Shubhankar Sahai

Email: ssahai (at) ucsd (dot) edu
Office: HSS 3073

I am a graduate student in the math department at UCSD.

I work in arithmetic and algebraic geometry. More specifically, my work is in the intersection of derived algebraic geometry, p-adic Hodge theory and algebraic K-theory. I also have some orthogonal ongoing work in the duality theory of Grothendieck differential operators (in the smooth proper case).

My advisor is Kiran Kedlaya.

Current Seminars

In Spring 2026, we have a seminar on alterations.

Research

I am broadly interested in algebraic and arithmetic geometry. My current work is aimed towards setting up some theory in order to study the coefficients for syntomic cohomology of Frobenius liftable schemes. In particular, in the list below, the titles which either have variants of the words 'prismatic' or 'syntomic', are all aimed towards this program. The papers titled 'differential operators...' are orthogonal to the above and are aimed at setting up a duality theory for Grothendieck differential operators. Perhaps the common underlying theme is 'de Rham cohomology'.

Preprints

• Duality of differential operators and algebraic de Rham cohomology arXiv — Preprint
  with Caleb Ji, Casimir Kothari, Oliver Li, Sveta Makarova and Sridhar Venkatesh.
• De Rham affineness of the Nygaard filtered prismatization in positive characteristic arXiv · PDF — Submitted
  Notes

  The PDF version has a more readable layout. Comments very welcome!

  The key idea in this paper is that the Nygaard filtered prismatization of an animated ring in positive characteristic is determined by its Nygaard filtered prismatic cohomology, functorially in the ring. The idea is not new and is already in the second paper of Bhatt-Lurie for unfiltered prismatization, but the observation that this is an intrinsic property of the functors in question (and not just the associated derived stacks) is. We name this property de Rham affineness. This concept will be useful in my future papers. Note that positive characteristic is not necessary and the same proof works mutatis mutandis relative to a perfectoid. It is also true relative to a bounded prism, provided one works with relative prismatic cohomology, but that's the subject of future work.
• Derived algebras on formal stacks and prismatic gauges arXiv · PDF — Preprint
  Notes

  The PDF version has corrected some obvious typos. I will update the arxiv version soon. Comments very welcome!

  A paper studying the theory of derived algebras (in the sense of Bhatt-Mathew and Raksit) in quasi-coherent sheaves on non-trivially formal derived stacks. The main goal is to develop a good working theory and prove classification theorems for derived algebras in prismatic gauges. Along the way we prove some basic theorems in (non-connective) formal derived geometry which seem to be missing from the literature.
• Relative Poincaré duality for differential complexes — Draft available on request
  Joint with Caleb Ji.
  Notes

  Generalises our previous joint work to the case of bounded differential complexes i.e. complexes of locally free sheaves whose differentials are finite order linear differential operators over the base and for coefficients.
• The syntomification of Frobenius liftable schemes — In preparation
  Notes

  This is the final paper in the series on Frobenius liftable schemes. The aim is to describe the syntomificaiton of Frobenius liftable schemes as Fontaine-Laffaille modules on the lift. Among other things, we achieve a stacky explanation of the (relatively) recent results of Ogus, and study some new ring stacks which (to our knowledge) weren't known before. Note that earlier I had announced a paper titled 'The relative Nygaard filtered prismatization', however I intend that to appear as a section of this longer paper and not as a different paper in itself.

Teaching

UCSD

• Math 184—Enumerative combinatorics: Spring 2026
• Math 154—Discrete mathematics and graph theory: Winter 2026
• Math 187A—Introduction to Cryptography: Winter 2023, Spring 2023, Winter 2024, Spring 2024, Fall 2024, Spring 2025, Fall 2025
• Math 109—Introduction to Mathematical Reasoning: Winter 2025
• Math 20D—Introduction to Differential Equations: Fall 2022
• Math 20E—Vector Calculus: Fall 2021, Winter 2022, Spring 2022

Some notes

Notes that I wrote at some point during my education. Please use with caution and note that there is no claim of originality. Please let me know of errors.

• Notes on Drinfeld's proof of Serre-Tate: Notes explaining Drinfeld's proof of the Serre-Tate theorem on how to control deformations of abelian schemes via their p-divisible group (or, more conceptually, why the obvious map from the moduli of abelian varieties to the stack of Barsotti-Tate groups is formally étale). Essentially taken without much modification from Katz's paper. Written for a seminar on Katz-Mazur.
• The stack of coherent sheaves: Informal notes filling in the details of Laumon and Moret-Bailly's proof of the algebraicity of the stack of coherent sheaves. Might be useful to beginners. Note that the mysterious references to Illusie refer to his notes on formal geometry.
• Flat descent for the cotangent complex: A somewhat relaxed exposition of the flat descent result of Bhatt. Written for a seminar on BMS II. Note that an analogous argument works mutatis mutandis in the animated setting. I will update the notes in the future.
• Crystals in fibered categories: A slightly terse explanation of how to view crystals as cartesian sections of a certain fibered category over the category of schemes. Written a long time ago.
• An informal introduction to descent: Slides introducing a general math audience to the ideas of descent theory. Written as an advertisement for a more advanced seminar on algebraic stacks which ran soon after.

Miscellaneous

Service

• I served as the MGSC Mentorship Chair for the academic year 2024-2025.
• I have been a mentor for the DRP program at UCSD twice. The topics covered included local class field theory in the first iteration and some advanced commutative algebra in the second, including Huneke-Lyubeznik's proof of the existence of big Cohen-Macaulay algebras in positive characteristic.

Discord server

Undergrad research

• Composition Tableaux basis for Schur functors and the Plücker algebra: Gives a representation theoretic interpretation of certain combinatorial objects called Row-Strict Composition Tableaux (RSCT) and variants. This has been pending revision, after referee-review, since 2021. I am hopeful that this can be carried out in finite time.

Some resources

Here are some resources which have been personally useful to me but may not be well known.

• Faisceaux pervers aka BBDG. A copy of the 2018 re-edition of BBDG's tome 'Faisceaux pervers'. Uploaded here so that one doesn't have to struggle with the unique scanned copy on Deligne's webpage. Note that you can download your own copy off the SMF website for free, provided you have institutional access.
• On the (co-)homology of commutative rings and Homology of commutative rings. Quillen's manuscripts introducing the cotangent complex.
• Logarithmic spaces (According to K. Kato). Illusie's succinct but hard to find survey on log geometry.
• Deformation theory MSRI. Videos for the 2007 MSRI (now SLMath) workshop on deformation theory. They can, unfortunately, not be found on the SLMath website anymore.
• The de Rham-Witt complex (after Bhatt-Lurie-Mathew). Illusie's lectures at the Chennai Mathematical Institute on the new approach to de Rham-Witt complexes.
• Quillen Notebooks.