The Math Behind Gerrymandering, Fair Representation, and Politics with Karen Saxe

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Vi har inredning för hela arbetsplatsen. Välkommen till AGR. What if I told you that some of the most political debates in America hinge on extremely mathematical ideas? Like what it even means for a shape to be compact? how you measure fairness, and why there might not be a single best way to do apportionment at all. Today on Breaking Math, we're joined by Dr. Karen Sack.

director of government relations at the American Mathematical Society, Karen has spent her career living at the intersection of math, policy, and public service. From thinking deeply about redistricting and representation to helping the mathematics community navigate the realities of federal funding, research priorities,

Visa's education policy and the fast moving world of AI. Before we begin, The views shared in this conversation are Karen's own and do not reflect the views of the American Mathematical Society or any agency, organization, or institution she works with or is affiliated with.

Now, I'm Autumn FNAF and I'm joined by my co host, Noah John Saracusa, and this is Breaking Math. All right, so Karen, earlier in your career you worked on gerrymandering, which I think of as a political topic, but you're a math professor, so what does that have to do with math? Yeah, a great question.

So gerrymandering is actually the sort of messing about with what's really called redistricting. Redistricting is the third step in this process that happens every ten years and all three are mathematical. So there's the census. that happens and then the reapportionment of congressional seats and then redistricting. And then that's where gerrymandering comes in. Every ten years since seventeen ninety, we perform the census. and these n uh numbers are then used to

um distribute the 400 the now 435 House of Representative seats out to the states through reapportionment. The census is required in the Constitution precisely for this reason of apportionment. So those two things have happened every ten years. Um, except for one time in nineteen twenty, which I could talk about why if you want. It didn't happen then and people's guesses always wrong about why it didn't happen. In any case, those things happen every ten years and then um

redistricting happened but since eighteen forty two it's been mandated in law that it has to happen. And um That's to decide which which people in the state, you know, said representative actually represents. So cutting up the state into these geometric shapes. And that brings me to what math has to do with it and what you know, what I kind of worked on.

So when I was uh I was trained in operator theory and that's what I did my first mathematical work in and I was studying I know this just sounds so weird now in a way, but like, okay, the spectrum of an operator, which for people listening

if you don't know the spectrum of a an operator is the analogue of the set of eigenvalues for a infinite dimensional operator. And that's a set in the plane. And I was studying sort of like how many holes that set could have, its convexity properties and so on. And th then I was for some reason in around the early two thousands I was reading about redistricting and measures of compactness. So since like the nineteen sixties

certain things have had to happen in redistricting to guard against gerrymandering. You might be looking for does the shape look not natural. It doesn't have long arms, does it have holes in it? And there's there's rules about that in laws. Some some are laws, some are guidelines for states. Now let me just ask here. I'm guessing when political people, judges are talking about whether a district looks compact.

They're not talking about infinite open covers having finite subcovers, right? Oh correct. Yes. Thank you. Yes, compactness means something completely different in this context. And um yeah, no open covers, none of that. It kind of is more what we would think about as convexity and simply connectedness. So pathwise connectedness.

So it's still mathy, it's just not attached to the same word that we learn in like an undergraduate real analysis class. Mathematically the the districts in the state have to have equal population. Okay, that is a math thing. Um'cause, you know, we we understand what equal numbers are, where some people don't.

And then also there's contiguity, which is the path connectedness part, and then c what they call compactness, which is more like well, basically they kind of appeal to or fetishize the isoparametric theorem, which is pretty interesting. That theorem is the theorem that states that the shape and the plane with a given perimeter that has the largest area is a circle. So in some ways the ideal congressional district shape is a circle. These quote measures of compactness

measure kind of many of them, not all of them, but many of them measure how far you are from being a circle. Well, that's kind of funny to a mathematician if you think about tiling any of our states with circles, that that's just not happening. Um, but you know, the the theorem's an interesting theorem and I love math, so I you know, like thinking about this. The theorem was was uh known about two thous over two thousand years ago and it well, took it was first

mentioned, I guess, in about two hundred B. C. It wasn't proved rigorously until the mid 1800s. So it took almost two thousand years to to get approved. So and then I just thought I'd mention so there's these three steps, the census, the reapportionment, and then redistricting. Reapportionment is actually an interesting math problem, probably more interesting in some ways than

redistricting and gerrymandering. And there's many, many different methods to do it. There's not a best method, which is one of the problems. There's no best method. I think that that's okay. What's the general n name for those in which? Now didn't didn't Alexander Hamilton of Broadway fame work on this reapportionment problem? Hamilton of Broadway fame had one of the first proposed apportionment methods. That's right. But it was not accepted and then

A different method was proposed. So in the first set uh in the first apportionment he his method was proposed. It was Uh vetoed by George Washington. I should check on that. And they didn't put this in the musical. Yeah, they missed out on this. It was the first presidential veto. Over a mathematical matter.

We need to write our own musical now. I know. Right. Oh boy. Yeah. I know. Yeah, they miss so many important things. But then mathematicians, we got our due because the method used now is due to two people, Hill, Joseph Hill and Edward Huntington and Edward Joseph Hill was the chief statistician for the Census Bureau at the time, and Huntington was a mathematician, he taught at Williams College, he taught at Harvard. He also served as MAA president.

So Math Association of America and the AMS vice president. So th you know, he was really involved in the the math communities. So that's maybe enough of a math lecture.

For the memortic. So anyways gerrymandering has to do with looking at the shapes in a plane and thinking about how weird or not they are. So maybe a way to kind of succinctly kind of tie this together is that the constitution demands that certain things happen and it turns out to achieve those or to achieve them fairly, a lot of mathematics is involved. Right. Yeah, and the term I was trying to think of before is an impossibility theorem. I couldn't think of that.

So like reapportionment is like an impossibility theorem and there's not one best way. Now thinking about this gerrymandering and we have this growing community of mathematicians working on these problems. What impacts do they have and do politicians actually listen to? Yeah, so good question. Um

They you're right, there's been a huge growing community, you know, is very popular now even in undergraduate curriculum to learn about this stuff. Several mathematicians have played roles in court cases, like serious roles, either acting as expert witnesses, providing reports, providing testimony, and that has had impact.

Um, and then do politicians listen to them? There are also in Congress various bills introduced to change how we do redistricting in the US. There are certain politicians who introduce these every Congress. Typically the thing is to get it out of the hands of politicians so they work on who does the redistricting, which it's up to ev each state to do it.

to decide. And in many of them it's the sitting state legislators, which of course the districts work for them. And it's not just US congressional seats. These districts are drawn for the state legislatures, water districts, school districts. There's a lot of places where redistricting is done in some form or other.

So speaking of mathematicians having an impact in political arenas, you spent part of your career working for Senator Al Franken's office. This is amazing for our younger viewers. Al Franken was a comedian on SNL, I believe. And then became a senator. So he was kind of like a celebrity senator, not just any senator. Yeah. So that's an amazing setting. Here's this math professor who's not only working in the Senate, but working for a professional comedian in the Senate.

I don't even know where to begin with questions. So just tell us what was that like? What were you doing? What was that? Well it was super fun. And so I um in twenty th this was in twenty thirteen, fourteen and I was the AMS Congressional Fellow. which maybe I'll get to later talking about what that position is. There's one per year. The AMS places one mathematician in Congress and I got to be that person.

And then you go through an interview process with various offices to see where you wanna work. And because I was from Minnesota, I didn't really want to work for one of my delegation because I felt I mean I I was pretty old when I did this. I was in my fees and like I knew Minnesota and I wanted to learn something new. This was gonna be like uh maybe we'll get on to this, but like what I was thinking about my career and and so on. So I was in my mid phase.

And I was looking to branch out and I thought, Oh yeah, I wanna learn something new. So in any case, but when I met with this office, it was they convinced me that I had something to offer them, which is that I was gonna work on a higher education policy and I knew the higher education landscape in Minnesota. very well. And and the high school

situation. So for example, if if Senator Franken was going to a certain high school, I could tell him what the demographic of that community is like, you know, so that kind of stuff. But also I had I developed this whole theory of trickle down personality when I when I worked for him, which is that, you know, there's the member of Congress, they hire their chief of staff.

Then their chief of staff does a lot of hiring. And there's the legislative team, which is what I worked on, and then there's the communications. team and then there's some other people who are sort of operational, I guess you would say. But um in Franken's office everybody was super smart, super funny.

good writers. You know, the kind of stuff that he valued were were the apparent in the office. So I really liked the office. The legislative director was a um a He'd been a political science professor, um, at the University of Minnesota. So like it was very um intellectual office and sorry to every other office on the hill. But you know, they they th they value different things, let's say. And so it was really it was really

Du skulle ju köpa några nya allställt i lagret. Det kanske blev lite mer grejer. De hade ju allt. Man hade skrivbord, jag köpte en sån här, och kontorstolar, och sen hade de en skit snygg till. Vi har inredning för hela arbetsplatsen. Välkommen till AI-produkten! Interesting. Overall working with the Senate, what has surprised you the most about your work? Yeah. So

In terms of like the the sort of day-to-day work, I would say the pace of the work is what strikes you at first, like coming from academia. It is very fast paced. things need to be done in hours or possibly days. You do not have time to like look up every source and do do what a mathematician would consider like a a complete job on something. You just have to say, okay, I'm gonna look at these sources

and you learn what those are and it's gotta be done by four PM tomorrow and that's just what it's gonna be. Oh, and by the way, they also only want it to be three pages long. So y you've gotta you've gotta curb yourself in in a in a lot of ways and sort of accept something that's a good product and done in time. So there was that and again on the sort of workplace The teamwork is very strong. You work

with people all day long every day in different variety of teams. So that's quite different also. than academia, I found. And then also just the staff is much more diverse than than from an academic setting. And then sort of something more about the work that you actually did, I found that working on legislation and working on math actually had some shared

properties. So both so when you're when when I was a c when you're a congressional fellow in this program I was in, there are about thirty-five of you and you come from all the different sciences. And so this would be compared to the other sciences. So in in math and law, the there's very highly structured writing styles that are used.

And both start with giving definitions. So this is kind of funny. You know, a law might start it starts with a def first it says who's introducing it in the name of the law, then it'll have definitions. And it might say something like a minority serving institution and what that means, that there's a legal definition, so something like that. The other thing is that w the we both write, so lawyer bill, laws and math papers are written in a very kind of terse

and no nonsense prose, you know, short sentences. And then the process takes years. It's slow. Both take years. And so I think mathematicians appreciate how long it can take bills to pass and law to get done. So th I think those are the things I found most surprising.

One time a few years ago I was at a conference, I think it was about like information, journalism, news system, and so not a not a math conference at all. And at one point someone made a comment that I don't think I'll ever forget. They came up to me and said You're the first mathematician I've ever met in my life.

And that seems so it was so striking because when you go to grad school in math, all you see is mathematicians. And then when you become a professor, you're just around them, you take it for granted. But this person has heard of like Isaac Newton and, you know, famous old mathematicians.

Maybe they knew that there are still living mathematicians, but they never met one. So I'm curious. Do you ever ask your class like name some mathematicians and they don't even name you or any of your colleagues? Like they name like So I'm curious if when you were in the Senate floors, did you have any stereotypes that people were like expecting a mathematician to look or behave a certain way, or did they just treat you like any other professional?

There wa there was one thing that really struck me and it was actually a great experience. So we I do these congressional briefings and I was doing one with a with a female mathematician and Believe it or not, Senator Schumer had wanted to come, and he couldn't, and he asked me if I could fly her back. And he could meet with her. So of course I made that happen.

we go to his office in the which was in the Capitol building then, and he had a couple other senators there with him. Senator Blumenthal, I think Senator Shaheen. In any case So it was me this female mathematician and then I also had the

president of the AMS at the time with us, and that was a female. So there's three of us. And we go in and Senator Schumer said, Before I start, I just have to say that when I saw that I was having a meeting with three mathematicians, I really did not expect three women to walk in here.

And he said that it was really eye opening to him and great. So that was like the one you know, that meant a lot to all three of us actually. You know, we all talked about it after and we were like, Wow, he noticed and so we made a big thing of then talking about that he asked us they they asked us some questions about the demographics of mathematics. Which was not what we were there to talk about. We were there to talk about privacy and security issues. But anyways, and we did get onto that.

But um that was striking. Other than that, not really, to be honest. You know, most staff on the hill are I mean, they're all college educated. One thing that's they're smart. They're not Scared of math typically. So no, that experience with Senator Schumer and those other senators was really the only time where That that really happened.

It's funny, in in my experiences, I've almost found more affinity for lawyers, I feel like the logic that's involved in law than with scientists, because scientists do experiments and collect data and analyze it. I've never done any of that as a mathematician. But lawyers, you know, they define things and they use logic. So I I do feel like law and math are very close. Yeah, and there's precedent in law. There's precedent in law. So whether you agree with a

prior law or not, you've got to assume I mean, it's the law. You know, you've got to build from there. In the same way with we have or where you're just like, okay, that's my starting point. That's what I'm gonna do and yeah, the That's right. Science is really quite different in that way. Because oftentimes there's some sort of additional variable or a temperature change or whatever fracture point in a material.

that happens and that is not the same. We're looking at concrete facts here versus whatever you have as a almost a dynamical system. So I now I do all this advocating and I do work with science funding, so I work with other science people. And one thing that's nice walking into offices is that a lot of offices don't have an expectation for what a mathematician's gonna be in the same way that they do for certain kind of scientists. And they

And they don't know if we're gonna have biases. So like I mean, this would be an extreme example, but if I go in with like a an atmospheric scientist, they think they're gonna be biased in in certain ways about climate issues. say. They don't know what to make of mathematicians in that way. Which is totally understandable. Yeah. But with your current position as head of government relations for the AMS, What are some things that listeners might actually be surprised to hear?

that math has any government relations to head. So Can you tell us a little bit about your job and what you do? So I think about my job in two ways, really. So I'm the director of government relations for the American Mathematical Society. And I think about it's a communications job and it's communicating to the math community what's going on in Congress and at executive branch agencies.

like National Science Foundation, Department of Energy, and so on, that they need to know about or I think they should know about. And then it's communicating back the other way from the math community to Congress and the executive branch agencies. about what the what's important in the math community. And the topics are of course on research funding. Uh you know, what's the v why should the federal government and taxpayers pay to fund math research and education initiatives in

math. But then there's a lot of other things. So, you know, I tried to make a list of like the other things that over the years and I've worked on and these wax and wane with the different administrations and different congresses. So right now we talk a lot about visas. and reduction of internet travel and brain drain. So that kind of

thing, what's happening to American science, the evisceration of the Department of Education is another topic, and the collection of education statistics. A topic that was bigger in the In the prior two administrations, so the first Trump and then the Biden administration and Obama. except for I wasn't here then. Open access policies publishing publishing things. Right now there's this heavy focus on research security and competition with China in particular. And then

I I'm sure we'll talk about AI at some point, but there's a lot about AI going on. We also deal with policies that support grad students and like one small success I mean it's big in the math world, but it's probably small. Like if I go out in the street and talk about it, people are like whatever. But um you're if you're a grad student, you know, you get your stipend and you also get a tuition waiver. That tuition waiver is not considered part of your income for tax.

But the Trump administration and the first Trump administration wanted that to be part of your taxable income. And we reversed that. So that was like a huge win for graduate students. And then of course there's like In the past we've worked very hard on supporting on legislation and policies that support women and other um people who haven't ha had good access to to math in the past. So any kind of DEI work, which we still work on.

It's not a lot happening in Congress on that right now. So those are those are some of the things. Well you you brought it up, so let's grab the bull by the horn. So everyone sees articles and news about AI, it's impacting all of our work, and there's no way your job is any exception. So how does AI come up in your work? Yeah. Well, th I guess I think of two ways. There's like as an AMS staff member and thinking about it in our our workplace.

So at the AMS we're talking about how to use it for making our workflow more efficient and then for you know, writing things like I could use it to write summaries of of laws or something like that. You know, things like that. So there's that kind of side of things. And then there's the policy in And what I do what I work with here's there's talking with here here are just some examples, I guess that's what I'll say. So talking with uh National Science Foundation, national academies about what

uh w what funding opportunities should exist for mathematicians in that space, like by the NSF. So over the past couple of years they've had to develop programs that you can apply to to get money for doing research in that area. So that's that's one example. Congressional members are introducing legislation right and left. There was something introduced today, probably yesterday. I got the announcement this morning, so probably yesterday, about telling the NSF.

to fund some education initiatives in AI. And I haven't read it yet, but what I don't know is is that talking about my guess is it's talking about AI literacy? So, you know, to me there's two like when people talk about AI in education and math, you know, I think about two things. AI literacy. Maybe I think about three things. AI literacy generally. AI for like learning from the student's perspective and teaching from the professors perspective. And then there's sort of

the real math in AI. You know, how is how to develop the math to advance AI and how AI can advance math research as we're Seeing, for example, in things like first proof. And congressional members will ask me for feedback from the community. these bills. And then, you know, of course they're really interested. There's a lot going on with like regulation of AI. We don't

have so much to do with that. I mean we could, but um we don't exactly. I mean, Noah, you've been involved with that over the you know, in the past a bit too. Dej, jag skulle ju köpa några nya palpställa till lagret. Det kanske blev lite mer grejer. De hade ju allt, man hade en skribord, jag köpte en sån här, och kontorstolar, och sen hade de en skit snygg till. Vi har inredning för hela arbetsplatsen. Välkommen till AI.

I not a lot of regulation is coming about, coming to fruition. Yeah. Yeah. So what wha speaking of news timely events, um, you mentioned one that you were just reading this morning. This morning while I was driving my kids to school, I was listening to an article, I think it was in science. And it was talking about a sort of reprioritization at the NSF to basically really drive towards AI and quantum research.

And I'm just kinda wondering, on the one hand, this is almost like a personal advice question. On the one hand, it feels so exciting for mathematicians to move into AI to use it in the research, to maybe try to develop foundational theories of the math of AI. On the other hand If we don't want to do that, are we

being getting left behind. What do you feel? Should we all jump on the bandwagon or should we keep doing what we're doing? What what do you feel? Yeah. So th I've got a lot to say on this. So first of all, for th that question, um I think You're right. Math needs to be a part of it. I don't think everybody needs to jump on the bandwagon. I'm glad to see there's groups.

as for example, first proof group, you know, who are doing this. And you might have seen the interview with Terry Tow in the Atlantic last week. So there are there are definitely, you know, many mathematicians who are jumping on and I think that great. I don't think everybody needs to or should for that matter. in terms of our research. Now, for the funding thing, okay, I'm not gonna defend the administration, but I will say that every administration has its priorities and so that's not unusual.

there are unusual things in this with this particular administration and those two priorities, which is they're really directing the the NSF and other agencies to spend much more in those areas.

So this is right now, so in late January the fiscal year twenty twenty six budget was passed for the National Science Foundation and many other agencies and they have to come up with what they call their spend plan. now and they're working on that right now. That means how are they going to actually spend that money? And we will see how much the administration's putting a heavy hand there and with those priorities.

So it's not on you every administration, every August they put out their R and D research and development priorities, like this is an annual thing that happens. The the thing about the National Science Foundation, one reason I really love that agency is because it's the only science funding agency that doesn't have a certain end to its research, like the NIH or the Department of Defense, like any research funded by the NIH.

has to do with health. The, you know, Department of Energy, they fund a lot of research, but you have to tie into their mission. National Science Foundation just funds good science across all fields, wherever it is. And so that's

a pro like this it feels like it's being told that to have this mission of AI and and uh quantum.

More than with past administrations. Now, do you think the community is ready, essentially, for these transformations in AI? The the bigger ones, some people yes and some people no. I mean, there are people doing great stuff in terms of well, there's the math research we already talked about. We're starting to see a lot of curriculum development at a high level about how to train mathematicians to do that research.

Then we're also seeing curriculum development about sort of the AI literacy thing, uh um, which is different. So in some ways and then I think there's some people who are ignoring it altogether. I mean, but you know, this happened with Computers. So, Karen, one thing I find fascinating about your career is you've there's this interesting symmetry where you've worked on the math of very political things, but also the politics of mathy things.

So I'm curious, are there any lessons that you've learned along the way on either side of that fence? I think on both sides there are people who are curious, you know, and fun to talk to and explore ideas with and that exists no matter no matter what field you're in and that includes mathematics. the broader higher ed world and then also working on policy. Now talking about policy, what actually made you switch from academia to your current

So it's kind of a I've always loved politics. While I got kind of into this looking at the redistricting stuff like I was talking about at the very beginning, I took a sabbatical and went to Berkeley and took some graduate political science classes. And then I got on in twenty ten I was on the Minnesota Citizens Redistricting Commission. So I was sort of getting involved in politics. I'd had my eye on this

fellowship at the AMS for a long time, applied, got it, came to work for Senator Franken. I figured that all that the year in Congress would enable me to move to a position as a dean or provost. That's where I saw my career going. I love teaching, but I'd done everything I could at my small college. I w had been department chair for a long time. I had

chaired all the all campus committees and I'd done a lot of leadership. I had developed a program for new chairs, new faculty development, dear friend of mine, another female who when we started having children, you know, there was no leave policy at all. So we developed a leave policy that our college adopted for parenting. So I'd done a lot of stuff like that.

And I wanted to figure out how to support the careers of other people, basically. Other mathematicians, but also other early career faculty and then women, uh, in all of sciences probably and all of academia really. So I thought I was gonna be a dean or provost. And then this job came open and this seemed perfect because I didn't want to leave math and academia and so this way I left academia, but I interact with more academics than I ever did before at my small college.

through the AMS leadership, the volunteer network, the membership. I interact with a lot of them. And then I also get to run three fellowship programs where I have four either grad students or other mathematicians each year that I get to sort of mentor and get to know over the years. And I actually do a lot of mentoring still too.

various peop various mathematicians typically, but even other other scientists and now science policy people in DC, which is kind of So one message that I hear in your career story is that I think when we're younger, I don't know if it's career advice or self imposed, we often feel a pressure to kind of keep our blinders on, stay on the the straight ahead path and don't deviate too much.

That doing the kinds of things he did like on a sabbatical to take graduate courses in a different discipline in political science rather than just publishing another paper would feel almost a nevma to you know what our PhD advisors might want. And now I was a full professor already on that medical. If you saw my transcripts, I took stuff in education, I took stuff in program evaluation, I took things in other trajectories. You won't learn it later in your career.

Let me try a positive spin on that, which is I suspect. All those digressions in your education, Autumn, and and Karen in your career ended up really being beneficial in the career path that you chose, that these things all paid off. It did. And I'm a huge liberal arts proponent generally, I think.

And I think we can't predict what's gonna be useful. So I think you just have to try things and do what you enjoy that you can do. Yeah, so back to the AI thing a second. Like, you know, I remember students you know, you decades ago being like, I have to learn Python coding or something and it'd be like, sure, but that you know, by the time you're working in a decade, I mean, who knows? I mean that

That was a good choice. But like with AI right now, you know, if I hear a student or someone else obsessed with learning a very particular I think it's misguided. Yeah, you don't I mean, it sounds hyperbolic to say, but like don't just follow the crowd, you know, create your own path and see where it leads. Yeah. And I think all three of us have done this and I encourage our listeners, just trust your instincts. If something is fascinating, which it sounds like politics has been fascinating.

throughout your career, even as a mathematician, you'll find a way to bridge those worlds and combine them. Now, usually we ask at the end of every interview one message that they'd like to share with listeners that they want them to take away. But In your case, I wanna change this just slightly. Is there one message concerning math that you would like to give to political leaders? Now part of this is colored by the times, but I would say that math is

fundamental in almost everything that goes on in the modern world now. And everybody really needs to learn it and we need to invest more in our teachers. to make that happen in this country. Karen showed us that math isn't just a classroom subject. It shapes representation, resources, and the decisions that structure public If this episode shifted how you think about math in the real world, share it with one person.

You can find Breaking Math on Substack, Apple Podcasts, Spotify, YouTube, and most other platforms. And as always, stay curious of the world around you. Med 226 gram saftigt kött, 100% nötkött och den perfekta syrliga Big Arch-såsen. Nya Big Arch för riktig McDonaldshunger. Nya Big Arch. Så mäktig. Med 226 gram saftigt kött. 100% nötkött. Och den perfekta syrliga Big Arch-såsen. Nya Big Arch för riktig McDonaldshunger.