Andrew John Casson 1943 — 2025
Andrew Casson died on September 5, 2025 in Fairfield, Connecticut. He was a major figure in geometric topology and a beloved member of the Yale mathematics department.
Andrew’s ideas changed the field of topology. His early work contributed to the Hauptvermutung problem in manifold theory. In 4 dimensions he introduced “Casson handles” which were instrumental in the proof of the topological Poincare conjecture. In 3 dimensions he defined the “Casson invariant” on homology 3-spheres. With his student Jungreis (and concurrently with Gabai) he proved the Seifert Fibred Space Conjecture. Jointly with Gordon he introduced the Casson-Gordon invariant in knot theory. He was awarded the Veblen Prize in Geometry in 1991, and elected to the Fellowship of the Royal Society in 1998.
He influenced and mentored many young mathematicians. His 27 students have carried on his legacy, many becoming active researchers in their own right with deep contributions in topology and geometric group theory, and students of their own.
At Yale he has been a valued colleague since his arrival in 2000 (following positions in Cambridge, UT Austin, and UC Berkeley). He served as department chair, and as Director of Undergraduate Studies for many years. In 2012 he was awarded the Dylan Hixon Prize for Teaching Excellence in the Natural Sciences.
A conference in memory of Andrew is planned for spring or summer of 2027. Please stay tuned for details.
Andrew will be missed by all those who knew him. Below are a few reminiscences from Andrew Casson’s students and colleagues.
Cameron Gordon:
Andrew was a very good friend for many years. I had the highest admiration for him, both as a mathematician and as a generous and principled individual who did things his own way. It was a privilege to have collaborated with him.
At the University of Texas at Austin around 1980, aware of the fact that R.H. Bing would be retiring in a few years, we were looking to hire a senior topologist. It was widely assumed that Andrew would never leave Cambridge, but when I was back there in 1980 he at least agreed that if he were offered a position at Austin he would not automatically turn it down. With this minimal encouragement, in November 1980 the department voted to offer him a full professorship. Of course this had to be approved by the administration, and given the idiosyncratic nature of Andrew’s file that was by no means certain. Indeed, in his letter to Andrew informing him of the departmental vote, the chairman pointed out that “the University is not accustomed to hiring full professors with few publications and no Ph.D.” At any rate, Andrew visited Austin in early 1981, one of the purposes of the visit being to meet the Dean of the College of Natural Sciences. I was not present at that meeting, but our math department colleague Jim Vick, who at the time was an Associate Dean of the College, was, and apparently it was a disaster. Next day I was sitting in my office with Andrew, feeling very despondent. I had dragged him over to Texas to put him through this uncomfortable experience, and probably to no avail. I didn’t know what to say, so we sat in silence for a long time. Finally Andrew spoke up: “Suppose you have a fibered ribbon knot …”
To their credit the administration did approve the offer to Andrew, and he joined the department in Fall 1981.
Greg Kuperberg:
Andrew Casson was a great mathematician and a very nice man. I had an independent streak as his student, which I think worked fine for both of us. Over time in my own career, I have come to realize more and more how valuable and impressive his mathematics was in his understated style. It is something to learn from, even though my own style is quite different from that.
Linda Green:
I met Dr. Casson in 1993, the summer before my third year as a graduate student at Princeton, a few weeks before he started there as a visiting faculty member for the year. I had already fallen in love with 3-manifolds, but I hadn’t been able to find anyone to work with at Princeton or at any nearby universities. I asked Dr. Casson if I could come talk to him about 3-manifolds while he was visiting at Princeton, and he agreed. So I started stopping by his office every Friday morning, as if we had a standing appointment. He always seemed slightly startled to see me, but he welcomed me in, listened to my ideas, and offered suggestions. His generosity continued – he helped me get permission to attend UC Berkeley the next year as a visiting student, filled out forms for me to get housing and library privileges, and continued to meet with me weekly, eventually reading and critiquing my thesis. Not everyone would have taken in a stray student from a different university like this. I appreciate his generosity even more now that I realize how much time and effort it takes to advise a graduate student, read their thesis, and write them letters of recommendation, and how little credit (zero) he got for giving all that time to me. Because of his help, I was able to finish my PhD in the area of math that I loved and go on to a math teaching career.
Sometimes when I went by Dr. Casson’s office, I saw him writing out a Calculus 1 lecture on the board – rehearsing for class. I was impressed by how conscientious he was, and how humble he was, to practice every lecture in a subject that he already knew by heart. His attention to teaching also showed in his seminars, which were so clear and well organized that they could be a book. I have tried to follow his example in my own teaching over the years.
Dr. Casson showed me many beautiful ideas. One of them I’ve turned into a math circle problem that I’ve shared with many high school students. I call it Planet Casson:
Planet Casson is entirely dry land, except for a system of canals. One canal runs around the equator, and three canals run from three different spots on the equator to join up at the north pole. (So the canals form a shape like the edges of a tetrahedron.) The canals separate the land into 4 land masses. There is one ferry boat for each land mass that circles its land mass continuously in a counterclockwise direction, traveling along the bounding canals. Unfortunately, the canals are too narrow for two ferries to pass each other, and there have been many crashes. Can you devise a ferry schedule with no crashes? (Ferries are not allowed to travel backwards.) What if you use a different arrangement of ferries … for example, an arrangement in the shape of the edges of a cube, or an octahedron?
Delightfully, Euler characteristic forms an obstruction to a crash-free schedule on any network of canals on any spherical planet, but crash-free canal systems on torus-shaped planets can be made with no problem.
Thank you, Dr. Casson. I will remember you and share my respect for you with the people I meet for the rest of my life.
Mahan Mj:
I was Andrew’s PhD student essentially from the beginning of my grad school.
His advising style was rather hands off and it took quite a while to understand that his intent was to groom independent researchers who would strike out on their own. I remember vividly two occasions where his key advice was a single sentence each time. The first time, I was thinking of proving the existence of Cannon-Thurston maps for some examples, and he told me to look at some other examples that we had discussed the previous week. These latter examples arose from small-cancellation theory. The second time, I told him that I was trying to work out the existence of Cannon-Thurston maps for normal hyperbolic subgroups of hyperbolic groups; but that even the simple example of a hyperbolic free-by-cyclic group was not clear. He encouraged me to focus on the hyperbolic free-by-cyclic case. In both cases, the effect of his suggestion was dramatic and similar. It made me focus on the simplest special case that was not known and work out a solution. Perhaps not unexpectedly, the special cases led to more general cases very soon.
On another occasion, he had suggested looking at Sela’s canonical representatives for hyperbolic groups in the framework of Gromov-Thurston’s proof of Mostow Rigidity. It is only very recently, while working with Nir Lazarovich in work following up on Nir’s finite index rigidity theorem, that I understood the depth and prescience of Andrew’s suggestion.
Many of us know that Andrew meticulously prepared for his lectures. One particular semester (in my third year), he was teaching two courses: one a freshman calculus course and another a topics course on hyperbolic 3-manifolds starting with Gabai’s then recent proof of `homotopy hyperbolic implies virtually hyperbolic’. For both courses, he would prepare in exactly the same manner: he would write the full lecture on his blackboard in the privacy of his office, and then proceed to the classroom and deliver the lecture on the board without any notes.
Andrew never ceases to inspire, both as a research advisor and as a mathematics expositor.
Saul Schleimer:
First just a bit about me – I was an undergraduate at Berkeley from 1991 to 1995, and then a graduate student there from 1995 to 2001. I was told (by Professor Jenny Harrison) that I was a topologist – so I started taking topology classes (from Justin Roberts). These agreed with me very well, and I asked Andrew to be my advisor before taking my qual (in May 1997).
I have several memories (airbrushed by time, surely) that consist of me, basically, telling Andrew what to do. At the very beginning of the 1998-1999 school year I informed him that we needed to meet weekly (because that was what advisors and students did). After meeting for a while, (half a year?) I informed Andrew that he needed to give me a thesis problem.
Meeting weekly turned out very well for me. But perhaps not so much for Andrew? He was very very very patient with me. He was far more patient with me than I am now with my students! Early on, he would give me very easy problems to think about. I solved some of these, but perhaps not in the level of detail (or understanding) that Andrew wanted. I made some progress, and eventually got to the point where I could sometimes ask a question that he needed to think about. He would then sit in his chair, pull at the hair of his beard under his chin, and think for an infinite amount of time… It was agony (for me!) to try and sit still and just let him think… On most occasions, he would then give a concise answer. In one instance he had obviously used the time to compose a lecture on small cancellation theory, which he then gave while I eagerly took notes.
Asking Andrew for a thesis problem turned out less well for me. He asked me to prove that all finite volume cusped hyperbolic three-manifolds have geometric triangulations. This was a well-known open problem (to the experts) at the time, and is still open today. I spent just two weeks thinking about the problem, went back in, and told Andrew that the problem was too hard! I then asked him for another problem. I unfortunately don’t recall his reply.
One more thing. I took all of the graduate classes Casson lectured from1997 to 2001. He was an amazing expositor. He spoke easily and clearly, had lovely board work, included examples with content, got through the material, and ended on time. And he did so without notes. Over the years I learned a little bit about how he managed this: A few times we had meetings immediately after class. On these occasions the blackboard in his office was covered in tiny handwriting with the exact content of the lecture he had just given.
Joan Licata:
Andrew didn’t say much, but I learned early on that everything he did say was worth holding onto. One of my favorites was a test for quality: good work should prove something about an object it didn’t define. I have passed this litmus test to my own students and I think about it often. However, one of my lasting memories of Andrew was as an educator rather than a researcher. When I visited his office, I would often spot drafts of his upcoming lectures stretched across the board. These meticulous notes were prepared not only for advanced topics, but also for calculus classes. I was just starting to teach my own classes at that point and this made quite an impression: here was a world-class mathematician who gave the same careful attention to undergrads as to research peers. I remain grateful to have had a role model who approached all his professional activities with such a high standard.
Miki Havlickova
I worked with Andrew for many years, and couldn’t have asked for a mentor more thoughtful and kind.
My memory isn’t actually mine, it’s a story from our colleague Michael and his wife Jean, who have been wonderful friends to me and to Andrew.
It reminds me of Andrew’s mild manner and wisdom in all circumstances.
Andrew was sitting in Jean’s and Michael’s house, with Bopper the cat on his lap. (So far, this isn’t surprising, every guest in the house ends up with a cat or two on their lap.)
At one point, Bopper decided that now would be a good time to stick his claws into Andrew’s kneecap, and proceeded to do so. The result was an unprecedented explosion from Andrew: His voice volume went up by 20%, and he said “eep”.
Having known Andrew for a while, Bopper appreciated the significance of this and wisely left the scene.
Michael Frame:
One day early in 2012, Andrew appeared in my office doorway. I knew he had to teach soon, so clearly he hadn’t stopped by for a chat. He uttered one sentence, “I have no idea how this happened.” Then he was gone. How what happened? I expected that I’d find out eventually.
A couple of hours later, Andrew again was in my office doorway. “I really have no idea how this happened,” and he turned to go.
“Hold on. How what happened?”
“Well, er, somehow I’ve been awarded the Hixon Prize.” This is one of the most prestigious teaching prizes Yale College awards. If you’ve seen Andrew lecture — research seminar, graduate course, advanced undergraduate course, or beginning calculus – the award is no surprise. Andrew prepared carefully; his lectures were models of economic presentation, celebrations of the architecture of ideas. He showed not only how to get from step A to step B, he showed why B was a natural next step after A. His lectures were a series of haikus.
My wife Jean and I knew Andrew fairly well. For over 20 years, every five or six weeks we had dinner at an Indian restaurant, came back to our house for dessert and relaxed, wide-ranging conversation, then watched a movie, often with a cat – and our cats adored Andrew – on Andrew’s lap. We talked about books, music, movies, ideas, but never much about Andrew’s life. He didn’t volunteer, and we thought it impolite to ask.
After Andrew died, his sister Barbara came from England to get his house ready to sell. Jean and I helped a bit, and we and Barbara traded stories about Andrew. Barbara did not know that Andrew had won the Hixon Prize; we did not know that Andrew had won analogous prizes at Austin and at Berkeley. Barbara had found the certificates among his papers. This sweet, gentle person never boasted of his achievements though he well might have.
Andrew’s absence has sharp edges, edges that hurt. Eventually the sharpness will wear away as the incandescence of loss is replaced by the gentler warmth of “Do you remember when Andrew tried to convince the waiter that he truly wanted his chana masala to be ‘Indian hot’?” I know that eventually smiles will outnumber tears, but now it’s still mostly tears.
Yair Minsky:
As a colleague, Andrew was gentle and dedicated, but also fierce when necessary. I particularly remember, during his time as DUS, his principled insistence in meeting after meeting that we carefully vet the incoming postdocs for teaching experience. Nobody would be surprised to know that as department chair he was thorough and detail oriented, but it is more surprising that he was able to take a firm stand with the administration on important points. Andrew is also the one who recruited me to Yale; I remember sitting with him at lunch at some conference in New York, and he asked me if I would like to come to Yale. Not getting his meaning, I asked — for how long? A day? A week? He sheepishly smiled and said “forever.” I remain grateful to him for that.
