Ayan Nath

About

I am a PhD student at MIT since Fall 2024.

My advisors are Bjorn Poonen and Keerthi Madapusi.

Previously, I was an undergraduate student at Chennai Mathematical Institute.

I am interested in number theory.

Here's my CV.

Research

  • Finite Generation in Cohomology of Artin Stacks (in progress)
  • Perfectoid Shimura Varieties of Exceptional Type (draft)
    We show that all minimally compactified Shimura varieties at infinite level are perfectoid spaces, including those of exceptional type for sufficiently large primes.
  • Compactification of Reductive Group Schemes pdf arXiv
    We prove that any isotrivial reductive group scheme over an arbitrary base admits a smooth projective equivariant compactification, verifying a conjecture of Česnavičius. In the adjoint case, we recover fiberwise the wonderful compactification. Finally, we give an example of a non-isotrivial torus admitting no equivariant compactification.

Pre-grad school

  • The Moments of Split Greatest Common Divisors (with Abhishek Jha and Emanuele Tron)
    To appear in Acta Arithmetica. arXiv
  • On the Least Common Multiple of Polynomial Sequences at Prime Arguments (with Abhishek Jha)
    International Journal of Number Theory, 18(6) (2022). journal arXiv
  • On the divisibility $a!+b!\mid (a+b)!$
    The American Mathematical Monthly, 129(3) (2022). journal
  • On Quotients of Values of Euler’s Function on Factorials (with Abhishek Jha)
    Bulletin of the Australian Mathematical Society, 105(3) (2022). journal arXiv

Seminars

Notes

  • Almost Purity pdf
  • Tilting equivalence for perfectoid algebras pdf
  • Special cycles on unitary Shimura varieties pdf
  • Hodge-Tate decomposition for abelian varieties with good reduction pdf
  • Moduli schemes of elliptic curves pdf
  • Deligne's construction of $\ell$-adic Galois representations attached to normalized cuspidal newforms of weight $k\ge 2$ pdf
  • Ribet's converse to Herbrand's theorem pdf
  • Galois representations attached to modular forms of weight 1 pdf
  • Tate uniformization of Drinfeld modules pdf
  • An introduction to the mod $p$ local Langlands correspondence for $\text{GL}_2(\mathbb Q_p)$ pdf slides
  • Alterations pdf slides
  • The Hochster-Roberts theorem pdf
  • (old) Olympiad handout on analytic number theory pdf AoPS Page

Misc