I’m pleased this week to present my interview with Alejandro Adem, Canada Research Chair in Algebraic Topology at University of British Columbia. Alejandro was born in Mexico City and completed his doctorate at Princeton University. Not only is he a stellar mathematician, but he is the CEO and Scientific Director of Mitacs Canada. Mitacs is a not-for-profit which began as a Network of Excellence, and it is devoted to supporting research and training of students and post-docs through partnership grants with industry.

Alejandro is an award-winning mathematician. A few of his honors include the Jeffery-Williams Prize from the Canadian Mathematical Society, appointment as a Fellow of the American Mathematical Society, and a Vilas Associate Award from the University of Wisconson-Madison. He is also a managing editor of the highly impactful journal Transactions of the American Mathematical Society.

My chat was Alejandro was inspiring and I found him open and generous with his answers. The breadth of his experience and knowledge was impressive. He talks about the K-theory of orbifold cohomology in one part of the interview and then Canada’s innovation ecosystem in the next.

Enjoy!

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**AB**: What were your early mathematical influences? Did anyone, prior to your studies at university, inspire your studies?

**AA**: I’m in a special situation because both my father and my uncle were mathematical scientists. My father had a PhD in Applied Mathematics from Brown University and my uncle had a PhD in Mathematics from Princeton University. They were both very distinguished scientists in Mexico, where I was born. They were the first generation in that side of the family to go to college. They were always inspiring examples to me.

On the other hand, teachers inspire much of what we do in mathematics. I did have some inspirational teachers, especially one in middle school who used the “new math” which I found fascinating. He taught me a love for the conceptual aspects of mathematics that carried me through for several years. When I was choosing career options, mathematics was on the table because of my family background, and my teachers motivated me. So going into the field was a natural decision for me.

When I was in school, I realized that if you understood the underlying method in mathematics, then you didn’t have to memorize facts like in other subjects such as history or biology. In those subjects, at least at that time, there was a huge emphasis on knowledge acquisition and memorization. While in mathematics, if you understood something, you could remember it. For me, it is the same in life: if you understand the process, you can remember it. For me, it was powerful. It unlocked a part of my brain, one that was not necessarily conscious.

**AB**: How did you come to work on your doctorate at Princeton University and what was your experience like there?

**AA**: I did my undergraduate at the National University of Mexico, and when I got to Princeton it was rather intimidating. My classmates included top-ranked students from a top places in the US. My main objective was to listen and learn. At Princeton, you learn as much from your classmates as you do from your professors. There are so many brilliant people there.

However, you can also understand what it means to have mathematical insight, which you have to separate from sheer intellectual power. It is great if you have both. There were individuals there who had great intellectual power but little insight. They weren’t natural mathematicians. It was a revelation for me when I held my own and did well, based on mathematical intuition and hard work.

We had very inspiring teachers there, including my own advisor Bill Browder. The professors were gentle and they unlocked the mysteries of higher mathematics for us in a nice way. There were no grades in the courses. It was all about learning and doing your research and supporting you through that process.

**AB**: How did you get to work with Browder?

**AA**: They have a tradition there of tea time every day around 3:30 pm. The professors would come down to the common room, and the protocol back then was that you would wait for them to come to tea and then approach them. I remember meeting him there around the time of my qualifying exams and he was very friendly. He remembered my application to the program and invited me to his office to talk mathematics. We had a discussion about things I could work on, and I went on my own to study the topics he gave me.

**AB**: You are an expert in algebraic topology and the cohomology of groups. Would you explain what these areas in simple language?

**AA**: Algebraic topology is a part of mathematics that tries to understand intrinsic geometric properties of objects that appear in the mathematical world, for example in physics, and in everyday life. We may try to measure, for example, the number of holes in an object, or identify if one object can be deformed into another. As you know, there is a famous analogy between the donut and coffee cup.

In areas of physics we may naturally encounter objects arising from topology and geometry, so topology is a fertile ground for connecting higher mathematics and ideas from the natural sciences. We also study objects such as knots, and it can be seen for example that the knotting of DNA connects topology with biology. There are also interesting applications of topology in economics.

We can separate out algebraic invariants that will determine these fuzzier, more complicated geometric structures. If you can derive a number that tells you that you can deform one object to another, then you are in great shape. The most basic example of that is the Euler characteristic, which many times can characterize a space. For example, for orientable surfaces, that number contains a lot of information. You can derive even finer invariants. These algebraic and topological invariants can, in many cases, help describe the original space and the geometric problems that it encodes.

Cohomology of groups is a bit more technical and harder to explain, but the idea is to move from groups to topological spaces. There are nice correspondences, which assigns to a particular group a space with geometry that fleshes out properties of the group. There is a rich direction there called geometric group theory which is quite important now. The spaces built out of group have a much greater impact than you would expect. Cohomology of groups computes invariants associated to these spaces, which are quite general. You can go from a group to a space, and then compute algebraic invariants of it (in this case, the cohomology ring). So it becomes a fine science in its own right, with fantastic contributions from people like Daniel Quillen who laid the foundations of the subject for finite groups. There many applications of group cohomology which connect areas of algebra and topology in unexpected ways.

**AB**: What are the research directions you are working on right now?

**AA**: One of the basic topics I have worked on in my career is topological symmetries or group actions. Given a manifold, can you describe or characterize groups that act on the space in a suitable way, such as free actions (that is, actions without fixed points). There is a classical problem that topologists solved, which is understanding the finite groups acting freely on a sphere; that is the so-called Spherical Space Form problem. That leads to many interesting questions surrounding the groups and the algebraic topology involved. Many sophisticated equivariant techniques have been developed to study these questions.. With more complex groups and structures, the problems become quite challenging.

One of the most interesting results I proved (joint was Jeff Smith) was characterizing spaces with periodic cohomology: one that repeats itself in sufficiently high degrees. That can be described very effectively using methods from algebraic topology modelled on techniques in group cohomology.

I’ve also worked on topics arising from physics and that led to very fruitful collaborations; for example, with my former colleague Yongbin Ruan who works in symplectic geometry and mathematical physics. We wrote a book on orbifolds and developed models for K-theoretic versions of orbifold cohomology. That ties into questions about resolutions occurring in string theory.

Through that, I also became interested in mathematics arising from Lie groups, such as structures associated to their commuting elements. With my collaborators, we developed commutative K-theory, which comes from the commuting elements in the unitary groups. There are also infinite loop spaces involved.

In my work, there is usually a group involved, symmetries, and functors from a group to a space. There are very sophisticated tools available to us in algebraic topology. But to me what is most important is to have an output that can be appreciated by a mathematician at large.

**AB**: You were at PIMS as Deputy Director from 2005 to 2008 and then as Director from 2008 to 2015. Would you us about PIMS and how you came to those positions?

**AA**: I was a professor at the University of Wisconsin, and I saw an ad for the Deputy Director at PIMS and also one for a Canada Research Chair at UBC. I sent an e-mail inquiring. I always liked Canada and BC, as my father would spend a lot of time here in the summers. It was a dream of mine to live here. They were interested and wanted me to send my CV. It was fascinating to meet with my colleagues here as well as with the then director at PIMS who was Ivar Ekeland (who is a very inspiring figure). What I liked about my interview was that we talked about the Serre spectral sequence, and not about bureaucratic detail. He wanted to know about it! I was impressed by the mathematical level of PIMS. I took both positions. I should mention that I knew a lot about mathematical institutes based on my role at MSRI in Berkeley as Chair of the Scientific Advisory Committee for four years.

I moved to Vancouver with my family and I love it here in Canada and BC. Working at PIMS was a special experience as it is a consortium of western universities in Canada and Washington state. PIMS was founded under the inspiration and guidance of people like Nassif Ghoussoub, Ed Perkins, and Peter Borwein who are all important Canadian mathematicians.

Part of my job was to visit institutions in the consortium and help create a critical mass of research strength that would rival concentrations in places like Toronto and Montreal, which have excellent strength in mathematics. PIMS is a grassroots organization, including collaborative research groups that tie together researchers at different universities to work on central problems in their field and also directly impact on training of personnel. It was a great pleasure to work with my colleagues in western Canada, and a big responsibility. There are always challenges with the changing nature of funding mechanisms in Canada.

**AB**: You’re in your second year as CEO and Scientific Director of Mitacs. Would are the aims of Mitacs and what your role is there?

**AA**: I was invited to apply for my present position at Mitacs, which as you may know was established by the three mathematics institutes of Canada: CRM, Fields, and PIMS. It was a product of the imagination and the boldness of the mathematics community, and built from the bottom up by my colleague Arvind Gupta.

Mitacs has evolved into an organization that now goes well beyond mathematics. It involves all the disciplines of knowledge. The key thing Mitacs does is connect universities with industry. The biggest program we have is called Accelerate, and it provides internships in companies for Canadian graduate students. Students go to a company for four months and work on a research project with direct economic impact. We have a high-level research committee that adjudicates proposals. The students work on research that will enhance the products and services at the company. Typically, a company has a problem that has to be solved with a view towards commercialization. Industry has to pay 50% of the internship. There is a huge buy-in from industry to this program, this past year we have delivered close to 4,000 internship units across the country.

We have people on the ground who make these connections. If you don’t have people creating links between professors and industry, then it’s almost impossible to do. We are a national organization now, and it has grown quite a bit over the last ten years.

We also offer an international program called Globalink that brings in talented students from a targeted list of countries for research projects in the summers. And we offer scholarships to bring them back as graduate students. The point is to use research in a targeted way to bring the best minds to Canada. In addition we have reciprocal programs to send students abroad – it is important to send Canadian students out as we lag behind other countries such as the USA in this process of global mobility and internationalization.

At Ryerson University, you see all around you that innovation and globalization are coming together. The knowledge-based economy has been going into hyperdrive. We see the Globalink program as a way of connecting the innovation ecosystem in Canada with that in other countries such as China, India, France, Brazil and Germany among others. Our programs are very much focused on addressing the challenges faced by this country.

As CEO and Scientific Director it’s been fascinating to learn about a completely different structure than at a university or institute. Mitacs is a not-for-profit company. We have around sixty member universities. We deal with the private sector and governments (both federal and provincial). Our programs are focused on graduate students and provides them with experiential education. My view is that every graduate student should have the opportunity of an internship; not mandatory or prescriptive, but at least given the opportunity. We have instances of programs that embed our internships in their curriculum.

Mitacs originally stood for Mathematics of Information Technology and Complex Systems. Now it is just an acronym! I always remind the staff that the “M” in Mitacs is for mathematics. Mathematics is at the heart of innovation. The more I work in other areas, and talk to companies and government, the more I am convinced that mathematics is at the centre of it all. What we do is of core relevance to society and everything around us.

**AB**: That is a very powerful message. I speak with many intelligent people who don’t understand the central role of mathematics in science and other disciplines.

I asked this next question to Nassif Ghoussoub when I interviewed him. What do you think the role of mathematicians in shaping public policy? For example, on matters pertaining to public math education and in agencies like Mitacs or NSERC?

**AA**: The knowledge that mathematical scientists bring, especially to the handling of data, is highly important. We will naturally be contributing through our work, especially given that we think clearly and logically about data as part of our research. Recently some of the top people in Canada working in artificial intelligence have been recruited to places such as Microsoft and Google, so it’s an ongoing challenge for Canada to maintain intellectual leadership. Mathematical scientists form a vital backbone for data science not only through our research in topics such as probability and theory of networks but also through our essential role in the training of university students.

Mathematicians are less involved in the political sphere than they could be. I would urge them to get more involved, like what Nassif, myself and Arvind Gupta have done. I also understand that mathematics is a quiet affair and we need time to focus on our research. However, I am convinced that we need to be smart about communicating what we do to other scientists and governments, in order to ensure it receives the attention and support it undoubtedly deserves.

Education is enormously important. Since my kids have been in school I hear a lot about that. My wife Melania works at UBC doing mathematics outreach. She has run a very successful summer program for indigenous high school students and also works extensively with teachers. Students often fail to learn the mathematical concepts and skills required for success in STEM disciplines and the mathematics awareness in society is not where we want it to be. It affects the kinds of students we see at university. We talk now about “fake news”, but “fake numbers” have been around for much longer. That can lead to poor decision-making at all levels.

I would like to see mathematicians involved at the local level, involved with schools.. I would like to see our colleagues more involved in the teaching of teachers. Some do it now and do a great job. We many times criticize what is taught in the schools. But have we reached out to schools to provide support or mentorship? For example, UBC Math has a program to send undergraduates to schools and serve as a resource. It’s not easy, but it is worthwhile and we have a big role to play.

**AB**: You are active on social media on both Facebook and Twitter. What do you think is the importance of social media for mathematicians in the 21st century?

**AA**: Absolutely it is important. Kids live and die by social media! They are more likely to watch a YouTube video on a topic then pick up a book. Social media is their first point of contact.

Some mathematicians like Ken Ono and Jordan Ellenberg already do that, and you do an excellent job of that. Social media can be more than your personal view or political view. If there is a source connecting people on social media to mathematical ideas, opening up people’s minds, then that can be very influential. Things you pick up on Twitter or Facebook can plant a seed in your mind which afterwards can blossom in an unexpected way.

I hope we have an expanding presence on social media with mathematicians. A mathematics and media program or training our colleagues on that would be interesting. At Mitacs, we have recently launched a program called Science Policy Fellows where we seek to embed academics in government departments. That is a way to send academics to influence policy. Mathematicians applying for that could create opportunities and make new connections.

We have to organize to create conditions and programs to empower people to do original research and connect mathematics with society. It’s like a mathematics problem: we have to use all tools at our disposal.

**AB**: What advice would you give to young people thinking about studying mathematics?

**AA**: I think that when you study mathematics you need to bite the bullet and take the hard courses. You have to understand what mathematics is about and the incredible knowledge there. If it is not for you, then you shouldn’t do it. Mathematics is an enabler, not a bottleneck or gatekeeper. You have to be honest with yourself and get the necessary background.

When I was younger, we focused on a few areas and were guided by traditional mentors. Mathematics is much broader now and with many things going on between pure and applied mathematics. For example, my area of topology is now broadened to applied topology and topological statistics. Many classical areas of applications can be leapfrogged by other areas of mathematics that were historically theoretical. The excitement that gives and the potential impact on society is very important. The monastic view of a mathematician should be a thing of the past. Mathematicians should contribute to their subject, their students, and society. It is a challenge, but if you have a faculty position then that should be treated as a rare privilege with an obligation to be a real contributor

Young people should also understand that what you are doing now is not what you will do in ten years. If you are doing the same thing, then there is something wrong. There will be times when you will teach more, do more research, or do more administrative work, or even outreach or industrial connections. If you are open-minded, then it can be enriching and gives you more mathematical ideas and keep you intellectually awake.

**AB**: I always finish by asking about the future. Through your work at PIMS and now at Mitacs, you must be exposed to so many different trends in mathematics. What is on the horizon for mathematics and where do you think the subject is going?

**AA**: We are in a golden era of mathematics. Huge conjectures have been solved over the last twenty-five years such as Fermat’s Last Theorem and the Poincaré conjecture. These are problems that were open for hundreds of years. The power of mathematics is evident and its beauty is shining as bright as ever. We are attracting extraordinarily talented people into mathematics.

There is concern among young people about finding academic jobs and post-docs. Is the model of the university sustainable? Are we a technological breakthrough away from a substantial shrinking of the faculty base? These are questions we have to wrestle with. You want to do what a robot cannot. Mathematicians know how to think and synthesize data and they are open to connecting structures and seeing patterns and recognizing them. I think mathematics will play an important role in our society where data and thinking are combined.

Mathematics has a track record of its ideas being important after their discovery, even in the commercial world. It also possesses ideas that are extraordinarily beautiful. If we get the proper tools and are open-minded, then mathematics is sure to be applied to all areas of knowledge. The use of quantitative methods in the natural sciences is becoming highly prevalent. The existing views we have in our head of what a mathematician is may change, and people will most likely be doing much more of it in the private sector than in universities. That may be a positive evolution and we should welcome change.

Anthony Bonato