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: 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: 🦆🐤🐤🐤