傳播數學知識．促進數學教育

Interview with Prof. Maria Chudnovsky

**Interview Editorial Consultant: **Tai-Ping Liu

**Interviewers: **Tai-Ping Liu (TPL), Ko-Wei Lih( KWL)

**Interviewee:** Maria Chudnovsky(MC)

**Date: **November 13th, 2017

**Venue: **Institute of Mathematics, Academia Sinica

Prof. Maria Chudnovsky was born at Leningrad, the Soviet Union on January 6, 1977 and later moved to Israel. She received her PhD in 2003 from Princeton University and then held teaching positions at Princeton University and Columbia University. In 2015, she returned to Princeton University as a professor of mathematics. For her outstanding contributions to the graph theory, she was awarded a MacArthur Fellow in 2012.

TPL: Thank you for coming. Ko-Wei reminded me that you stayed in Russia until 13 years old.

MC: That's true.

TPL: So that is a very important part of one’s life, right? Now you're in Princeton. Is this journey rather nonlinear? How did this happen?

MC: I was born in Russia and then when I was 13, my family moved to Israel because that time we were allowed to leave Russia, so it was a good move. And then I went to high school in Israel, and I studied math in Israel. And then I decided I wanted to do PhD in math, and so I applied to the places that I thought would be best for the kind of math I like doing, and Princeton accepted me. Then, I moved to Princeton and got my PhD. I was a postdoc there for 3 years, and then I thought I got to get a job, I moved to Columbia and I was very happy there for 9 years, and then I moved back to Princeton.

TPL: I'm also interested in your time in Russia. Did you get interested in Mathematics, so in somewhat serious way in Russia?

MC: When we lived in Russia, I went to a special math school, and it was sort of ... the atmosphere and the culture were very nice. Basically, being good at math was the best thing you could be. Everybody in the school as far as I know wanted to be good at math. So, it's true, by the time I was 13 I thought if I'm very lucky I'll be doing math for the rest of my life. So, the roots that are probably in that school.

KWL: Were you selected to enter that special school?

MC: So, for the way it is for the first 6 grades of the school, There was some selection entering grade 1, but I don't think it a math biased selection, just a general selection. Then at the end of 6th grade we had to take a very serious exam to be allowed to go into a math 7th grade which I did, and I passed it. And I remember it was the first sort of serious exam I had ever taken in my life. It was three hours and I was 12 years old, and I remember walking out of this exam just thinking “I have not been so tired ever in my life”. I just didn't think it was possible for a human being to be so tired.

KWL: What topics were covered in that exam?

MC: I don't remember anymore.

KWL: Oh.

MC: A very tiring one.

KWL: Did you have geometric problems in that exam?

MC: Probably. There must have been. But I mean, honestly I'm just saying it from sort of common sense, like what kind of questions can you ask a 12-yr old? Well, there's some geometry problems, some equations, some algebra. It wasn't like a math competition type of thing. It was the kind of thing you just study at school, just hard problems.

TPL: More like an IQ test?

MC: No, no. Like a school math test, just really hard and really long. It wasn't a standardized math test. It wasn't a math competition test. It was just a hard school test.

TPL: That kind of school is everywhere in Russia? Or not?

MC: You know, I'm not really sure. I think probably more so in Moscow and St. Petersburg. But you know, we are talking about 1988 or so. Now, I really have no idea. I don't really know, even back then I don't really know. I think there was at least one other high school like that, that I knew about in St. Petersburg, and most mathematicians I meet today who grew up in St. Petersburg went either to my high school or to that one.

KWL: So, when your family moved to Israel that was after the collapse of the Soviet?

MC: No. It was before

KWL: Before. Ah, okay.

MC: It was a couple of years before. We could already leave, but the Union still existed.

TPL: The first few years in the West maybe you were too small to see the culture shock and so forth?

MC: Well I could certainly see there were differences. I was very eager to learn them and embrace them.

KWL: What language did you speak when you moved to Israel?

MC: I learned Hebrew when I moved to Israel.

KWL: So, when you were young you would learn Hebrew in your family?

MC: No. Before we moved, I got a book. You know, a self-studied study book in Hebrew, and I started during the summer and then when we came to Israel, I could speak some. Because there are so many people coming to Israel from different places, they have excellent programs for learning Hebrew and so on, so that was not a problem. A year later, basically, I was comfortable with Hebrew.

TPL: Hebrew cannot be an easy language to learn.

MC: It's actually a very easy language. It's very, very mathematical. There are very few exceptions. There are very clear rules. For example, all verbs fit into a matrix, indexed by two attributes. The position in the matrix tells you exactly how to conjugate the verb.There's none of this funny business like in English. And there are other things. The language is built from roots and then there is a finite number of ways to create a word. So, if you know the root even if you don't know the word, you can tell what it means. It's actually, it's kind of fun language to learn if you're into math.

KWL: So, is it written from right to left?

MC: It's written from right to left, that’s right. But you get used to it. It's also written without vowels, but again you know they say now that when you read any language you don't actually read every letter. You just kind of glance at a word, and you guess what it is. So, in Hebrews, it's kind of built in because the vowels are not there.

KWL: So, vowels never appear in print?

MC: No.

KWL: Okay.

MC: I mean in a kid's books they do.

KWL: I see, I see.

MC: But as a result, all books are very short. There's something psychologically pleasant about getting to read a short book instead of a long book.

TPL: It's implicitly clear that this vowel has to be there.

MC: Yeah. And just like, you know, let's say, you read in English and instead of "cat", you have "ct" everywhere in the story, you could guess what it is.

KWL: You got your master's degree in Israel.

MC: That's right.

KWL: Aharoni was your advisor. He's done some work on the infinite combinatorics using the version of Menger theorem.

MC: That's right. He works in combinatorics; he worked on a Menger’s Theorem; he also worked on hypergraphs. And actually, when I worked with him we were working on hypergraph matchings.

KWL: Okay. So, you were only working on hypergraphs with him.

MC: Yes. At that point I had never done anything else in my life

KWL: He also used some theoretical methods, right? So, infinite coloring or things like that.

MC: I think he's very interested in infinite stuff, but for the couple of years that I was around he took a break from infinite graphs.

KWL: I also know he published a book 'Arithmetic for Parents'.

MC: That's right.

KWL: He taught in elementary schools.

MC: I don't think he taught in elementary school but he --- But he felt that the way that mathematics was taught in Israel was unacceptable. He's friendly, very energetic; he went and talked to everybody about how it shouldn't be done like this, it should be done like that, and he succeeded. They changed the way math is taught. I could not believe it, that one individual could change it.

KWL: Were you involved in that project though?

MC: No, I ‘m just in awe of his ability to do it.

KWL: Would you like to describe a little bit how discrete math was developed in Israel.

MC: It was already there when I started. I think it's a very strong field in Israel. I don't know how it became like that.

KWL: So, Erdős must be a very big influence for that, right?

MC: I'm sure he is. He visited and had connections to Israel. Do you know that there's now a children's book about Erdős?

KWL: Yeah. I heard about that. That's a cartoon book or something?

MC: No, just like a book for kids. There was this little boy. His name was Paul. He likes numbers. It talks a lot about how he just did math with his friends. And there's an illustration there, you know, it’s like a cartoon illustration of Erdős and his friends sitting around the table. But everybody is a real person. Fan Chung is there, Ron Graham is there, Béla Bollobás is there, it's really--- I have this picture on my phone if I remember to show you. But you know, I actually, I saw Erdős when I was in Israel, but he didn't meet me. I think I saw him at 2 conferences or 2 talks. And you know, it's an incredible experience for a 17-year old.

KWL: Okay, I see. Once in, maybe 94 or a little bit before that, I went to a conference in San Francisco. At that occasion, Erdős even wrote something on transparencies. He actually usually used blackboards. So, when that was over, everybody left, I thought this was very precious. So, I collected his transparencies.

MC: Do you have them?

KWL: Yeah right. Yeah.

MC: It's very cool.

KWL: It's a very rare item. So when you went to Princeton, you just got interested in perfect graph conjecture immediately?

MC: When I arrived, I learned that Paul Seymour was working on it. I guess I was very bold. I did not know where the limits were. So, I went to Paul and said, "I would like to work with you on this paper of perfect graph conjecture, would that be okay?” And I guess he was so surprised, shocked, and that he said 'Yes.'

KWL: At that time, did he finish his project on graph minors?

MC: I think so. They finished it a long time ago, but it was not written down. In fact, the last paper only appeared a few years ago.

KWL: Number of?

MC: 22

KWL: 22. Okay.

MC: People were like--- There was no talk about graph minors anymore. But you know, so for myself, I became interested in discrete math when I was in college. I must just have taken some very good courses, because I started math and wanted to do math and some math was more appealing, and some Math was less appealing. The math I like most was discrete math. It's always a question: "Is it something in your brain that makes you better in one kind of math than another, or is it just the teachers you had that made you like it?"

KWL: So, before you went to him, did you take his courses?

MC: Paul’s?

KWL: Paul’s.

MC: I knew he was working in graph theory on that. I knew something about graph theory and so on. Because I already had a master's and so, yeah. I came to Princeton to work with Paul and that was my plan. And he knew it, we corresponded before I saw him, so I showed up, and said "Hi, I'm here, can I work with you?"

KWL: Besides Paul, was there anyone else who worked on graph theory in Princeton?

MC: There was Benny Sudakov who's more probabilistic.

KWL: Now, he moved to Switzerland?

MC: He's in Switzerland. Yeah. He was in UCLA for several years.

KWL: Was there a price offered for the solution of the strong perfect graph conjecture?

MC: That's actually a very interesting question. I often talk about it during my talks. So, there's this mathematician Gérard Cornuéjols, and he has a very precious book. It's a $\$$90,000-book that contains 18 problems and for the solution of each of them, you'll get $\$$5,000.

KWL: Really? It's a private book?

MC: Yeah. But you have to do it by 2020, so we're almost out of time. But Cornuéjols points out that this is not an efficient way to make money - to solve problems from his book. But anyway, the strong perfect graph conjecture is there, and there's another conjecture there called the skew partition conjecture, and we proved both, but we were worried. The skew partition conjecture says that something cannot happen in the minimum counterexample to the strong perfect graph conjecture theory. So, if you prove the strong perfect graph conjecture, the skew partition conjecture follows, but there was a concern. Are we going to get paid for one conjecture or two conjectures? I'm pleased to report we got paid for both.

TPL: What do you see, just in your personal view, the future of graph theory? What would you like to be able to do or what the picture you envision?

MC: I would like to see connections with other fields, because I think it would be good to have some tools in graph theory. On the one hand, it’s the feeling about graph theory. You just sit down and you do it. You don't need to study for many years. You need to learn the techniques, you need to learn the tricks, but you don't need to learn a lot of backgrounds. And it's fun, it's nice. But on the other hand I think somehow we've solved all the easy problems. Many papers on graph theory now are just technically very, very, involved, and it would be nice to have some tools that would somehow bring some more---

KWL: Kind of a feeling of some kind of theories?

MC: Yeah, yeah.

KWL: Don’t you regard your structure theory of graph as a part of theory?

MC: I do. But it's hard to say “we develop this and now you can use it for that.” I don't…For example, we developed all these theorems and we proved the strong perfect graph conjecture, and now we can change parts of them and prove other things. But it's not that the next generation can just say by the theory developed by those we can do that. So, it would be nice to have some, something like that. So, for example, at one point I was talking with David Bayer in Columbia and he had some idea of proving the 4 color theorem by algebraic geometry which I don't think quite works. And I think, other people have had similar ideas but so far I think none of them worked, but something like that you can sort of connect from other places.

KWL: Have you ever touched the Hadwiger conjecture?

MC: I have, it's a beautiful conjecture. I have some results.

KWL: I see. You see that, suppose once it is solved, it will shade some great light on the four color theorem.

MC: I suspect that if it is solved, the proof will use the four color theorem.

TPL: I know there's a computer-aided proof, is there a computer free-proof?

MC: No, there's not a computer-free proof.

TPL: You said that you would like to have a tool or develop a tool. You must have in mind that the other part of mathematics has a tool, a developed tool.

MC: I don't unfortunately. It always seems like in another part of math, they have all these general ways to approach things. I don't know, maybe I'm fooling myself, maybe nobody has them. I guess what I mean is, for example, when I worked with Ron Aharoni, we used topological methods. So you have a graph theory problem or hypergraph problem, and then you encode it as a problem about simplicial complexes. And then there's already known theory of simplicial complexes, you can just use homology and so on and you can deduce conclusions on the combinatorial problems. So, things like that where you can sort of say, "Oh you know, this is like that, because it's actually a special case of something, something much more general……..''

TPL: The great harmonic analyst Lennart Carleson once said that, instead of a general theory, general tool, he reduces the problem and then at the core it is a combinatorial problem you just have to compute it out.

MC: I must say while I think that having tools would be good, what I do is completely tool-free. Start at the start and you develop things you need for this problem, and then solve the problem. And then you go to the next problem and... I mean again, it's not exactly true, some ideas transfer, but there may be a better way to say it. It doesn't have to be a tool from another part of math. All these ideas that we have, and then we teach our students and then they know them. It would be nice to somehow turn them into something more cohesive. You can just quote something rather than tweak, take a theory that basically exist and tweak it to work in another setting.

KWL: Because our magazine is just for general readers, so probably just add a little more about the basic definitions of perfect graphs and the strong perfect graph conjecture. Just for the sake of the reader.

MC: In graph theory, we have a notion of graph coloring. That means assigning colors to vertices so that adjacent vertices receive different colors. The minimum number of colors that makes such an assignment possible is the “chromatic number” of the graph. There is an obvious lower bound on the chromatic number, that is the maximum number of pairwise adjacent vertices. This second parameter is called the “clique number” of the graph. The first fact is that the chromatic number is at least the clique numbers but there is no bound in the opposite direction. There are graph with clique number 2 and arbitrarily large chromatic numbers. Then the question becomes: when can you upper bound the chromatic number as a function of the clique number? The first step forward this question is to understand when the clique number and the chromatic number are the same. You need to be a little more technical here, but basically a graph is “perfect” if the two numbers are equal. And there was a conjecture, made by a French mathematician Claude Berge that gave a structural characterization of perfect graphs. It listed all minimal graphs that are not perfect. It was open for 40 years, Robin Thomas, Neil Robertson, Paul Seymour and myself proved it. This was the Strong Perfect Graph Conjecture.

KWL: So, did you ever meet Berge?

MC: I did not meet Berge. I missed by a little bit. Just at the time that we proved the theorem, it became clear that he was dying. We don't know if he knew that the problem was solved. He was told, but it's not clear if he was able to understand.

KWL: I met him once in the Philippines. There was a conference around 91 or something. I went there. He was much welcomed by the Philippines. And he also collected all kinds of anthropological items.

MC: That's right. That's right. He must have been a very interesting person. I'm sorry I didn't get a chance to meet him.

KWL: After you proved the strong perfect graph theorem, you developed the structured graph theory in other directions, right?

MC: Yes. We worked on solving the strong perfect graph conjecture theorem, and there was a question of designing an efficient algorithm to test if a graph is perfect; and a group of us found such an algorithm. So that was done. But in a sense perfect graphs are not done. There's this general question, "How do you build the most general perfect graph?", and we don't know the answer for that. The proof of a strong perfect graph theorem, started out in that direction, but then as soon as we knew enough to deduce the strong perfect graph conjecture, we stopped. Because we did not know how to go any further. But that's still a very interesting open question. Actually, I'll talk about that in my talk a little bit tomorrow. So, that was 2002, the structural characterization of perfect graphs in terms of something called induced subgraphs. Like I said, the conjecture was what the minimal non-perfect graphs are. But to say minimal, you need an order. And so, being an induced subgraph is to one of several containment relations you can think about. But that's what I've been working on ever since. Problems related to excluding induced subgraphs. For example, there's a very active area now: so perfect graphs have chromatic number and clique numbers the same; but what about chromatic number being at most of a function of the clique number, when is that true? There has been a lot of progress in that area, and I've been part of that also. There was a question "What other graphs you can exclude as induced subgraphs and understand the structure?" There are some theorems like that, but that seems too hard. It's interesting. So there's the excluded minor theorem, that was before my time. I say okay, my contribution to the world will be an excluded induced subgraph theorem. But it doesn't work somehow. It's, maybe I can do something for small graphs, but there's not a general construction, it's not clear what it could possibly be. It seems now that maybe just, we need to kind of redefining --- to adjust our expectations. Maybe there is something good you could say about excluding a general induced subgraph, but it just won’t be as explicit as when you exclude minors. Maybe something algorithmic.

KWL: Would you please add a little bit more, why the perfect graphs are important?

MC: Why perfect graphs are important? To be completely honest, I think they are mostly important because in 1961, Berge defined them and a million people worked on them, and they couldn't get anywhere. So, by the time it was 2001, they were important. That's sort of how math is. If it's solved right away, then nobody hears about it. If it's so hard that nobody can get anywhere, and then nobody hears about it. But if it's something that people keep working on, by the time it's been open for 30-40 years, suddenly it is important. It has connections to various things. It generalizes a lot of graph theory that was known at the that time. It has connections to combinatorial optimization. I can tell some story about how it's related to things, but I think the real reason why it is an important theorem is because there are were so many people trying to solve it. And it's pretty. Basically it says the world is not chaotic, the world is beautiful because there's only two obstructions to this nice property. And again, that's what math is. It's not chaos, it's structured like this.

KWL: You developed a structure theorem for claw-free graphs and so called bull-free graphs?

MC: Yes. Bull-free graphs, that's right.

KWL: Why those two kinds of graphs are important graphs?

MC: I'm trying to remember. Both for perfect bull-free graphs and for claw-free graphs, there are various optimization problems that could be solved in those classes in polynomial-time, but in general they are NP-complete. And so, you know, you always ask, why is that? Why is this problem so easy? And the chances are it's because there is some underlying structure. Maybe the people who developed the algorithms didn’t understand the structure yet, but these algorithms work because there's a structure. And this is sort of a standard passage. We understand the structure, and then an algorithm comes out for free. For claw-free graphs, it was like that. Actually, I remember exactly Bruce Shepherd kept coming to Paul Seymour's office and saying, "But claw-free graphs, but claw-free graphs. You should do claw-free graphs". And maybe there were other people saying that, but I distinctly remember Bruce Shepherd---

KWL: Are there common structures between claw-free graphs and the perfect graphs?

MC: It's a very interesting question. There is actually a structure that comes up in both. And for a while it seemed that maybe it is the magic. And it's fairly clear how to make it more general. It seemed that maybe it is like the concept of surfaces in graph minor theory. But now it seems that it was just an illusion.

KWL: Please just first give us the definition of bull-free graphs. How come the bull comes in?

MC: A claw is just vertex with three pairwise non-adjacent neighbors. To be claw-free means to not have a vertex that has three non-adjacent neighbors. Now, a bull is a triangle with two pendant vertices. And if you draw it correctly, then the triangle the face of the bull. So, I'm trying to remember how I started working on bull-free graphs.

There were theorems known about both bull-free and claw-free perfect graphs. They were similar theorems by similar authors. And it seemed that if we push harder we can describe the structure of these graphs completely. And then it turned out we could get rid of the perfection assumption too. Can I just say something else about bull-free graphs? There's the structure theorem which is very hard. It's nice to have it, but I think basically it’s too hard to use. But something very nice came out of it. There's a conjecture called the Erdös-Hajnal conjecture. This conjecture is known to be true for very few graphs. There's an operation where if you know the conjecture for these graphs, you can put them together, and you'll know it for bigger graphs. But if I’m talking about prime graphs, it doesn't matter exactly what it means, but ones that you cannot get by this operation, then it's known for very few. And the biggest one for which is known is the bull. I think, of my work on bull-free graphs this is probably the prettiest part. Definitely the prettiest part, but also the most important part. And that's nice. It's like a 15-page paper. Just magically everything is beautiful.

KWL: But you know, your structured graph theorem, your proofs are very long.

MC: And the theorems are very long too. Like for example this structure of claw-free graphs. The theorem is that you can get them all by gluing in certain ways basic graphs, that are called strips, and there are 15 kinds of strips. It's kind of amazing that it's 15 and not infinity.

KWL: Right.

MC: So, as you know, it is worth writing them down, publishing, but on the other hand the theorem is 5-pages long, because you have to describe 15 kinds of strips.

TPL: In Princeton, there's a good size group of finite math — not counting number theory, right?

MC: Right, right. It's a great place for discrete math. We have senior people, junior people, people rotating, and students; a perfect place to do it.

TPL: Where else the finite math is located in the US?

MC: CMU, San Diego, I mean Princeton, CMU, Stanford, UC San Diego, UCLA

KWL: MIT?

MC: MIT. Where else? There are a couple of places, Urbana-Champaign, probably others that I am forgetting.

TPL: Now, I always have this vague impression that finite math is important for computer science. They are quite related. Is that so?

MC: I think it is because of the algorithmic side of things; we both are interested in algorithms. So, one set of problems I worked on is problems related to complexity, to designing efficient algorithms. You exclude an induced subgraph and you want a polynomial-time algorithm to three color a graph. Generally three coloring is NP- complete, but if you exclude some induced subgraphs, then it’s polynomial. One reason something can have a polynomials-time algorithm is because all obstructions are finite. And that's again, if the world is pretty, that's how it is. But the world is not so pretty, so sometimes, the number of obstructions are infinite, but there's still finite, there's still an efficient algorithm. And so, with a group of collaborators, we were working on designing efficient algorithms, then we did something with that, and then we started thinking about finite obstructions structures which you know, as a mathematician, to me that is so much nicer a problem, and one problem we solved. We found exactly all the cases wherein the number of obstructions is finite. We submitted it to a computer science conference. And they rejected our paper and they said, "Yeah, this is unfortunately just talks about the number of obstructions and not about the complexity of the algorithms." which you know from their perspective I'm sure that what it looks like. From my perspective, it was like so much more of an interesting and fundamental question.

KWL: You are also affiliated with the program of Applied and Computational Mathematics. What's the nature of that?

MC: That's just how it's structured at Princeton. Everybody who does discrete math is half math, half PACM.

KWL: So, that's not a department or something-

MC: No, there's a Program in Applied and Computational Math. But this I am there because of this.

KWL: Did you have a joint work with the people from engineering?

MC: I did when I was at Columbia. At Columbia, I was in the engineering school. I actually had a few papers with electrical engineers. I worked a lot together with someone from computer science, we had a joint student, that's even more than a joint paper. So yeah, that was sort of the fun part of being in the engineering school. People would knock on my door and ask me a question I could answer.

KWL: Is Dr. Liu Chun-Hung, is he a member of your group?

MC: He is.

KWL: You're collaborating with him?

MC: We wrote a paper together, yes.

KWL: I see. Good. He was a graduate student; he was graduated from National Taiwan University.

MC: That's right. He told me to come here. I said, Chun- Hung I got this invitation from Taiwan, what is it? He said "I don’t know what it is, but you just go".

KWL: He's very good, yeah.

MC: He's very good. He's very good.

KWL: He was a master's student of Gerard Jennhwa Chang. He went to Georgia Tech, right?

MC: That's right. That's where he got his PhD.

KWL: With Thomas.

TPL: What are you working on right now?

MC: So, I'm working on a couple of things. I'm working on problems related to the Erdös-Hajnal conjecture that says that excluding one induced subgraph gives a big clique or a big stable set. I'm working on problems related to bounding the chromatic number as a function of the clique number, and I'm also working on this question, “What do you need to exclude to make coloring polynomial?” And it's almost done. Not by me, but by the world. Basically, for almost everything we know when it is and when it isn't polynomial and almost always it isn't. But there are still a few cases left you might try to exclude and I'm working on that, and it's fun. Somehow it's a general problem, but there's not so much left. It's kind of a fun direction to work in.

KWL: That's a big problem, right? A big result if you finish.

MC: I hope so.

TPL: You worked with quite a number of people, and then your first one was you went to this Paul Seymour. The other day, an Australian female mathematician called Nalini Joshi came here for a conference on gender gap. For you, there's no gender gap?

MC: Let me tell you a story instead. Because I grew up in Israel, I had to serve in the army. So, my first day in the army, I went up to my boss, I said, "I would like to take courses at the university, so is it okay if on Thursdays I come late and stay late?". And I think he was just so shocked, he said "Okay". Later it turned out that in fact you could do that but only if you had been there for a couple of years. But luckily I didn't check the rules. I think not dwelling too much on what you're supposed to do and not supposed to do is helpful.

KWL: So maybe you could say that if you don't feel the gender gap then you're okay.

MC: You know, I don't really want to talk about that because I think it's not that simple.

TPL: I know. I know.

MC: There's like how bold you can be, and then there's what kind of push back you're going to get. It helps to kind of pushing as hard as you can, but on the other hand you know---

TPL: Let's just not pretend this does not exist. Because it exists.

MC: It shouldn't be used as an excuse not to push as hard as you can, but on the other hand not everything is perfect. You know, I'm happy to say something, but I don't want to make a blanket statement about gender gaps. Yeah, it's all complicated. Compared to that, math is easy. Well, “easy” maybe is not the right word. (TPL:Math is more “simple”.) Yeah,yeah. That's true because most people cannot imagine unless you are in that position, right?

TPL: For people in the majority to say that we can feel the pain of the minority, usually that is not so real?

MC: Well, you can be aware. I think there's a difference between feeling the pain versus trying to minimize this pain in others. You don't need to be a monkey to be a veterinarian. Not everything has to be a firsthand experience.

KWL: Do you know the international mathematical union joins with the international union of chemistry. They sponsored the kind of the gender gap survey and this project was led by a female professor here in National Taiwan Normal University. They just had some kind of working meeting this year.

TPL: That's why Joshi came here.

KWL: That's why she came here.

TPL: It's not very clear which culture has more gender gap than which other culture.

MC: I agree.

TPL: Like here in Taiwan, there's gender gap. But in some aspect, I think, it is less than the US?

MC: That's what my guidebook said.

TPL: Enjoy your time here and your trip to Palace Museum.

MC: Yeah, sounds good. That's my plan.

TPL: Okay, thank you.

MC: Thank you very much.

- Tai-Ping Liu and Ko-Wei Lih are faculty members at the Institute of Mathematics, Academia Sinica.