Emily Riehl is an incredibly accomplished early-career mathematician, working at the interface of category theory and homotopy theory. She is also an amazing amount of other things, such as a creative scholar who studies many subjects, a working musician, and a high-level athlete. She did her undergraduate work at Harvard and her graduate work at Cambridge and the University of Chicago. From 2011 to 2015, she was an NSF and Benjamin Peirce Postdoctoral Fellow at Harvard and is now an Assistant Professor at Johns Hopkins University. Emily has been awarded an NSF standard grant and a CAREER award to support her work. She has written 21 research papers, two books (Categorical Homotopy Theory and Category Theory in Context), and a lot of other papers that explain things. she does all of this and plays rock and alternative bass for a living and on the US women’s national Australian Rules football team. I just found out about Emily’s work and profile while looking for women mathematicians to interview for the newsletter of the Association for Women in Mathematics. I thought that maybe people who read PhD epsilon would also be interested in hearing about some really cool things that a young mathematician is doing. The interview below is a collection of emails and Skype calls that Emily had in August 2017 while she was in Australia for the AFL International Cup.
The Association for Women in Mathematics will soon send out a newsletter with a longer version of this interview.
We want to know how and why you became interested in category theory. Is there a basic result you can share that shows why you like it so much?
I put off going to graduate school at the University of Chicago for a year so I could do what they call a “Part III” at Cambridge. Class theory was one of the classes they offered at Cambridge, and I loved it right away. I feel like it chose me as much as I chose it. It was because the proofs seemed like the right way to think about math, which is why I think everyone chooses their field finally: I felt right away that this is the sort of argument that I wanted to delve into.
Category theory can sound intimidating because it’s highly abstract, but it’s actually not that hard. It doesn’t take long to state and prove the theorems, and some of the most important definitions are pretty simple. In fact, a common belief in category theory is that if you understand the statement of the theorem, you can probably come up with your own proof. Identifying the correct definitions is really the harder thing. You usually don’t learn category theory until you’re in graduate school because you need to know a lot about math to understand what it’s for.
One of my favorite theorems in category theory says that left adjoints preserve colimits and right adjoints preserve limits. In category theory, you can get a dual theorem by just “turning all the arrows around.” Inverse s preserves intersections and unions while direct s only preserves unions. This result explains why tensor products spread over direct sums, why quotients of topological spaces are made by first finding the right points and then topologizing this quotient set. Maybe it’s because I like having one proof instead of having to make the same argument over and over, but I think the category theoretic proof is the right one. It uses the fact that limits have a “mapping in” universal property and colimits have a “mapping out” universal property.
What kind of work have you done so far? A: You have written a lot of papers, two books, and shorter pieces that explain things, like posts on the n-category cafe. How do you do so much stuff? Do you have any insights into how/why you are so productive?.
ER: I read Hardy’s “A Mathematician’s Apology” in high school. The most important thing I learned was from the introduction by C.P. Snow, who wrote about Hardy’s typical day: he did math for four hours in the morning, from 8 to 12, and then watched cricket in the afternoon. Because I thought it was a great way to live, I’ve always focused more on working well than on working long hours. To get the most done in the least amount of time, I start working on the most important task last, when I’ll be fully focused. And so on. If I have to turn in a referee report in three months, I don’t start reading the paper until almost three months have passed. I also get ready to teach an hour or an hour and a half before class. It often feels like a race to figure out how to prove all the theorems before I have to run across campus. This sometimes causes me problems, like when I was trying to set up a transfinite induction over the reals and couldn’t figure out why all the intermediate stages were “countable.” (As an aside, I now strongly believe that the axiom of choice is true and the well-ordering principle is false.) But this approach is very effective at reserving time for research and other long-term projects.
ER: The worst thing is how intellectually alone we all are and how few people, even other mathematicians, we can share the most interesting ideas with. For me, it’s very frustrating that most of the people I care about can’t see a big part of my emotional life.
My favorite part of my job has always been giving talks. Because of the reasons given above, research talks are my favorite. But I also enjoy giving colloquia and even teaching. Even in high school, I enjoyed the performative aspects of lecturing. When I ran for student body president, the only thing that really interested me was giving the campaign speech in front of the whole school.
William Thurston wrote in “On Proof and Progress in Mathematics” that “what we are doing is finding ways for people to understand and think about mathematics.” You start your book Categorical Homotopy Theory with this quote. In what ways has Thurston’s view of mathematics as a community activity centered on human understanding changed the way you do math?
ER: There have been times when I’ve wondered if I should be worried about how much time I spend on expository projects like the books, since they do take up research time. This is one of many times Thurston’s essay, which I’ve read more than once, has helped me keep these kinds of projects in perspective. The passage you quoted is his idea of mathematical progress, which he sees as being more than just proving theorems. I really like explaining math, so I think it makes sense—or as economists would say, is a comparative advantage—for me to play that role in the community as a whole.
At the opening meeting for the AMS-sponsored Mathematics Research Community workshop in Homotopy Type Theory that I co-organized this past June, I read from a different part of this essay that talks about how hard it is for mathematicians to talk to each other. This was done to set the tone for the week and give people who aren’t already part of the community (e.g. g. to get in (because they’re getting their PhD somewhere that doesn’t have a faculty member working in that area)
ER: One thing I love about working in academia is that the job changes all the time, or it can if you want it to. At the moment, I’m working on a few long-term research projects and growing the category theory group at Johns Hopkins. I’d like to finish these before the 2020 MSRI semester on Higher Categories and Categorification. I hope to be working on projects I can’t even think of now in ten years and have found a way to be a part of bigger mathematical and public conversations. This entry was posted in.
Homotopy theory is a complex and abstract field of mathematics that studies the topological properties of spaces by considering continuous deformations between mappings. It underpins many areas of modern mathematics and has important applications in physics computer science and engineering.
If you have an interview coming up for a role involving homotopy theory you can expect to face some challenging technical questions to assess your conceptual knowledge and problem-solving abilities. With thorough preparation you can enter your interview with confidence and give compelling responses.
In this article, we walk through some of the most common and tricky homotopy theory interview questions and provide tips to help you nail your answers.
Explaining the Basic Concepts
Interviewers often start by asking candidates to explain some of the fundamental ideas and definitions of homotopy theory. This tests your understanding of the core concepts.
Q What is a homotopy between two continuous functions f and g from a topological space X to a topological space Y?
A: A homotopy between f and g is a continuous function H from the product space X x [0,1] to Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. Intuitively, H continuously deforms f into g as the second parameter moves from 0 to 1.
Q: What is meant by saying two spaces X and Y have the same homotopy type?
A: Two spaces X and Y have the same homotopy type if there exist continuous functions f: X → Y and g: Y → X such that g o f is homotopic to the identity map on X and f o g is homotopic to the identity map on Y. This means X and Y are homotopy equivalent – their topological structures can be continuously deformed into one another.
Q: Explain what the fundamental group of a topological space represents.
A: The fundamental group of a topological space X captures information about the non-contractible loops in X. It is defined as the group whose elements are homotopy classes of loops in X based at a point x0, with the group operation being concatenation of loops. Intuitively, it records the different ways loops can wrap around holes or other non-simply connected features of the space. The fundamental group is a key invariant in homotopy theory.
Solving Homotopy Theory Problems
You should expect several questions that require working through homotopy theory problems and explaining your reasoning. This tests your technical skill in applying concepts.
Q: Let f be a constant map from the unit circle S1 to the real line R that sends every point to 0. Is f nullhomotopic? Explain.
A: Yes, f is nullhomotopic. We can construct a homotopy H from f to a constant map c that sends every point to 0 as follows. Let H(x,t) = (1-t)f(x) + tc(x) = (1-t)0 + t0 = 0. Then Hcontinuously deforms f to c over t in [0,1], so f is nullhomotopic.
Q: Calculate the fundamental group of the wedge sum of two circles.
A: Let X be the wedge sum of two circles S1 ∨ S1. The fundamental group is the free product of the fundamental groups of each S1. Since π1(S1) = Z, the integers under addition, we have:
π1(X) = π1(S1) * π1(S1) = Z * Z ≅ Z * Z
where Z * Z is the free group on two generators.
Q: Show that if f: X → Y is a homotopy equivalence, then f : π1(X) → π1(Y) induced on fundamental groups is an isomorphism.*
A: Since f is a homotopy equivalence, there exists g: Y → X such that f o g ≃ 1Y and g o f ≃ 1X.
Consider the induced maps on fundamental groups. We have:
f* o g* = (f o g)* ≃ 1*Y = 1π1(Y)
and
g* o f* = (g o f)* ≃ 1*X = 1π1(X)
Thus, f* is both injective and surjective, so it is an isomorphism.
Answering Conceptual Questions
You may also face abstract conceptual questions that test your deeper understanding of homotopy theory ideas. Take time to think through your responses.
Q: What does it mean mathematically when we say two spaces are weakly homotopy equivalent but not homotopy equivalent?
A: If two spaces X and Y are weakly homotopy equivalent, it means there exist maps f: X → Y and g: Y → X such that f o g is homotopic to the identity and g o f is homotopically trivial (but not necessarily homotopic to the identity). So f and g induce isomorphisms on all homotopy groups, but are not actual homotopy equivalences.
Q: Explain why homotopy groups are important invariants in homotopy theory.
A: Homotopy groups capture deeper topological information than just the fundamental group. πn(X) measures the different ways an n-sphere can wrap around a space X. So homotopy groups record higher-dimensional holes and obstructions to contractibility. Furthermore, homotopy groups are homotopy invariants – spaces with equivalent homotopy groups must have the same homotopy type. This makes homotopy groups powerful algebraic tools for distinguishing and relating topological spaces.
Q: Describe how the relative homotopy groups πn(X,A) generalize the notion of homotopy groups.
A: For a space X with subspace A, the relative homotopy groups πn(X,A) describe spheres that wrap around X but are constrained to lie in A at the basepoint. Intuitively, πn(X,A) detects holes in X that are “filled in” when looking just at A. So relative homotopy gives finer topological information by considering a pair of spaces. Algebraically, πn(X,A) is the nth homotopy group of the quotient space X/A.
Talking About Applications
Since interviewers want to know you can connect theory to practice, expect questions about uses and applications of homotopy theory. Discuss specific examples relevant to the role.
Q: How is homotopy theory used in physics and topological quantum field theory?
A: In physics, homotopy theory is important for studying topological phases of matter and quantum field theories. Quantum states or field configurations can be represented as mappings from spacetime to a target space of states. Homotopies between these mappings characterize different quantum phases. Physicists use homotopy groups and homotopy equivalence of target spaces to distinguish phase transitions.
Q: What are some ways homotopy theory arises in robotics and motion planning?
A: In robotics, configuration spaces of robot arms and mechanisms have non-trivial topology determined by joints and obstacles. Homotopy groups and fundamental groups of these configuration spaces capture the distinct ways a robot can move or deform. Homotopies represent valid paths for robot motion planning. Homotopy also provides tools for analyzing configuration space connectivity and coverage for planning algorithms.
Q: How does homotopy theory connect to machine learning?
A: Topological data analysis uses homotopy theory to study the “shape” of high-dimensional data sets. Persistent homology computes topological invariants like Betti numbers to characterize holes, clusters, and other patterns in data. Homotopy also provides tools for analyzing neural networks, which can be viewed as parametrized mappings between topological spaces. Techniques from homotopy theory are being applied to understand the topology of loss landscapes for training machine learning models.
Key Tips for Acing Homotopy Theory Interview Questions
Here are some key strategies for mastering homotopy theory interview questions:
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Thoroughly review core concepts like homotopy groups, fundamental groups, homotopy equivalence, and induced maps on homotopy groups.
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Practice working through concrete examples and sample problems to improve technical skills. Verbalize your reasoning.
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Connect abstract ideas back to visual intuition about deforming and wrapping spaces. Use pictures and diagrams if helpful.
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Anticipate definitions, comparisons (e.g. homotopy vs homeomorphism), and conceptual questions.
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Read up on applications in physics, robotics, data science, etc. Have examples ready to discuss.
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Ask clarifying questions if you get stumped – this shows thought process.
With diligent preparation, you can demonstrate deep knowledge of homotopy theory and pass your interview with flying colors. Master the key concepts, practice solving problems, and connect the ideas back to applications. This will help you stand out as a strong candidate for any role involving this complex and powerful field of mathematics.
What is…homotopy?
FAQ
What is an example of a homotopy function?
How do you prove something is a homotopy?
What is the theory of homotopy?
Why are homotopy groups of spheres important?
What is homotopy theory?
In the beginning, homotopy theory dealt with what happens when you define an equivalence relation (“homotopy”) on maps. Focusing on weak equivalences is an entirely different perspective: we are picking out a collection of maps that will be regarded as “equivalences.” They are to become the isomorphism in the homotopy category.
What is a pointed homotopy?
Pointed homotopy is again an equivalence relation, and we have the pointed homotopy category, or, more properly, the homotopy category of pointed spaces HoTop . We’ll write [X; Y ] for the set of maps in this category. Definition 3.4. Let (X; ) be a pointed space and n a positive integer. The nth homotopy group of X is
What is fiberwise homotopy theory?
Fiberwise homotopy theory studies the category S=B of spaces over B. Let O be an operad and B be a space with the structure of an O-algebra. Then S=B has the structure of a strongly O-monoidal cate-gory in the following way. For spaces X1; : : : ; Xd and Y over B, the space of multimorphisms is the pullback
Why are cohomology operations constrained in homotopy theory?
The combined structure of these cohomology operations is very e ective in homotopy theory because of three critical properties. These operations are natural. We can exclude the possibility of certain maps between spaces because they would not respect these operations. These operations are constrained.
