the AMS to create small grants that could have an outsized impact on somebody.

And what was incredible was that this idea, while I was president came to fruition

because of a huge number of people in the community working together

to actually make it happen.

And that's what the AMS can do.

And what it does is the president can have an idea,

but that doesn't turn into an actual program that funds 50 people a year or something like that.

Which is what it turned into.

So 150 people are something in a steady state because they give three year grants now.

So it's the AMS, when it sits on a huge number of committees.

I, the number of committees at some point, I made a list and I looked at it and I was like, over 20.

Each of them actually meets.

It's not, you know, something just on paper.

It was about 15 trips a year, various places, mostly to Providence and DC.

But I also got to go to Oslo to the Abel Prize ceremony.

And I went to New Zealand to join AMS New Zealand and Australia Math Society's meeting

and met tons of people that were, that was an amazing part of it.

Is there anything that surprised you the most in this position?

Well, I think that one of the things that surprised me is how many misconceptions people have

about what the AMS does.

And I was continually surprised by how many things it really does.

Which is much more than anybody knows.

Little things that mathematicians take for granted as the resources are there.

Often the AMS is behind it.

Not only the AMS, AMS with collaborations like so.

AMS and Duke University offer math jobs.

AMS has math sign-et people don't always realize that the huge databases entirely the AMS.

AMS has all sorts of grants and programs and fellowships and awards.

AMS provides latex support.

It provides support for accessibility in all these changing guidelines.

It runs webinars and seminars for people to how to do these things.

So the breadth of this is what surprised me the most.

So the message I'm getting from you to, in answer to my question,

is it's not so much that you're leading mathematicians.

It's more like you're leading efforts and resources to assist mathematicians.

Is that accurate?

I think that's a good way to put it.

By itself, the AMS couldn't exist without the huge number of volunteers and people working

to basically give a ton of their time to help this.

And it's all trying to provide mathematicians wherever they are with the resources that they need to succeed.

So let me ask, when I'm very excited to ask you about,

I've been actually looking forward to asking you this for several months.

Back in January, we were both at the joint math meetings in DC.

This is the big annual math conference for those who aren't aware.

And you gave the farewell presidential address.

There's a tradition every year.

The outgoing president gives a speech.

The incoming president gives a speech.

Fortunately, I had a scheduling conflict and I missed yours.

But I bumped into you the hall into you in the hallway.

And you said, normally, the outgoing president gives this talk and it's about research.

And you did talk about research, but you couldn't resist.

And you also talked about some broader issues about the mathematics community,

mathematics culture, the direction of mathematics.

And I was so tantalized, but off you went to a meeting and off I went to a talk.

So for months, I've been waiting to hear.

So can you share with our audience a little bit,

what did you say that went outside the norm of this research talk?

And why did you feel it necessary?

Or why were you motivated to go beyond the tradition?

Yeah, so I strongly feel that research is not just about doing research in your silo,

but it's also about communicating your research and sharing ideas

and working with others and learning from others.

And that doesn't happen enough in our community.

And so this was an opportunity for me to make a case that we need to communicate more.

And the debt is one of the issues facing our, one of the challenges facing our community.

So I believe there's huge numbers of challenges facing mathematicians.

And those challenges are ramping up.

They come from tightening budgets, you know, resources are becoming scarcer and scarcer.

Our teaching is changing rapidly.

AI is disrupting how we do things, both in the classroom and in our research.

And I think math departments have to evolve or be swallowed by other entities.

And it's in our interest both not just for our own sake.

I don't mean it that way.

It's in the interest of the future because mathematicians I think need to be involved

in training of the future generation of scientists and engineers.

And mathematicians need to be involved in AI and where it's going

and what it's used for and how it's used and how it developed, you know,

the math is behind it, it's development.

But right now what I'm seeing is a lot of people just saying,

these changes are not going to affect me or it's too much.

I'm just not going to, I'm going to close my eyes.

So to me, this was an opportunity, a stage on which to share that.

And so I titled my talk of mathematicians challenge.

And actually it'll be online any day because AMS posts these.

And I wanted it to be provocative to get people to think about it.

But at the same time, I did not want to give up the opportunity to share my research,

especially the research that's happened in the last couple of years

while I was president of the AMS because being president of the AMS

did not slow my, well, it did slow down my research.

But it affected my research, but I really made an effort to continue working.

And I did a lot of work on in the last few years

that some long term projects really came to fruition

exactly while I was president of the AMS.

So this is an opportunity to both share that research and to explain the challenges.

And so what I did was, well, you can be the judge if it was successful or not.

But I started off with what I view as five challenges to the mass community.

And then I gave what I view as my answers, how we have to address those challenges.

It's all about communication.

And then I used that as a segue to explain how I communicate about my research.

And my, I give examples from public lectures that I have given

in the last few years of some slides just took them directly from those

so that it's explained and then segue into my research.

So you have two large roles in the mathematics community.

You're the chair of the math section for the National Academy of Sciences.

And you're also the chair for the National Committee of Mathematics.

Tell us what that means in both cases.

And how do you see math fitting into a larger scientific community?

This fits in the larger theme that I think mathematicians need to communicate more with each other

and with the outside world, meaning non-mathematicians.

So one of the opportunities that mathematicians have to communicate with each other

is through the International Congress of Mathematicians, which is once every four years.

And the next one happens to be coming July 2026 in Philadelphia.

That is preceded by something called the General Assembly,

which is a meeting of all of the members of the International Mathematics Union.

And that will be a two-day meeting that happens in New York beforehand.

And the representatives to that come from something called the US National Committee for Mathematics.

So each of the major professions in the US has a US National Committee.

And I happen to chair that one, the one for mathematics right now.

When I agreed to chair this, it was, I was told that this is a really easy thing

and you get to go to this amazing meeting, the General Assembly, represent the US

and vote on important topics.

It was a bit of a different world back then when I agreed to do this a couple of years ago

because current, up until, well, a week ago, the US NCM, the National Committee,

was represented, housed, sort of, by in the National Academy of Sciences,

meaning that the National Academy of Sciences was our adhering organization

to this International Math Union.

Budgetary cuts, including cutting up a grant that funded this,

meant that the National Academies could no longer be providing the support for this.

And so what happened to me as chair of the US NCM was we had to find a new home for this

as of last week, which are academies withdrew as being the adhering organization.

We're currently in limbo waiting for the International Math Union to accept the proposal

that came for a new adhering organization.

We expect it'll be accepted, but I won't have that news for another couple of weeks.

But something that happened in this is that the various professional societies that are

represented in the International Math Union joined together and found a new way to support

the National Committee. And it will see. It's a little bit of a turbulent time,

but we will be at the General Assembly. We will have five people representing the US National

Math community, and we'll be there in New York. So it's interesting. We'll see.

Looks like it should be interesting. Exciting.

So, Brian, when we were chatting a little bit before we started recording,

we mentioned, oh, it's good timing that your episode is coming out in March because this is

Women's History Month, and it's nice to see women in math, and you corrected us and said,

you don't see yourself as a woman in math. You see yourself as a mathematician who happens to be a

woman. So, I want to ask you to elaborate a little bit on that. I can kind of guess what you

mean, but I'd like to hear in your own words and let our audience hear. But also, I want to ask,

you've done a lot of leadership around mentoring women in math or mathematicians who happen to be

women, however you want to word it. So, can you tell us a little bit about what this means to you,

why you're careful with the wording, what your mentorship and leadership activities have been,

just let's go in that direction a little bit. Yeah. So, I like to view mathematicians as mathematicians

first, and then they have other characteristics. Many of those characteristics make them better

mathematicians, but I always want to view mathematicians as first and foremost that I view them as my

professional colleagues. And that said, clearly there are very few women in math still, even with

lots and lots of efforts. The number of women has not increased dramatically. There are some fields

where it's done very well, but other fields, there are still women are anomalous creatures, shall we

say. Can we expand on that? Why do you think some fields are doing better than others in this regard?

Yes. So, some fields, when there are many women in a field, other women go into field, the same

field. This is, so it's not good. It has to happen organically, I think. One cannot force it,

but I think there are things one can do to nudge things along. And that is what I have tried to do,

is create more opportunities, and not just for women, but also for historically underrepresented

groups, a minoritized groups in mathematics. I've really tried to create more pathways. And to

me, this is something actually that even goes broader than women or underrepresented groups is we

ought to be creating more pathways for more people in mathematics to enter. That's part of the

challenges that I alluded to earlier. So, as far as women's particularly, I always want to have more

women around me in math. And since I was starting at Penn State, where I was tenure track, I

founded a group, women in math there, just to have informal social and mentoring opportunities.

That group is still going strong. I was there last week, and they invited me to lunch, and it was

really delightful more than 20, 25 years later. And then when I moved to Northwestern, which is over

20 years ago, first thing I did was found another women in math group. And again, that's a local

mentoring. They're not exclusionary. We invite anybody who wants to come. But the idea is just a

space where people can talk about things that they might be feeling or have asked questions in a

safe place that they feel like they want to know answers. Our conversation is often about research.

It's not, it could be a conversation among any group of mathematicians, but somehow having it

with a majority female group, often people feel freer, women feel freer to ask questions.

And sometimes our conversations about juggling, kids and family, and how you deal with a

two-body problem in math, and all the things that women face more than men,

at typically in the field. And then I went on to found, I looked at the demographics.

The number of women applying to grad school kept dropping, looking at the grad applications that

we'd see here, increasing number of applications and declining percentage of women applying.

And so I founded something called Grow, which is a conference, a weekend-long conference called

the Graduate Research Opportunities for Women, Graduate Opportunities. Graduate research opportunities

for women, but now it's called the Graduate Research Opportunities Workshop. It's changed names

a little bit. But the goals are the same, to increase pathways for people into math. And it's a

weekend of research talks, mentoring sessions, and panels on how-to, like all the nitty gritty.

And the reason I founded this is, well, to help the young women who come, you get 80 people in a

room, a majority of whom are women, and it changes the conversation. But it also has an effect on

the organizers. And for me, a principle from the beginning was that organizers to change the number

of women in math, it can't just be the women changing it, it has to be the women and the men.

And so organizers are mixed, and speakers are mixed, women and men.

Always a lot of women represented, but a mixture. And the people, the men who go through it say,

that had an amazing effect on me. It changed the way I think about these issues. So to me, that's

what we have to do is change not just the women, but the men, because we won't get there without that.

And I've also been involved with other projects that try to address other small groups,

even much smaller than women in math. Historically, there are very few people from underrepresented

groups in math. And I am one of the co-founders. Eric Zazlow was the leader of this, but I worked a

lot with him on this cause weight program, which was a post-becaloriate program aimed at historically

underrepresented groups in math to, again, singles get more people, more pathways going into mathematical

sciences, graduate school and mathematical sciences. Do you think it's not just a gender issue,

but it's also a resource issue? One can't really separate it, but it's, I don't know if it's a

gender issue, to be honest. I think it's just that historically, there are barriers to women and

other minorities going into mathematics. And we haven't done enough to break down those barriers.

It starts very early. I grew up with mathematics. So it's something I knew. But if you don't grow up

with it, how would you know that there is something called a career in mathematics? No matter how much

you love it in school, you might not know that that exists. So I think it's really a matter of

opening more doors to more people. Everything takes resources. It's how we choose to spend our

resources that matters. But now let me look a little bit more at your career, but zoomed out.

You know, I can't even list all your honors. They're just, you know, it would take the whole hour

of the show. But you know, you've done groundbreaking research. You've served in these incredible

leadership positions. One, all these awards. So I've two questions. One is, is there one thing that

you've done that you're most proud of? And you know, this could be being president AMS or could be

a paper you wrote or theorem could be anything. So is there one part of your career you're most proud

of? And the other part is, maybe this is naive of me, but you've done everything you've accomplished

at all. What are you going to set your sights on for the next five or 10 years? What left is

there, you know, in your ambitious career? So let's hear what you think about both of those.

I don't know how to single out one thing that I'm most proud of. There are research, research

pieces that, to me, I'm just very proud of, to be honest, that I, you know, still look back and

think, wow, that's kind of nice that we did that. In some sense, maybe I'm proud of some of the

softer things of trying to get people to change how they think about interacting and communicating.

Because I do think, you know, this communication at all levels, all stages and different audiences

is something that I have pushed a lot. But I, yeah, so I find that really hard to single out one

thing that I'm really proud of. I, I'm, I'm very happy, you know, every time I convince somebody

else that they should think about these issues, that makes me, you know, really happy.

Well, the, the secret is you don't have to pick one. We just wanted to hear your thoughts and I

think we gave, got a nice window into that. And now what about looking to the future? What's next?

Yeah, what's next is a really good question. I do have some research projects that I'm pretty

excited about right now. Don't know if they'll come to fruition the way I want them to. That's

true for every research project, I think. There's a huge number of young people in my field now. When

I went into the field, the number of young people was quite few. And now there's this next generation

of students and postdocs and tenure track assistants and professors who are just amazing. So

I've been continuing working on research, but I, I'm also ready to take a little bit of a step back

from that and spend more time exactly communicating and working. And I don't know exactly where that will

lead. That's, to be honest, it, I have interests on other, other types of roles, but it's not clear.

I guess I'll have to wait and find out. Yeah, I'll have to wait and find out. I'm not sure if I will

find the right shit for me because I always want to have a finger in math still and in keeping

track of what's going on. Now out of curiosity, is there one piece of advice that you would give to

someone who wants to study mathematics and go into research? Well, it depends on their level. There's

younger people I tell them to be open to all types. I think people should be open to, you don't

know where the next research will be. One piece of advice that maybe I wish people had told me

when I was already getting involved in research and already in grad school is that if there's

something that you can think of that might make it easier for you to do your research or to advance,

just ask for it. You never know. You might just get it. The number of times that I think people

are afraid to just say, Hey, I can't teach at that time because of this reason or I could really use

this small amount of money to go to that conference or other things, whatever it is. Just ask,

like I feel like people are afraid to ask because how it would be viewed, but it's sort of never

harmful to have been viewed that way. I can give you a specific example if you want. I'm sure.

When I was tenure-track, there was no maternity leave policy at the institution that I was at.

I said, you know, I need some leave because I'm going to have a kid and this kid's going to be born

in the summer, but there's no way I can teach, start teaching two months later. And the answer was,

okay, we'll figure something out. And would you like to chair a committee for the entire college to

figure out how to introduce a maternity leave policy? And I said, whoa, I'm tenure-track. Do I

really want to do that? And they said, well, we'd like you to do it. And I said, sure. And it was

adopted. So just ask. If one doesn't ask, it doesn't happen. You dedicate your life to understanding

how things evolve over time, how patterns reoccur, how structures stabilize, and how long-term

behavior emerges from simple rules. And then you realize the mathematics community itself is a

dynamical system. So maybe that's the final recurrence here. Ask, communicate, collaborate, and repeat.

Bryna, thank you so much for coming on the show and being a leader within the mathematics community.

And to everyone listening, whether you're a mathematician, a student, a teacher, or just someone

who likes asking dangerous questions, keep pushing the system forward. Until next time, stay

curious of the world around you. Until next time, stay curious of the world around you.