Do you have a question? Send an email to questions dot drakengren at gmail dot com (alternatively, if you wish to be anonymous, use this google form instead). If I answer your question, it will be posted here anonymously.
Q: What is geometry, for you? What does it mean to work with geometry? Do all fields that are called “geometry” have something in common, or is the word “geometry” just overused?
A: First, I would like to highlight some thoughtful reflections from the person who sent me the question:
“I used to have the following macrosight on geometry: geometry is an area of mathematics concerned with shapes and figures, and each subarea is, essentially, the use of specific tools to try to understand some properties of that shape. So, differential geometry is essentially using calculus on shapes, while algebraic geometry is the use of algebraic techniques, and so on.”
My comment: Shapes can also “arise” from a certain area. For instance, the study of solution sets to polynomial equations originates from algebraic geometry. Algebraic geometry is in some sense about properties of polynomials.
“I attended the Measure Theory course one year ago, with an excellent and very experienced teacher. Once I asked him about the applications of measure theory techniques in geometry, and he answered by providing some applications. But what really stood out was this: Well, geometry is more a way of thinking than an object of study. This phrase had a real impact on me. While the sentence makes a lot of sense, it raises some interesting questions. What is this “way of thinking” that fascinates me so much? What is to think geometrically?”
My comment: That is a nice quote!
“Algebraic geometry is a lot about algebra, and perhaps it involves some constructions even more abstract than the ones in algebraic topology. Additionally, people generally separate differential topology from differential geometry based on the concept of (Riemannian) metric. But in algebraic geometry, as far as I'm aware, there is no basic standard concept of metric, curvature, etc. Then there is not “metric” argument available here. So what makes algebraic geometry “geometric”?”
My comment: Let’s take complex schemes as examples; they are generalizations of usual “vanishing sets of polynomials” in C^n. The vanishing set of x^2 in C is just 0. But, for the same reason as the tangency of the curves y-x^2=0 and y=0 in C^2 can be seen as these curves intersecting in a double point at x^2=0, the subscheme of C given by x^2 is viewed as a double point. With some imagination, I would say that you can “see” a double point. Many objects that exist in differential geometry are also there in algebraic geometry, with the only difference that your objects need to be defined by polynomial equations in the latter case. The Chow ring is an analogue of cohomology, where the cycle classes instead come from algebraic varieties, and the equivalence relation is algebraic. Everything you do in geometry here can also be written down in algebraic terms. Doing a blowup can be described algebraically, even though you would think about it geometrically. Same with vector bundles, fibers of maps and intersections etc. For this reason, what appears on paper as abstract might have been formed in someone’s mind in more geometric terms.
“The year was 2024, and I attended the symplectic geometry classes as a listener. At one point, we were talking about the torus, and the teacher talked about we cannot see the Lie group torus in R^3. I think he was a little mistaken in his speech: what we can't see it is the torus as a Lie group with a compatible metric, the flat torus. But this is not the point, because I understood what he meant. But then I said: “That's sad, because I want to see it” Then the professor and my colleagues (already in graduate course) joked: “It's just that you are starting out, when you start going deeper you won't even think about seeing things anymore”. This was just a friendly joke between colleagues. But it raised a deep question for me. Because time has passed, and I still want to see things. Of course I don't expect to see a 14-dimensional torus, but for me visual intuition is just too important. I want to imagine. That's the point. Of course my friends didn't mean that visual intuition was not important. But that joke showed me just how important the visual intuition was for me to appreciate geometry. And I believe it's no different for most of us.”
My comment: I don’t agree with the quote “when you start going deeper you won't even think about seeing things anymore”; if you go deeper and still want to make sure you actually understand the concepts, so that you know why you work with them and what you can do with them, it certainly helps with geometric intuition, although you must work harder to obtain it! With some experience, you might even have a clear picture of what is happening before you have tried to write a proof! I can totally say, with your mindset, that geometric intuition will play an essential role throughout your career!
“I found this sentence on the book “Differential Geometry”, by Sharpe. It's a quotation from the philosopher René Guénon: “It must be agreed that 'hypergeometry' seems to have been devised in order to strike the imagination of people who have not enough mathematical knowledge to be aware of the true character of an algebraic construction expressed in geometrical terms, for that is really what 'hypergeometry' is.” This phrase is really deep. It's important to notice that, at least apparently, this sentence of Guénon (from The Reign of Quantity & the Signs of the Times) is not intended for mathematicians: actually, the book has some connections with mathematics (could be classified as a “philosophy of mathematics” book, at least partially), but its intentions are in another direction. But the phrase makes me wonder: is my wish to see things in geometry just this strike Guénon talked about? This would mean that perhaps I don't understand, at the core, what geometry is supposed to be. That is a terrifying thought.”
My comment: I am instead annoyed by the fact that there is so much geometry that you will never be able to understand, or see!
“What I'm trying to show is similar to the philosophical question of “what is time?”: it's a really hard question, but intuitively we all know what it is, and we live our lifes normally without know an exact answer that. On the same way, we mathematicians understand intuitively what is to think geometrically, most of the time we agree on it, and keep living our lifes. But as the previous examples and provacations show, once we start to try to actually understand what we try to grasp when we use the word “geometry”, things start to get messy. And as someone that likes geometry, and that would like to work with geometry, it is a little scary the thought of that, perhaps, I like and work with something that I don't really understand.”
My comment: The benefit of not understanding is that there is something to explore! I think the exploration is fascinating on its own. I guess I prefer that to everything being known, that surprises don’t exist, and that there were no problems to solve.
The difficulty in seeing can mean that the picture you try to construct is overly simple, or doesn’t convey the actual nature. This picture might instead make you realize what you don’t know! You might feel an urge to complete the picture with details.
Second, here are some more of my own thoughts:
For me, geometry is math that you can see. If you can see schemes or open sets, then it is geometry. If the dimension is so high that you cannot make a plausible visualization, I would no longer consider it as geometry (but someone who can visualize it is doing geometry).
On the other hand, simply drawing a painting is mostly not geometry; there has to be mathematics involved; you must accompany the picture with a choice of logical statements about the shapes/relations of the objects in the pictures, where these logical statements can be “seen”. I don’t consider it so much geometry to prove a geometric statement without using geometry. However, if your method of proof is completely algebraic, but you see what happens geometrically at the same time even if you don’t use the geometry to prove it, I tend to consider it as geometry. This is slightly contradictory; you can see this as an infinitesimal/continuous version of “using a non-geometric proof to prove something about geometry”. But mostly, the geometry comes from actively using the geometry to motivate your arguments. For instance, when working with cohomology classes (topological or algebraic), you can often visualize them and see how they should map/intersect by choosing suitable representatives.
Doing geometry for me is mostly cinematic rather than static; your picture is changing as your argument goes forward, and the argument and the picture support and complement each other mathematically.
As for “moving” and “static” geometry, the former involves deformations, mappings and blowups, whereas the latter might be e.g. counting lines, holes or components of some geometric object. Projective geometry feels more mobile than Euclidean geometry in this aspect. Euclidean geometry feels more like discovering more and more interesting objects in the picture, like collinearities, significant lines and angles, rather than transforming the picture (I still like Euclidean geometry from school days, though!).
I asked a friend, an arithmetic geometer, how he used geometry in his research despite working over characteristic 2. He said that, mostly he could get the intuition from characteristic 0 and then try to obtain similar results over other characteristics. But sometimes there is no analogy in characteristic 0, which makes it less visualisable.
Mathematics will not be the same for everyone, since much of what you see will come from what you find rather than what is already visible to everyone. Claire Voisin writes: “The reason why I find it very hard to speak about mathematics is that I cannot disconnect my activity as a mathematician from my whole life and personality. I do believe there is something narcissistic in the mathematical practice, because the question is not only: “what is true?” but also “what am I able to prove?” which is a question about myself. If I push this idea a little, I find that, when I am doing mathematics, as for many other human activities, my ultimate goal is to know more about myself, as advised by Socrates.”
Some people prefer not to think geometrically. I am glad that there are many ways of thinking about math and of finding your own place there. The richness of mathematics can allow anyone to shape themselves within the field.
Q: How do you spend most of your time (is there any work other than thinking non-stop?)
A: I really value the time when you can think uninterruptedly about interesting problems. On the other hand, it is good to intertwine other forms of mathematical activity to avoid getting exhausted. For instance, this can be going to seminars, reading expository notes or articles, preparing and giving exercise classes or discussing math with other people in the research group. We tend to have very long lunches together here, so the workday feels very social. I also like traveling for conferences!
I do spend much of my free time doing math, but this is entirely self-chosen. I have many other hobbies which I also like to spend time on.
In short: Your PhD is what you make it!
Q: How hard/busy is PhD for you?
A: In terms of mental exercise, I prefer it to be challenging, so quite hard in that aspect. My schedule is tight but I make time for what I want to do.
Q: How do you feel about your PhD journey so far?
A: Living my dream! I have visited so many beautiful sites in algebraic geometry. I am glad to be part of a large and supportive research group. Zürich is a fantastic place to live, especially if you like hiking and to explore Swiss, German, French and Italian cultures. ETH is good if you like taking the stairs four floors up to your office everyday.
Q: Do you have suggestions for people going to start a PhD?
A: Sure! Be prepared to learn a lot during your PhD. I can tell you which methods of learning I like:
Collaboration. I really like to discuss mathematics with others; you will be able to share different ideas and viewpoints, answer each others' questions and learn what is most interesting and relevant. Even if you are working on your own project, I recommend talking to others and ask questions. Conferences and talks are also good opportunities for this.
Quality time. The time spent, often a longer consecutive interval, on thinking about a problem/concepts related to your problem, with all of your attention - using only pen and paper. This way, I dedicate time to understand the deeper meaning of the constructions I work with. Some hours of dedication always takes you further in solving a problem.
Giving talks. This will solidify your background about the topic, and will allow you to share your research to a wider audience. It also gives you a sense of accomplishment.
Enjoy your PhD. Do things that you like. This will give you a positive attitude and courage to attach difficult math problems.
Q: Why study algebraic geometry?
A: In short, you will study the geometry of solutions to systems of equations (think curves, surfaces, 3-folds...), known as algebraic varieties. Among other tings, we can study limits and degenerations, look for invariants or combinatorial descriptions, search for rational points, or study how varieties of smaller dimensions intersect inside a surrounding variety.
You will be surprised by the cool constructions and exotic elements that allow us to translate back and forth between algebra and geometry; there is a high wow-factor!
For instance, an abelian variety is a compact variety which also has a group structure. Abelian varieties generalize elliptic curves with their chord and tangent operation to higher dimensions. Over the complex numbers, they can realized as complex tori, i.e. as quotients of a complex vector space by a lattice. There is a moduli space, denoted Ag, which parametrizes all the (principally polarized) abelian varieties of a fixed dimension g. Being a geometric object on its own, we can study how subvarieties intersect inside Ag (or any compactification Āg). We can form a ring CH*(Ag) out of the formal linear combinations of the subvarieties, imposing suitable equivalence relations and letting the product of two subvarieties (in good cases; this is not quite true in general!) correspond to their intersection. There is a natural subring, called the tautological ring and denoted by R*(Ag), which is isomorphic to the ring CH*(LGg-1) of the Lagrangian Grassmannian LGg-1, a compact smooth manifold of interest in symplectic topology. While much is still unknown about CH*(Ag), we can sometimes gain information about the ring by pulling back elements to the (better-known) moduli space Mg, parametrizing curves of genus g. Having said this, I barely mentioned the deep arithmetic theory behind abelian varieties. In other words, a single object can be subject to investigation from lots of different perspectives!
In algebraic geometry, you are free to choose your tools; with a glue stick and a pair of scissors you might be able to cut problems into pieces which you later can try to patch together, or you can use sheaf cohomology which encodes valuable invariants and is rather nice to work with.
I quite like the AG-community, too!
Q: Is algebraic geometry too difficult?
A: No (and yes)! The subject is so fascinating that you will become completely captivated and thereby surpass the boundaries of what you thought you were capable of achieving. The area is so wide that there is something for everyone; there are connections to physics, number theory, topology and more. You choose the level of challenge you want.
Q: Your favourite animal?
A: 🦆🐤🐤🐤