Levent Alpöge (pt&rl, anthropic, pbc / junior fellow, harvard sof, alpoge@fas, arXiv, cv, minor planet!! :D)



hi im levent!!!

alas and of course there's none of my cs work here (besides the really old stuff:)), but there are links below to my math work...!

the most recent paper is a solution of hilbert's tenth problem (posed in 1900) over all number fields

the second most recent one solves the problem of: assume whatever conjectures expected to be true you want --- can we then specify an algorithm which solves one of the "original problems" of number theory, namely to find all integral or rational solutions of a polynomial equation in two variables? (paper 15 gives an algorithm in special cases which doesn't assume anything.) i call it literally the world's worst algorithm

finally, for the third most recent one prob better to just link a quanta article

here's three more quanta articles: about the two hilbert ten papers that came out independently, an interview of Wei's in which she talks about some of our other work, and an article about new proofs of ultra old theorems, one of which i came up w (funny story which i def omit:)) in undergrad



2023 Nesin matematik köyündeki "integral points on elliptic curves" hakkında verdiğim işkencenin kursun öğrencileri (ve ben)

کور يوسف ضياءالدين پاشا

aka Pickle

炙り




Theses:

  1. Points on curves. (My PhD thesis at Princeton.)

  2. The average elliptic curve has few integral points. (My senior thesis at Harvard.)



note that the last solo paper of mine to be published in a journal is from the year i finished undergrad. that is certainly deliberate, though it hurt my career. i decided they must be defeated when i was a kid

Papers:

  1. Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field (with Manjul Bhargava, Wei Ho, and Ari Shnidman).

  2. Conditional algorithmic Mordell (with Brian Lawrence).

  3. Integers expressible as the sum of two rational cubes (with Manjul Bhargava and Ari Shnidman).

  4. Local systems and Suzuki groups (with Nick Katz, Gabriel Navarro, Eamonn O'Brien, and Pham Huu Tiep).

  5. Un peu d'effectivité pour les variétés modulaires de Hilbert-Blumenthal.

  6. Quadrics in arithmetic statistics.

  7. Note on a theorem of Professor X.

  8. Modularity and effective Mordell I.

  9. A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so (with Manjul Bhargava and Ari Shnidman).

  10. A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so (with Manjul Bhargava and Ari Shnidman).

  11. The second moment of the number of integral points on elliptic curves is bounded (with Wei Ho).

  12. The average number of rational points on odd genus two curves is bounded.

  13. Square-root cancellation for the signs of Latin squares.

  14. Nagell-Lutz, quickly.

  15. The average number of integral points on elliptic curves is bounded.

  16. van der Waerden and the primes.

  17. Analytic number theory and quadratic reciprocity.*

  18. Proof of a conjecture of Stanley-Zanello.

  19. Self-conjugate core partitions and modular forms.

  20. Low-lying zeroes of Maass form L-functions (with Steven J. Miller).

  21. Maass waveforms and low-lying zeros (with Nadine Amersi, Geoffrey Iyer, Oleg Lazarev, Steven J. Miller, and Liyang Zhang).

  22. Decidability and shortest strings in formal languages (with Thomas Ang, Luke Schaeffer, and Jeffrey Shallit).

  23. A vtk-based, CUDA-optimized non-parametric vessel detection method (with Alark Joshi, Dustin Scheinost, John Onofrey, Xiaoning Qian, and Xenios Papademetris). (My Intel STS project from high school.)

    *: (I once thought I'd discovered a new and "purely analytic" proof of quadratic reciprocity. In fact the argument was known to Dirichlet! So I turned it into a little expository article.
               See the fun!! section for a non-expository version, which amounts to a few lines.)

Some talks:

    recent recorded talks

"Book":

  1. Math 123 (= Algebra II) notes (with Dennis Gaitsgory and Gurbir Dhillon).

  2. Math 122 (= Algebra I) notes (with Dennis Gaitsgory).




fun!!




generals


some amusing math history links


human languages: english (native), turkish (native), spanish, french, russian, german

through my own laziness the last four are in varying states of disuse

i have started making real progress in ottoman and hope to learn (spoken) latin and (spoken) ancient / modern greek!

i also have character-colour synesthesia!

this includes numbers and other scripts e.g. cyrillic and greek, though not (yet?) ottoman (script or numbers)...i wonder if that'll change over time!