I'm currently on the market for tenure-track jobs in mathematics. The application process is somewhat formal -- I think it's hard, in a research statement, to get across my passion and excitement for mathematics (though hopefully this comes through in recommendation letters). This post is an informal pitch, and an explanation of what I'm thinking about and why I think I'll be a good colleague. Of course, my CV is probably the best place to see my formal qualifications.

### What I'm thinking about

I'm interested in pretty much all of algebraic and arithmetic geometry, and in many aspects of related fields (for example, parts of algebraic topology and representation theory). I think my colleagues will testify that I love talking about more or less anything they're thinking about, and that I'll happily think about any mathematical problem in a field related to algebraic geometry, very broadly construed. Thus far, I've written about:

- Galois actions on fundamental groups, and monodromy representations (here and here),
- A "non-abelian" Lefschetz hyperplane theorem (using positive-characteristic vanishing techniques),
- The fine classification of algebraic varieties (in particular, adjunction theory),
- Deligne-Illusie style vanishing theorems (improving the Bott-Danilov-Steenbrink vanishing theorem for toric varieties),
- Algebraic dynamics (joint with John Lesieutre), and
- The Grothendieck ring of varieties (here and here).

My current projects are mostly related to Galois actions on fundamental groups, which is incredibly fun -- I expect that the techniques I've been developing will have really cool applications to Iwasawa theory and to questions surrounding the geometric torsion conjecture, for example. I'm also working on a direct computation (along the lines of work of Deligne, Anderson, and Ihara) of Galois actions on certain fundamental groups, which reveals some really beautiful structure. If you're interested in more technical details of my work, you can read a draft of my research statement.

I'm thinking about a few more classical questions as well; for example, I'm working on a project with Alex Perry about the irrationality of low-degree hypersurfaces in projective space. I'm also thinking about some somewhat more speculative projects on Brauer groups, isomonodromic deformations, and p-adic Hodge theory.

### My favorite theorem

I want to tell you a bit about the theorem I'm proudest of proving in the two years since finishing my PhD, as I think it gives a somewhat better idea of the things I'm thinking about than the vague remarks above.

**Theorem** (L-, Theorem 1.1.2) Let \(X\) be a normal algebraic variety over a finitely-generated field \(k\) of characteristic zero, and let \(\ell\) be a prime. Then there exists an integer \(N=N(X, \ell)\) such that: if $$\rho: \pi_1^{\text{ét}}(X_{\bar k})\to GL_n(\mathbb{Z}_\ell)$$ is a non-trivial, semi-simple, continuous representation which extends to a representation of \(\pi_1^{\text{ét}}(X_{k'})\) for some \(k'/k\) finite, then \(\rho\) is non-trivial mod \(\ell^N\).

The \(N\) in the theorem is effective; in particular, if \(X=\mathbb{P}^1\setminus\{x_1, \cdots, x_m\}\), then \(N(X, \ell)=1\) for almost all \(\ell\).

So why should you care? The first reason is that the main examples of representations satisfying the hypotheses (i.e. those which extend from the geometric fundamental group to the arithmetic fundamental group) are *monodromy *representations, e.g. on the cohomology of a family of varieties over \(X\). So this tells us a new structural result about such monodromy representations, which I view as a global analogue of Grothendieck's quasi-unipotent local monodromy theorem, related to e.g. the geometric torsion conjecture. The second reason is that this is an *anabelian* result -- the statement is about the structure of the arithmetic fundamental group \(\pi_1^{\text{ét}}(X)\). So in fact it tells us something completely new about the Galois action on fundamental groups of arbitrary normal varieties.

I also think the proof is pretty awesome, but I won't test your patience by describing it here. And the use of arithmetic/anabelian techniques like these to deduce fairly classical consequences (i.e. the application to monodromy representations) is pretty cool as well, I think.

### Why I'm a good colleague

I probably shouldn't write this part, since I'm not my own colleague. But let me give it a shot. I take service to the department seriously -- in the two years I've been at Columbia, I've run two REUs (one with Daniel Halpern-Leistner and one with David Hansen). As a graduate student, I helped organize GAeL XXII and XXIII. I've sat on the graduate admissions committee, one thesis defense, three oral exam committees, helped organize student seminars, and the algebraic geometry research seminar. I am currently mentoring two undergraduate research projects, both of which are going swimmingly. I think I've been good at these mentorship activities, and I really enjoy them -- watching the students grow mathematically has been incredibly rewarding.

I also regularly give expository talks, both to undergraduate and graduate students. I take teaching seriously (if you'd like to read a more serious-minded document outlining my thoughts on teaching, check out a draft of my teaching statement), and I think the students like me and learn from my classes -- last year I taught Calculus III. Some entertaining excerpts from my evaluations:

Of course I've largely focused on mathematical matters and service to the department, but I also think there are less tangible, non-mathematical aspects of being a good colleague. It's hard to give concrete evidence that I possess these qualities, but here is a sample: I have many non-mathematical interests (for example, literature -- if you search this website sufficiently hard, you might find some non-mathematical writing of mine); I care a lot about the treatment and quality of life of graduate students; and I think that I've contributed positively to the social life of the department, both at Columbia and during my graduate student years at Stanford.

Please let me know if you want to hear more, and if you have a job opening! Of course I'll be applying via mathjobs in the next couple of months, so if you're on an admissions committee, you may see some of this in a much more formal way soon. Wish me luck!