Research

I am mostly interested in Arithmetic Geometry and Algebraic Number Theory. Most of my research time is spent thinking about torsion subgroups of elliptic curves over number fields. I am also interested in ranks of elliptic curves and ranks/torsion of abelian varieties generally. So broadly speaking, I am interested in $k$-rational points on abelian varieties. But of course, I am interested in a broad variety of Mathematics, which I have listed below, roughly in order of interest or how much time I spend thinking about it.

    • Torsion Subgroups of Elliptic Curves
    • Rational Points on Abelian Varieties
    • Galois Representations
    • (Nonabelian) Chabauty-Kim Methods
    • Perfectoid Spaces
    • Class Field Theory
    • Quantum Cryptography
    • Iwasawa Theory

If you are interested in learning Arithmetic Geometry, there is a wonderful ‘roadmap’ to learning Arithmetic Geometry by Matthew Emerton (posted on a post of Terence Tao). The typesetting of Emerton’s answer, insofar as I am aware, was due to Daniel Miller. If you are interested in the things I am researching now, the video below is a fantastic summary of the progress in understanding torsion subgroups of elliptic curves.



  

Further Details & References for the Curious

 

  • Elliptic Curves:

    • Elliptic curves are the most basic interesting examples of abelian varieties. They also serve as an interesting class of abelian varieties as they are genus 1 curves. In genus 0, if a variety has a rational point, it has infinitely many. If the genus is larger than 1, there will always be at most finitely many such points by Falting’s Theorem. But elliptic curves, i.e. the case of genus 1, can have either none, finitely many, or infinitely many points. By the Mordell-Weil Theorem, we know the basic structure of these groups. But researchers are interested in understanding the possibilities for the rank and torsion depending on the number field. If you are interested in reading more, there are a few resources below:

     

  • Arithmetic Geometry:

    • Arithmetic Geometry can be very difficult to define. But broadly, it is Algebraic Geometry which is applicable to Number Theory. Of course, Algebraic Geometry can mean many things as well, so when we say it here, we mean the kinds of things you will find in Ravi Vakil’s working book The Rising Sea. But below are a few references for those wanting to see a bit more about Arithmetic Geometry.

    • Wikipedia Page: For a very quick, extremely terse definition of Arithmetic Geometry, you can see the Wikipedia page for Arithmetic Geometry.
    • Kiran Kedlaya Conference List: What better way to see what Arithmetic Geometry is than going to a conference for yourself! Kiran Kedlaya maintains an amazing webpage listing upcoming conferences in Arithmetic Geometry. He also happens to be an amazing researcher and wonderful speaker as well!
    • Arizona Winter School: If you do not have the opportunity to go to a conference, there are many amazing lectures and materials for Arithmetic Geometry posted by the Arizona Winter School. I’ve had the amazing fortune of being at three of these workshops/conferences, and they have been some of the best mathematical experiences that I have had.
    • What I think of: Of course being such a large field, Arithmetic Geometry means many things to many people, depending on their interests. So what do I think about when I hear ‘Arithmetic Geometry’? If I’m asked about a paper, I would of course think of Barry Mazur’s groundbreaking paper, “Modular Curves and the Eisenstein Ideal.” If you instead wanted to see some videos that show what Arithmetic Geometry means to me, see Curves of Genus 1 by Andrew Wiles, Abelian varieties with maximal action on their torsion points by David Zywina, Locally symmetric spaces and their torsion classes by Ana Cariani, Recovering elliptic curves from their p-torsion by Benjamin Baker.

     

  • Algebraic Number Theory:

    • Algebraic Number Theory is approximately the study of number fields, i.e. finite extensions of $\mathbb{Q}$. More broadly, Algebraic Number Theory is the study of global and local fields. Most people that have heard of Fermat’s Last Theorem will be familiar with Algebraic Number Theory (as well as Arithmetic Geometry). For those that want to learn a bit more, here are a few more resources.

    • Wikipedia Page: This Wikipedia page actually does a decent job of explaining and summarizing Algebraic Number Theory.
    • Fermat’s Last Theorem: Most people will be familiar with Algebraic Number Theory via Fermat’s Last Theorem.
    • LMFBD: The L-function, modular form database website is a collection of data for L-functions and modular forms. But this website also has a lot of data on elliptic curves, global fields, and local fields, which is typically what I more use it for – explaining why it is here rather than under Arithmetic Geometry.
    • Course Notes: There are wonderful course notes available for learning Algebraic Number Theory. Two such examples which are freely available and that I think are good for the beginner are J.S. Milne’s Algebraic Number Theory and Introduction to Algebraic Number Theory.

     

  • Rational Points on Abelian Varieties:

    • Rational points on abelian varieties are not something just a curious person can follow. But if you do come with sufficient background, say most of an undergraduate degree in Mathematics, you could take a look at Bjorn Poonen’s Rational Points on Varieties, which he has posted for free for casual use. If you come with a stronger background, you may be interested in Andrew Snowden’s course MAT 679, which covers Mazur’s Theorem. If you come with more of an expert background, perhaps you would be interested in lectures from a conference titled, Reinventing Rational Points.
     

  • Galois Representations:

    • Galois representations are not something a casual reader can follow. However, if you have some amount of background, What is a Galois Representation by Mark Kisin is an excellent read. One may also read Richard Taylor’s great Galois Representations, or to see how I more use Galois Representations, a talk titled A brief introduction to Galois representations attached to Elliptic Curves by Alejandro Argáez-García. If you are curious how this ties in with Fermat’s Last Theorem, see Galois Representations and Modular Forms by Ken Ribet.
     

  • Chabauty-Kim Methods:

    • This topic is a recent mathematical development, especially its non-abelian methods. So there is much that is still be said about this topic. However, these techniques certainly are the method for finding rational points on higher genus varieties. If you are interested in reading more, I would first recommend An effective Chabauty-Kim theorem by Jennifer Balakrishnan and Netan Dogra. I would also recommend David Corwin’s From Chabauty’s Method to Kim’s Non-abelian Chabauty’s Method or Netan Dogra’s Unlikely intersections and the Chabauty-Kim method over number fields. There is also a great talk by Bjorn Poonen on the topic.
     

  • Perfectoid Spaces:

    • Perfectoid Spaces are a creation of Peter Scholze in an attempt to prove the Weight Monodromy Conjecture. Although Scholze has described this as a ‘failed theory’, because it failed to prove the conjecture in all cases, they are extremely useful, being used to in locally symmetric spaces, modular forms, Galois representations, and p-adic Hodge Theory. Moreover, perfectoid spaces were by Yves André to prove the Direct Summand Conjecture. Scholze won the Field’s Medal—the most prestigious medal in Mathematics—for his work. Below are a few references for the curious:

    • What is a Perfectoid Space?: This great article by Bhargav Bhatt summarizes what a perfectoid space is.
    • What are perfectoid spaces?: This question was asked on MathOverflow and was answered by none other than Scholze himself!
    • Perfectoid Spaces: No better place to learn something than from the original source, especially because Scholze is a fantastic writer!
    • Course on Perfectoid Spaces: Here are notes for a course on perfectoid spaces given by Bhargav Bhatt in 2017.
    • Arizona Winter School on Perfectoid Spaces: In 2017, the Arizona Winter School was on Perfectoid Spaces. The speakers included Scholze himself. I was lucky enough to be able to attend this and can affirm the quality of all the talks!
    • Scholze’s Talks: Peter Scholze is a wonderful speaker. His talks are remarkably understandable, and he has given many talks on perfectoid spaces, many of which you can find compiled here.

     

  • Class Field Theory:

    • Although this is really just a subset of Algebraic Number Theory, it is vast enough to be worthy of its own space. Class Field Theory was born out of the attempt to understand abelian extensions of $\mathbb{Q}$, i.e. finite extensions of $\mathbb{Q}$ with abelian Galois group. Class Field Theory, while very technical, is very useful in a variety of areas. In particular, Class Field Theory yields another interpretation of torsion subgroups of elliptic curves in the CM case. You can read more about the history of Class Field Theory in this nice historical overview by Keith Conrad, view an overview of the main statements of Class Field Theory by Bjorn Poonen, or if you’re deeply interested, read Class Field Theory by J.S. Milne.
     

  • Quantum Cryptography:

    • Most modern day cryptography is based of elliptic curve cryptography. However if in the future quantum computing becomes the norm, traditional methods of encrypting will no longer be secure. To see how quantum computers break traditional encryption, see this great explanation. So different techniques will be required for future encryption which are quantum secure. One of the proposals is isogeny-based encryption. To see how these methods work, watch this video, or if you’re looking for a more detailed description, this is the video you probably want to watch.
     

  • Iwasawa Theory:

    • This is another technical topic. Essentially, Iwasawa Theory studies arithmetic in towers of extensions. Iwasawa was intimately used in the proof of Fermat’s Last Theorem. However, Iwasawa Theory is also extremely useful in study growth in ranks of elliptic curves. If you are interested in reading more, the Wikipedia page does a nice summary. However, there is a better overview by Romyar Sharifi, who has also a nice collection of lectures on Iwasawa Theory. There was an Arizona Winter School on Iwasawa Theory that I was lucky enough to attend. To see a bit more on how these are used in elliptic curves, you can see this nice summary by Edray Goins. Of course, one of the best places for papers on Iwasawa Theory are Ralph Greenberg’s papers.