অর্ণব কুণ্ডু

Preprints

1. Compactification des groupes parahoriques via leurs réductifications, (in French), submitted, 2026.

   Using tame reductification proven in [6], I construct wonderful compactifications of parahoric group schemes in good residue characteristics.

2. Reductification of parahoric group schemes, submitted, 2026, (recorded talk from 2024).

   I establish that, after a suitable ramified base change, every parahoric group becomes reductive in an appropriate sense. As an application, I obtain an analogue of the Grothendieck–Serre conjecture.

3. 𝔸¹-connectivity of motivic spaces, (with Tess Bouis), submitted, 2025.

   Using the presentation lemma proved in [4], we extend the shifted unstable 𝔸¹-connectivity theorem of Morel and Ayoub to arbitrary qcqs base schemes.

4. Beilinson-Lichtenbaum phenomenon for motivic cohomology (with Tess Bouis), submitted, 2025, (recorded talk from 2024).

   We show that motivic cohomology recovers algebraic cycles. We also prove a purity result over deeply-ramified bases. The latter could provide a cohomological refinement of an ingredient in Nizioł's proof of the crystalline conjecture.

5. Isotropic torsors on smooth algebras over Prüfer rings, submitted, 2025.

   I demonstrate that a generically trivial torsor under a totally isotropic reductive group scheme on a smooth algebra over a valuation ring of rank one is trivial. This generalises the results of my thesis, which proved the same in the case of quasi-split groups.

6. Gersten's injectivity for smooth algebras over valuation rings, submitted, 2024.

   I prove that the K-groups of the local rings of smooth algebras over valuation rings inject inside the K-groups of their localisation at the generic point of the special fibre.

Publications

1. Tilting correspondences of perfectoid rings, J. Ramanujan Math. Soc. 38, No. 1 (2023) 23–32, journal.

   I establish Scholze's tilting equivalences of étale cohomology of perfectoid rings algebraically, i.e., without using tools from almost ring theory or adic spaces.

Thesis

• Torsors on Smooth Algebras over Valuation Rings, CNRS-HAL Id : tel-04129990 (2023).

  I demonstrate that a generically trivial torsor under a quasi-split reductive group scheme on a smooth algebra over a valuation ring of rank one is trivial.

Notes

• The Etale Fundamental Group of an Elliptic Curve - University of Chicago Math REU'17