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Alexander Kechris

Alekos Kechris
Alexander (Alekos) Kechris

Alexander (Alekos) Kechris

Professor of Mathematics

By David Zierler, Director of the Caltech Heritage Project

February 11, 2022

DAVID ZIERLER: This is David Zierler, Director of the Caltech Heritage Project. It's Friday, February 11th, 2022. I am delighted to be here with Professor Alexander (Alekos) Kechris. Alekos, it's great to be with you. Thank you for joining me today.


ZIERLER: Alekos, to start would you please tell me your title and affiliation here at Caltech?

KECHRIS: I'm Professor of Mathematics.

ZIERLER: Alekos, tell me a little bit about your main field of research in mathematics. There's so many different ways to pursue math. What would you say has been the central focus in your career?

KECHRIS: My general area is mathematical logic, including aspects of set theory, definability theory, computability and model theory, but in recent decades I've been working on connections of mathematical logic with other areas of math, especially analysis, dynamical systems and combinatorics.

ZIERLER: Alekos, some nomenclature, some name questions—what is set theory? What does that mean?

KECHRIS: Set theory is an area of math which investigates the concept of set, which is simply a collection of objects. Set theory goes back to the work of the 19th century mathematician Georg Cantor. Today it can be viewed as a basis for developing all of mathematics. In other words, you can base mathematics viewing in principle all mathematical objects as being sets. There are some basic laws about the universe of sets, which are called the axioms of set theory. Then you can develop mathematics within the axiomatic framework of set theory.

ZIERLER: Another theory question—definability theory. What is that?

KECHRIS: It is the study of mathematical objects through their definition. It is concerned with mathematical objects that can be described by formulas of some sort and investigates the connection between their definitions and their structural properties.

ZIERLER: Computability theory—does that mean something that simply can be computed?

KECHRIS: Yes. Computability deals with processes that can be computed.

ZIERLER: One last one that makes me smile is model theory. Of course, all theories have model components to them. What is model theory?

KECHRIS: It's an area of mathematical logic that is concerned with the connection between formal, syntactical expressions and their semantical interpretations, their models.

ZIERLER: To the broad audience of people who will enjoy this transcript who might not be mathematicians, I wonder if you can explain what mathematical logic means. Where it's involved in numbers and where it has elements of logic that philosophers, for example, or computer scientists might recognize.

KECHRIS: Traditionally logic was developed by philosophers but mathematical logic is the study of logic in the context and from the point of view of mathematics. Mathematical logic today encompasses a broad variety of subjects. It deals with the methods of reasoning used in mathematics, for example, proofs and axioms. It also includes the study of the foundations of mathematics, for instance through set theory, which provides a global framework for mathematics. Model theory is another part of logic as well as computability theory. Mathematical logic is also of course connected with computer science, since the theoretical background of computer science is the theory of computability.

ZIERLER: How do you look at mathematical logic in the way that it interacts with dynamical systems?

KECHRIS: Dynamical systems is an area that, broadly speaking, deals with groups acting on spaces and preserving certain structure in these spaces. There's a way of studying problems in dynamics from the point of view of an area of logic, called descriptive set theory, which is part of exactly what we discussed earlier, a study of objects that can be described in some way by formulas. You can bring some of the intuitions from descriptive set theory to study questions in dynamics. For example, mathematicians are interested, both in dynamics and many other areas, in questions concerning classifying objects. We have developed over the last 30 years or so a method of measuring the complexity of a classification, how difficult it is to classify some class of mathematical objects, involving ideas from descriptive set theory. For example, how difficult it would be to classify isomorphism between certain kinds of structures. If you look at it from the point of view of descriptive set theory it's a question of how complicated it is to define such a classification. This is one point of view for addressing some problems that occur in dynamical systems.

ZIERLER: Is topological dynamics a form of a dynamical system?

KECHRIS: Yes. It's a form where you are studying continuous actions on topological spaces. That's a subarea of dynamical systems.

ZIERLER: I've come to appreciate how large a field combinatorics is. In what ways does your research contribute to the broader study of combinatorics?

KECHRIS: My involvement in combinatorics is partly in an area called Ramsey theory. It turned out that we discovered an interesting connection between problems in Ramsey theory with questions in topological dynamics and this connection passed through an area of model theory. So one can combine three different areas: dynamics, model theory and Ramsey theory.

ZIERLER: Some even more technical questions—what are Borel equivalence relations?

KECHRIS: You can think of an equivalence relation as a partition of a set of objects into disjoint pieces. Two objects are equivalent if they belong to the same piece. This gives a relation between these objects. In the case where the objects you are studying form a nice topological space—this is usually called a Polish space—then you have a way of measuring the complexity of sets and relations on this space in something called the Borel hierarchy. The simplest sets are the so-called open sets and the closed sets, which are the complements of open sets. Then you can perform countable operations on those and create more complex sets, called the Borel sets. A Borel equivalence relation is an equivalence relation which as a set of pairs, because an equivalence relation is just a set of pairs, is a Borel set. We measure this way the complexity of a particular notion of equivalence. Intuitively a Borel equivalence relation is one in which one can determine the equivalence between two objects by employing only countable operations.

ZIERLER: Alekos, a more general question that comprises all of your research. I assume all of this is being done within the context of pure mathematics, not applied mathematics.

KECHRIS: That's correct.

ZIERLER: Do you see any areas of your research that do touch on applied mathematics or applications?

KECHRIS: Not directly in what I'm doing. I have worked in areas like computability theory, which as I mentioned forms the theoretical background of computer science. But I have not myself worked in what you can call applied areas.

ZIERLER: What about physics or mathematical physics? Is there anything in physics that's relevant for your work?

KECHRIS: No. Not in my work. Again, there are always connections. For instance, dynamical systems are related to problems that sometimes arise from questions in physics, but after a while they become part of pure mathematics. This is what I'm involved with. I'm certainly not involved in physics.

ZIERLER: What about computation? What role do computers play in your research? Or are you more pen and pad kind of mathematician?

KECHRIS: Do you mean whether I use computers in my research? Not really. Very rarely. There's an occasional problem where I can try to calculate a few things just to get some intuition about this problem but this is about it.

ZIERLER: Alekos, with all of your decades at Caltech some institutional history questions serving all the way back to when you joined the faculty in the mid-1970s. One is has the math department grown over the years? Has it roughly stayed the same size?

KECHRIS: It has fluctuated roughly between 12 to 17 faculty members. Right now we are at the upper range.

ZIERLER: What have been the areas of research in math at Caltech that have been a strong point for Caltech where the best prospective graduate students would say, "I want to come to Caltech because there's a strength in this aspect of the program."

KECHRIS: They have changed over the years. When I first arrived, I would say that in pure math there was emphasis more in algebra, combinatorics and certain areas of number theory and analysis. Later on some areas became less active and others much more active. Right now the faculty represents much broader areas of mathematics especially in analysis, number theory, geometry, topology, mathematical physics, combinatorics, probability and logic.

ZIERLER: Alekos, of course with the unique way that Caltech organizes itself at the divisional level with math being part of the division of physics, mathematics, and astronomy (PMA), what have been some of the challenges or opportunities for mathematicians at Caltech? In other words, there's never been for example a division chair in PMA who's been a mathematician.

KECHRIS: That's correct. Not since the time I came here.

ZIERLER: What have been some of the challenges or opportunities of being in a larger division as opposed to being in a department of mathematics like you'll see at so many other universities?

KECHRIS: There are opportunities in being in close contact with physicists and astronomers but some of the challenges come from our small size which has always been a bit of a problem for us.

ZIERLER: You mean small relative to physics and astronomy?

KECHRIS: I'm talking about comparing us with other leading math departments in the U.S. We are the smallest among the top 10 departments in the U.S. This creates various problems of retaining, for example, faculty who sometimes want to have a bigger group to work with. Our small size has been something of a problem over the years.

ZIERLER: For you generally do you work on your own or do you generally work in collaboration with others?

KECHRIS: Both. In the department usually we have a postdoc in my area and graduate students and we certainly work together. We also have traditionally a very close relationship with the mathematical logicians at UCLA, including a joint seminar. Unfortunately, it hasn't been running the last two years because of COVID, but before that we had this joint seminar every Friday basically since the mid-70s. It's been a long tradition. Of course, I also collaborate with many colleagues in other departments in the U.S. and abroad.

ZIERLER: Alekos, a question about undergraduates. You've been at Caltech long enough where you've witnessed where earlier in your career physics was the dominant major among undergraduates. Today of course, it's computer science. What has been the impact for math at Caltech in the kinds of things students are interested in when they're pursuing math?

KECHRIS: It's true that a lot of the students that pursue math have also an interest in computer science but several of them have an interest in physics and other areas as well.

ZIERLER: Just as a snapshot in time circa February 2022, what are you currently working on?

KECHRIS: Right now I'm working on some problems that are within the context of developing a theory of complexity of classification problems. Part of it is what you referred to before as the Borel equivalence relations. These are ways of rigorously studying problems related to classifications of objects. A classification is basically dealing with some equivalence relation and we study them from the point of view of set theory. This is very closely connected with a lot of other subjects. For example, dynamical systems, including ergodic theory and topological dynamics, but also combinatorics, especially graph theory, group theory, etc.

ZIERLER: This has been a great high level overview of your research career. Let's now go back and do some personal narrative history. Let's start first with your parents. Tell me about them.

KECHRIS: I was born in Athens, Greece. My father was an army officer and my mother a homemaker. My father was born in the town of Chalkis, which is not too far from Athens, about an hour and a half by train. My mother was born in Istanbul, Turkey, where there was a substantial Greek community in the early decades of the 20th century. Her family moved to Greece in the 1920s, in a city in called Thessaloniki, which is the second biggest city in Greece.

ZIERLER: What were their experiences during World War II?

KECHRIS: As I mentioned, my father was an army officer, specifically in the quartermaster and transportation corps, so certainly he was involved in the earlier stages of the war. There was first the war with Italy. Then Greece was invaded by Germany, who occupied Greece until October 1944.

ZIERLER: What was his work after the war?

KECHRIS: The Greek Army was of course dissolved after Germany occupied Greece for several years. Then there was a civil war in Greece from 1946 to 1949. After that my father returned to the army. He did similar work until he retired. Then he spent a few years working in management for a company.

ZIERLER: You grew up in Athens?


ZIERLER: In an apartment, a house—what was it like where you lived?

KECHRIS: It was originally a house, a two story house. Eventually most of these houses in Athens were sold to developers who usually built several story apartment buildings. After that we lived in one of those apartments.

ZIERLER: What were some of the major holidays or cultural observances from your childhood?

KECHRIS: In Greece the major holidays are Christmas/New Year and Easter. There are also a few other holidays throughout the year. Of course there are also longer vacation periods in the summer.

ZIERLER: What kinds of schools did you go to?

KECHRIS: I went to a public elementary school close to my house, about three blocks away. When I finished, I went to a school called Varvakio, which is a very well-known public high school. I was there for six years. This school was further away from my home, in downtown Athens. I usually took a bus or a tram to go there, a ride that took about 45 minutes.

ZIERLER: Were you always interested in math? Did you always have special mathematical abilities that your teachers and parents saw?

KECHRIS: I really became more interested in math around the last two or three years of high school.

ZIERLER: Alekos, there was so much political upheaval in Greece in the 1960s. What stands out in your memory? What were some particularly intense times in Greece?

KECHRIS: Obviously the most intense one was the military coup in 1967. At that time I was at the university. (The military dictatorship lasted from '67 to '74.) I still remember walking in the streets in the morning of the coup. They were empty and the only thing you could see there were tanks. Just tanks in the street and practically nobody else walking around. It was very eerie.

ZIERLER: Was the plan for you when you started undergraduate to pursue math? Was that what you wanted to do?

KECHRIS: Actually I went to the National Technical University of Athens, an engineering school. I was studying towards a diploma in electrical and mechanical engineering, some kind of a combined major, although my emphasis was more on the electrical side. This is a five year program, you can compare it to a master's program. This diploma also gave you the right to practice as an engineer. I of course took a lot of engineering courses but there were also quite a few math ones. Basic courses like calculus, differential equations, etc., but also more specialized courses in applied math and mathematical physics. Personally I wasn't much interested in the engineering side of that. I really liked the courses in mathematics and the many theoretical courses in classical physics, including mechanics, electricity and magnetism, thermodynamics, etc. They were taught at a fairly theoretical level. I really enjoyed them very much. I didn't really care too much about other aspects, like labs, design courses, or machine shop.

ZIERLER: Were you politically active at all yourself during college?


ZIERLER: Two questions. When you graduated how did you make the determination to pursue mathematics and not the more applied work you were doing as an undergraduate? When did you start to think about leaving Greece and pursuing education elsewhere?

KECHRIS: During that time at the university I was studying more and more mathematics. In particular I became interested in the foundations of mathematics and mathematical logic. I decided then that when I graduate this is what I wanted to do, study mathematical logic. It happened that during my last year at the university, in late '68 and early '69, I met a Greek mathematician, a leading expert in mathematical logic. His name is Yiannis Moschovakis and he was a professor at UCLA (he retired from there a few years ago). I had a professor at the National Technical University of Athens, Pantelis Rokos, who earlier taught mathematics in the high school in Athens where Moschovakis was a student, so they knew each other well. Rokos got in touch with Yiannis Moschovakis and we exchanged a couple of letters at this time. When Moschovakis came to Greece on a sabbatical, I started meeting in person with him. He gave me advice on what to do and what to read. When I finished the university there was no opportunity to pursue studies in mathematical logic in Greece and I decided to go to the U.S. I applied to various universities and was accepted in particular at UCLA, where Yiannis Moschovakis was a professor, and he became my Ph.D. advisor.

ZIERLER: How was your English before you got to the United States?

KECHRIS: In high school we learned French as a foreign language. We had six years of French. I didn't really know English. I could just pick up a few things here and there. Then, because towards the end of my university studies I knew that I'd go to the U.S., I had a tutor teach me English. This is how I learned English, basically.

ZIERLER: Had you ever been to the United States or anywhere else abroad prior to this?

KECHRIS: No. This was the first time I left Greece.

ZIERLER: What were your impressions both of Los Angeles and UCLA when you first arrived?

KECHRIS: [laugh] That was quite interesting. LA was very different from Athens in any way you can imagine. UCLA was a very friendly place. People were very nice to me, both the faculty and the administrators in the math department, and they were also extremely helpful. I was married and my wife, Olympia, who is an architect, came with me. We lived in the married student housing of UCLA which is in West L.A., not far from where I live right now actually. Of course in LA you need to have a car to move around. We didn't have a car until I graduated from UCLA and so I was going to school by public transportation. There was a bus from where we lived in the married student housing going to the campus. It was not a problem, but you couldn't really go too far out. Sometimes a friend helped us go somewhere. It was kind of limited, what we could do without a car.

ZIERLER: Alekos, not being a math major undergraduate—was there any catchup work that you had to do for graduate school?

KECHRIS: I had some preparation because I had math courses at the National Technical University of Athens and that gave me some good background in mathematics. Of course, I was also studying on my own. When I started graduate school, I had to take some basic graduate level courses in algebra, analysis and topology. In some of them I had a better background than in others. Then I had to take the qualifying exams. But the background that I had helped me a lot. There was some additional studying to catch up in certain areas, but it was OK. It wasn't particularly stressful.

ZIERLER: Tell me about your thesis research. What did you decide to work on?

KECHRIS: My thesis was in the area of descriptive set theory, which is the study of definable sets and functions in well behaved topological spaces. At that time there was a lot of work in exploring the consequences of a new theory, extending the classical framework of set theory, based on the so-called Axiom of Determinacy, and my thesis was in this subject. I worked in this area for another 10 years or so and then I started exploring connections of descriptive set theory with other areas of mathematics.

ZIERLER: Looking back what were your principal conclusions or contributions with your graduate thesis work?

KECHRIS: The development of classical set theory is based on certain basic laws, the standard axioms of set theory. They're called the Zermelo-Fraenkel axioms with the Axiom of Choice. You can essentially base almost all of mathematics on these, i.e., they can serve as a global foundation for mathematics. However there is the famous Gödel's Incompleteness Theorem that asserts that no matter what reasonable axiomatic theory you have, there will be always problems that you cannot solve in the framework of that theory. The axioms would not be able to show that the answer is yes or no. You have unsolvable problems within the context of that theory. One of the most famous such problems is the Continuum Hypothesis—a question about measuring the size of the continuum, the set of real numbers. In the area of descriptive set theory, which goes back to the beginning of the 20th century and originally was developed through the work of many mathematicians in France, Russia, Germany and Poland, it turned out that there were some basic problems on which by the 1930s mathematicians couldn't make any progress whatsoever.

ZIERLER: What was the problem? Why could they not make progress?

KECHRIS: It turned out that they couldn't solve them because these problems were unsolvable within the framework they were working—which is the framework of classical set theory based on the standard axioms of set theory, that I mentioned earlier.

Of course, this led to a major foundational problem in set theory. What do you do about such problems? How do you resolve them? One approach is to develop new laws of the mathematical universe. New principles that go beyond the classical theory. The hope is that with these new principles, with these new laws, you will be able to resolve these problems. By the ‘60s, there were two main approaches towards that. One approach involved the so-called large cardinal axioms. which postulate the existence of extremely large sets. The second approach involved infinite games. An infinite game involves a payoff set which is a set of infinite sequences of integers. In a run of the game, you have two players who take turns in choosing integers in infinitely many rounds. This way they produce an infinite sequence of integers. If this infinite sequence is in the playoff set the first player wins. If it's not, then the second player wins. The question is whether in such a game one of the players has a winning strategy, i.e., a rule that the player can use in every run of the game to win no matter what the other player does. Such a game is called determined. For example it is a classical fact that a finite game is determined. But for infinite games it's a different story.

It turns out that a new principle was postulated that certain types of games—definable games to go back to that—are determined. This is again something you cannot prove on the basis of the classical theory. It is a new law of the universe of sets.

Mathematicians then developed the consequences of large cardinals and determinacy. What can you do with these new principles? It turned out that many of the old problems in descriptive set theory from the ‘20s and ‘30s could be solved on the basis of them. That was part of what I was doing during my thesis—investigating the consequences of these new laws.

One final comment: As I just mentioned, one had two proposals for new theories in set theory, large cardinals and determinacy. These are of a very different nature but by the 1980's it was discovered that they are, in an appropriate sense, equivalent, so one has now a unified theory.

ZIERLER: Alekos, on the personal front after you defended your Ph.D. did you think about returning to Greece? Did you know at that point you wanted to pursue a career in the United States?

KECHRIS: After I finished my Ph.D., I got a postdoc position at MIT for two years. At that time there were not many opportunities for me to go back to Greece. So I applied for tenure- track jobs in the US and I came to Caltech. After that, in the late ‘70s, I had a sabbatical for a year. I spent most of the time in France, for about six months, and then I also visited Greece just to see how things were there. Again there was no one really working in my field of research and moving to Greece wasn't really something that I would consider seriously. By that time we also had two children that grew up in the U.S., so it wasn't really even practical or feasible to do something like that. It would be so difficult.

ZIERLER: What did you do during your time at MIT? What was your research focused on?

KECHRIS: I continued the research related to my thesis and I also started working on other problems, for instance in some areas of computability theory. MIT had at that time a very strong and large group in my area, including many excellent graduate students, and I had a very good and productive time there.

ZIERLER: You enjoyed your time at MIT.

KECHRIS: Very much. It was a very good place overall and especially for my own interests. We also very much enjoyed living in Boston, despite the cold and the snow. I'd never lived before in a place before which was so cold and had so much snow in the winter.

ZIERLER: Alekos, back home in 1974 the Turkish invasion of Cyprus and all of the tumult this caused in Greece—were you worried for your family at all? Was this a difficult time for you?

KECHRIS: Not personally for my family. Of course, I was worried about the overall situation, but there wasn't any immediate danger to my family, my father and my mother, or other relatives.

ZIERLER: When it was time to go on the job market where were you looking? What departments were you looking at?

KECHRIS: I was looking at places where there was some activity in my own area of research. At that time actually Caltech did not have any regular faculty member in math working in logic. For me though, it was very attractive because of the proximity to UCLA which had a strong group in research close to my own area. Also, we lived in Southern California before so we knew well life here.

ZIERLER: Caltech was strong in your area or your hire was part of an effort to build up that program?

KECHRIS: Yes, the second.

ZIERLER: Was anybody on the faculty doing at least relatedly what you were doing at that point?

KECHRIS: The closest person was Wim Luxemburg, who was actually the executive officer at that time. He was an analyst. He was primarily working in functional analysis, but he also had a strong interest in an area called nonstandard analysis, which applies ideas of logic, especially model theory, to analysis. It's a subject that started by developing a rigorous theory of infinitesimals. Infinitesimals were used in mathematics since the beginning of calculus but without any firm foundation. Mathematicians knew how to work with them, but there was no rigorous theory. Eventually in the 19th century people got away from infinitesimals by developing a firm foundation for analysis. In the 20th century, Abraham Robinson, who was a logician, found a way to apply some ideas of model theory to create a rigorous theory of infinitesimals and this was the beginning of non-standard analysis. Many analysts became interested in this theory and Luxemburg was one of them. He certainly had a connection with logic and he was very friendly to ideas from logic, so he pushed towards that direction to hire someone who was working in this field.

ZIERLER: What were your initial impressions of Caltech? Was it a good place?

KECHRIS: Yes. Absolutely. It was a very good place. It was a very small department compared to MIT and it was very nice that you got to know everybody. There were also amazing students.

ZIERLER: What new areas of research did you take on as a result of coming to Caltech?

KECHRIS: As I said, for a number of years myself and also my colleagues at UCLA developed a program related to these ideas that I mentioned before about new axioms of set theory. I spent roughly the first 8-10 years of my time here at Caltech pursuing this program. In the mid-1970's we started a joint seminar in logic with UCLA, which we called the Cabal Seminar. The name, Cabal Seminar, came originally from a joke that I will not get into. This seminar is very well known in the mathematical logic community and continues to this date (with the exception of the COVID period). We have published a series of proceedings of the seminar in the 70's and 80's, altogether four volumes. There is now an updated, also four volumes, version published by Cambridge University Press in the past 14 years. The last volume was published about a year ago. These new volumes collected all the papers that were in the old volumes together with many new papers that contain results and surveys that are more recent. The Cabal Seminar in its first ten years or so was primarily concerned with foundational issues related to these new axioms concerning determinacy of infinite games and large cardinals. This was the focus. After that it developed in different directions, including all these new connections of set theory with other areas in mathematics.

ZIERLER: Alekos, your promotion to associate professor in 1976—was your sense that that was a very fast promotion?

KECHRIS: I don't think so.

ZIERLER: Being a Sloan Foundation Fellow from '78 to‘80—what did that allow you to do?

KECHRIS: In 1978-79 I took a sabbatical and spent six months in France and several months in Greece after that. It was partly financed by the Sloan. It gave me the opportunity to have some free time to focus on my research.

ZIERLER: Alekos, tell me about your longstanding affiliation with the Association for Symbolic Logic (ASL). First of all, what is the ASL?

KECHRIS: The ASL is the main professional organization of logicians. This includes not only mathematical logicians like me, but also philosophers, computer scientists, and others who have an interest in logic, broadly understood. It is an organization that started in 1936. I have been a member of the association for a long time. I was also involved in the various activities and committees of the organization, including the council and the executive committee. Eventually I was elected president of the association for a three-year term.

ZIERLER: How big is the community of ASL? How many researchers worldwide are there roughly?

KECHRIS: At the time I was president it was about 1,500. This is an international organization, not just in the U.S.

ZIERLER: Is there an American component that's part of the American Mathematical Society (AMS) or it's a totally separate organization.

KECHRIS: It's a separate organization but there are some connections. For example, in the annual meeting of the AMS there's a joint meeting of the ASL.

ZIERLER: Moving into the 1980s, what were some of your key research areas at that point in your career?

KECHRIS: 1980s…let's see. This was the time where I transitioned from these more foundational issues of developing consequences of new axioms, into problems that connected my research in set theory with primarily analysis. Studying problems in analysis related to real functions, trigonometric series, harmonic analysis, measure theory, etc.

ZIERLER: Tell me about being awarded an honorary degree from University of Athens. That must've been very special for you.

KECHRIS: Yes, it was a great honor.

ZIERLER: Would you go back and visit Greece every once in a while? Did your kids get to see Greece?

KECHRIS: Yes. We visit there every summer except for the last two years, unfortunately, because of COVID.

ZIERLER: Of course.

KECHRIS: But we are planning to go this summer. Our children and grandchildren always come with us and they really enjoy it. We have several relatives, especially in Thessaloniki, so we spend some time on the coast an hour and a half south of Thessaloniki.

ZIERLER: Alekos, what were some of your responsibilities as executive officer for math in the 1990s?

KECHRIS: Everything that has to do with the math department: hiring, assigning courses, dealing with the administration, the faculty, the students, as well as the various committees in the math department.

ZIERLER: Between being a visiting research professor in 1998 and the lectures named in honor of Alfred Tarski—tell me about some of your work at Berkeley.

KECHRIS: At Berkeley I spent some time in the Fall of 1998 as a Visiting Miller Professor. I just basically did research there and I gave a series of seminar talks. The Tarski Lectures is something completely different. These are named lectures that are given once a year at Berkeley, named after Alfred Tarski, who was one of the major figures in mathematical logic. He was a professor for a long time in Berkeley. I gave in 2004 a series of three lectures on the connection between dynamical systems, model theory and finite combinatorics.

ZIERLER: When you were a Guggenheim Memorial Foundation Fellow in 2003 was that also a sabbatical?

KECHRIS: It partly financed a sabbatical.

ZIERLER: Where did you go? How did you spend that time?

KECHRIS: I took a sabbatical for the Fall Term. Part of it I used for a research visit of about a month in Barcelona.

ZIERLER: That same year you were the recipient of the Carol Karp Prize of the ASL. First, who was Carol Karp?

KECHRIS: Carol Karp was a mathematical logician at the University of Maryland. This is an award named in her honor. It is awarded every five years by the Association for Symbolic Logic

ZIERLER: In 1999 you wrote about new directions in descriptive set theory.

KECHRIS: Yes. This was the content of my Gödel Lecture of the ASL in 1998.

ZIERLER: What were some of the new directions? What were you thinking of there?

KECHRIS: The new directions were more or less what I discussed before, namely how you connect some of the ideas in the pure aspects of set theory, especially in descriptive set theory, with problems that come up in other areas of mathematics. In particular I talked then about classification problems. How you approach problems of classification of mathematical objects of various sorts from the point of view of logic and set theory. That was the main focus of my lecture, which then became an article.

ZIERLER: In your book Classical Descriptive Set Theory—was that a textbook? Was that an introductory text?

KECHRIS: It is a graduate level textbook. It grew out of a graduate level course that I gave at Caltech. I had notes from the course and then I added some more material for the book.

ZIERLER: It's always an interesting question with textbooks. Was your sense that the field needed to be updated? Was there not a good textbook available at that time?

KECHRIS: There were textbooks by various authors. There's a classical textbook by Kuratowski. It is a textbook in topology, consisting of two volumes, but a large part of the first volume was devoted to descriptive set theory. That was written a long time ago. There were other textbooks that dealt with some other, more recent aspects, of this area. At the time when I wrote my book there wasn't a textbook in the direction that I wanted to write about. It had some overlap with other textbooks, but it was going in a different direction.

ZIERLER: Moving into the 2000s. What were some of the major areas of research at that point?

KECHRIS: At that time I had moved towards the direction of connections of set theory with other areas of mathematics. Among others this involved problems of complexity of classification, connections with dynamical systems, combinatorics, group theory and other areas.

ZIERLER: How do you determine where those interfaces are with set theory and other aspects of math?

KECHRIS: For example, in one aspect, I wanted to understand the general idea of classification in mathematics and how this could be approached from a set theoretic point of view. This brought me into contact with many areas of mathematics dealing with classification problems of various kinds.

Other times it happened because I become interested in a problem, in some area, by reading something or talking to someone and then I thought "OK. That looks interesting. Maybe I can do something related to this problem." For example, when I worked in Ramsey theory it started with discussions with a colleague and then I realized that there are some interesting connections that I could pursue.

Sometimes it's just completely accidental. One research area in which I have been now working for some time just started because I was visiting Stanford in the 1980's to talk to a colleague about something I was working on at the time that was related to harmonic analysis. Then another mathematician dropped by whom I didn't know. He was a postdoc there. He started to talk to my colleague about something he had done in his thesis, which was completely unrelated, it was in ergodic theory, an area in which I had never worked before. I started listening to what he was explaining, and I thought "That's very interesting." Soon after I saw how it related to something I was interested in. That's how I started working in a subject related to ergodic theory and then it developed in all sorts of other directions. If I didn't happen to be there at that time maybe I would have never worked in that area because I wouldn't have connected some seemingly unrelated things I had been thinking about with what happened during this discussion.

ZIERLER: Alekos, I'm not sure if you remember. It was one specific panel, but in 2000 at the ASL you were on a panel called Prospects of Mathematical Logic in the 21st Century. What were those prospects and how they have played out in the past 22 years?

KECHRIS: This was a panel in a 2000 meeting of the Association for Symbolic Logic whose goal was to discuss prospects in mathematical logic for the new century. The panel had several speakers. Each one of us talked about one of the areas of logic. One speaker talked about computability, another about model theory, etc. I talked about set theory. Each one of us gave a short talk, around 20 minutes or so and then there was a period of questions by the audience and further discussion.

Set theory is a broad area and I could not possible discuss all of its aspects in a short talk. so I focused on the topics I was more interested in at the time. I discussed—this is related to what I said before—how these new principles in set theory could be used to solve some of the old problems in the area of descriptive set theory. Then I mentioned that this program has been completed to some extent and we have a very satisfactory theory on the basis of these new principles. I then said that we're now going more in the direction of connections of that theory with other areas of mathematics. Towards the end I also discussed the Continuum Hypothesis. This is outside the scope of descriptive set theory as it deals with the structure of arbitrary, not necessarily definable, sets of reals. We also know that it cannot be solved on the basis of the new theories of large cardinals and determinacy that I mentioned earlier and which were so successful in dealing with problems in descriptive set theory. There have been extensive investigations over the last 20-25 years towards formulating further new principles that could be used to settle the Continuum Hypothesis but the situation is far from settled yet.

ZIERLER: Was it around this time that you became more involved with Ramsey theory?

KECHRIS: No, that was a few years later. I happened to talk to a colleague who was working on some problems rather unrelated to what I was doing at that time but I realized that they were somehow connected to something I had thought about before. We started working on them and then a third colleague joined in. It took us a few years until we finally wrote a paper on that subject.

ZIERLER: Alekos, a general question. With all of your invited lectures and talks really all over the world, all over the globe, so many countries that you've spoken at—what is the research value of having that interaction with scholars the world over?

KECHRIS: It's of great value being able to talk to people and exchange ideas. As I mentioned a few examples before, some of my own research did start with such discussions and in fact even with discussions with people I didn't know I had any common interests. By being able to go to meetings or visit colleagues, you get to interact with people. Sometimes they are closely related to what you're doing; sometimes they are not. Often it happens that you talk about something new that you find interesting and then you try to pursue that. A lot of things do start like that. This is what we missed the last two years.


KECHRIS: We do as much as we can with Zoom meetings. The Caltech Logic Seminar was an in-person seminar with several participants at Caltech, including occasionally visitors. We now have a Zoom seminar, and we have people attending from all over the U.S. and Europe. (The time difference is a problem with colleagues from Asia.) That works actually quite well. It's about the only thing that works well when you're not meeting people in person. Hopefully we can try in person meetings next Fall, although I think we should also continue the online seminar.

ZIERLER: Hopefully. We said that before. Maybe this will actually be it now.

KECHRIS: I am planning to go to some meetings this summer, but we'll see.

ZIERLER: Alekos, similarly what is the value in participating in the celebration of colleagues either for a birthday or an anniversary? When you and your colleagues get together and focus specifically on the contributions of a particular mathematician—what's the value of that for the field?

KECHRIS: First of all you honor the person. You also have the chance to disseminate their work and put it in context and explain their ideas to a broader audience and maybe to people who are not very familiar with the extent of the work of this mathematician.

ZIERLER: I'm curious—the organizing committee work you've been a part of, does that set a curriculum for the field in terms of these are the things that we should be teaching students? Is curriculum development part of these committees?

KECHRIS: It depends. There are meetings in which there is a component of how to teach something at the undergraduate and graduate level. Other times they're just pure research meetings. People come and present their work. Of course, these are valuable also to students who attend these meetings.

ZIERLER: Tell me about being named an Inaugural Fellow of the AMS in 2012.

KECHRIS: The AMS started the Fellowship program in 2012. The inaugural Fellows where members of the AMS who satisfied certain criteria. From then on a cohort of new Fellows is selected every year. Proposals for new Fellows are solicited and a committee choses among them the new Fellows for that year.

ZIERLER: Over the last 10 years would you say that's when most of your work in ergodic theory has taken place?

KECHRIS: I would say the last 10 to 20 years.

ZIERLER: That gives the sense that there's just a tremendous amount of work to be done in this field.

KECHRIS: Oh, certainly. We have a lot of interesting problems and directions and many people, including many young ones, are working very much in this area and they're doing excellent work. It is a very active area which connects mathematicians working in logic and ergodic theory but also other areas like group theory, combinatorics, probability, etc. It's quite exciting.

ZIERLER: Alekos, bringing our conversation closer to the present. You were a coeditor of an anthology Nine Mathematical Challenges: An Elucidation. It's so intriguing. What are the nine mathematical challenges? [laugh]

KECHRIS: You have to look at the table of contents. [laugh] There was a meeting to inaugurate the renovation of the math department, the Linde Building. The meeting lasted a weekend and there were several invited talks by distinguished researchers in different areas of math. They all gave a talk in their area and then many of them wrote an article for this volume. There are papers in algebra, analysis, geometry, topology, number theory, mathematical physics and logic. Every paper deals with some challenge in an area of math. The authors explained the background, the history and the future prospects.

ZIERLER: These are challenges not just in the fields that you work on, but for all of math.

KECHRIS: Yes this is for all of math. This was a math conference and, as I mentioned, researchers that gave talks came from many different areas of math.

ZIERLER: Alekos, your publications over even the past few years have been quite significant. Have you found that working during the pandemic has been a time of particular productivity even if you can't be with your colleagues in person?

KECHRIS: I don't know if it's more or less than before but it's different. We don't have the same interactions that we had before. There were times that we had to stay at home where actually the only way we could communicate was through Zoom. There were other times where we actually met in person, but it was much more limited than usual. As I said, we had a joint seminar with UCLA on Fridays and we haven't been able to do that for two years now. I would say it was productive even under these circumstances. We had to adapt to new methods of communication.

ZIERLER: That's right. Alekos, now that we've worked right up to the present for the last part of our talk I'd like to ask you a few broadly retrospective questions about your career and then we'll end looking to the future. Again to go back to the idea that for this transcript many nonmathematicians will read from it and learn from it can you explain how you understand progress in the field? In other words how do you know when you're on the right track when you've made some mathematical achievement and that it's time to move on to something else? What does progress look like?

KECHRIS: Progress can take different forms. It could be, for example, the solution of a significant existing open problem or the creation and development of a new area of mathematics.

ZIERLER: Alekos, in the way that nature serves as an anchor point in science that you can do experiments to prove or disprove theories—what are the anchor points in your field to give you a sense of what is true and what is not?

KECHRIS: In my field we're dealing with these issues of extending the framework of classical mathematics, classical set theory, by new principles. We don't really have the ability to test these new theories with experiments. In my opinion, the justification of these new theories has to do with their mathematical structure. Do they have a coherent structure? How do they fit into the previously existing theories? Do they illuminate what we knew before, perhaps simplify it in some way or give a better understanding of it? Do they allow you to solve problems that you couldn't solve before?

ZIERLER: Alekos, in your career as a mentor to graduate students and postdocs what have been some of the major fields of research they have worked in and what kinds of careers have they gone into?

KECHRIS: Many of my graduate students and postdocs have worked in these areas that I mentioned before. Some of them later went on to work on different fields, e.g., number theory or computer science. Most of them went to an academic career but a few also work in government or industry.

ZIERLER: You mentioned previously some work earlier in the 20th century on unsolvable problems. For your own career have you ever come up on a problem that you determined was ultimately unsolvable?

KECHRIS: What I mentioned is that in the early decades of the 20th century people ran into problems that turned out to be unsolvable—again unsolvable not in an absolute sense. It means unsolvable within the framework in which they were working on at the time—which is the framework of classical set theory. We learned that later. In this area I was working on, we had these theories based on new principles, and many problems could be solved on the basis of them.

ZIERLER: In all of your research if you could think about it in broad terms what stands out in your memory as being most satisfying either because of its impact on the research or simply because it was so elegant or enjoyable to work on?

KECHRIS: Among several others, I could mention here my work in developing a theory of complexity of classification problems in mathematics, exploring the connections between logic, dynamical systems and Ramsey theory or studying the interactions of descriptive set theoryr with graph combinatorics.

ZIERLER: What about a so-called eureka moment? Do those happen where in a flash you can really make progress or is it always gradual?

KECHRIS: Both. I remember cases where there was such a moment but it did not come out of nowhere. It's something that you work on for some time and then suddenly one day, while you are thinking about this or about something else, you just get this idea and then it clicks. Yes, this has happened several times. Other times it's more gradual, you make a little more progress until you reach your goal.

ZIERLER: Finally, Alekos, on the question of progress. Last question looking to the future. What is the progress that you want to see? For as long as you want to be active in the field what's important for you to work on and more generally where do you see things headed?

KECHRIS: I have a number of projects that I'm thinking about. Also there are other projects on which I have worked on earlier in which other people are more actively involved now and I am extremely eager to see what progress is being made. For myself, there are some long-term projects that I am pursuing right now. For example, if you look at my recent work, I have written some long surveys in certain areas, including the theory of countable Borel equivalence relations and descriptive graph combinatorics, in which I do mention specifically many problems that I'm interested in and thinking about now. Trying to make some progress on these questions and develop methods to solve them is what occupies me right now.

ZIERLER: But it's always enjoyable. You always enjoy what you're doing.

KECHRIS: Yes. Definitely.

ZIERLER: Alekos, on that note I'd like to thank you so much for spending this time with me. It's been a great conversation. I appreciate it.

KECHRIS: Thank you very much.