Back to main

## Research

Currently, my research explores the use of categorical and geometric methods in modular representation theory. Topics I enjoy thinking about include:
• algebraic groups and Lie theory,
• categorification, diagrammatics, and (higher) category theory,
• (singular) Soergel bimodules and related combinatorial categories,
• Frobenius kernels and their representation theory,
• derived categories and perverse sheaves,
• the affine Grassmannian and geometric Satake,
• and almost anything else in algebraic geometry or algebraic topology!

The motivating problem for my PhD project (2018–present) has been the categorical conjecture of Riche–Williamson. Given a connected reductive group $$\textbf{G}$$ over a field of positive characteristic, the conjecture states that the associated affine Hecke category $$\mathscr{H}$$ acts on the principal block $$\text{Rep}_0(\textbf{G})$$ of finite-dimensional algebraic $$\textbf{G}$$-modules. Great consequences in representation theory follow from the existence of such an action, including new character formulas for simple and indecomposable tilting modules. Two methods of proof are now known for the categorical conjecture: one found by Bezrukavnikov–Riche, using coherent sheaves and Abe's bimodule-theoretic realisation of $$\mathscr{H}$$, and one found by me, using constructible sheaves and Smith–Treumann theory. I am now working on a singular extension of the result and its applications.

Previously (2016–2018), I completed a research master's degree under Konstantin Ardakov at Oxford. The subject of my research was rigid geometry, specifically D-module theory over Ardakov–Wadsley's sheaf of infinite-order differential operators on a rigid analytic space.

### Papers and preprints

1. Ciappara, J. Singular Hecke category actions. In preparation.
2. Ciappara, J. Actions of the Hecke category via Smith–Treumann theory. Submitted. Preprint available here.
3. Ciappara, J. and Williamson, G. Lectures on the geometry and modular representation theory of algebraic groups. Journal of the Australian Mathematical Society 110, 1–47 (2021).
4. Arunasalam, S. and Ciappara, J. and Nguyen, D.M.H., et al. A note on categorification and spherical harmonics. Algebras and Representation Theory 23, 1285–1295 (2020).
5. Invariants of D-modules. Master's thesis (2018).

### Talk notes

1. The Atiyah–Hirzebruch spectral sequence (IFS 29/10/21).
2. Introduction to model categories (IFS 17/09/21).
3. Hecke category actions via Smith–Treumann theory (Bonn 23/07/21, IFS 21/05/21).
4. Introduction to Kazhdan–Lusztig theory (SAS 25/05/21).
5. Derived categories and functors (SAS 24/03/21).
6. Geometric representations of Heisenberg algebras (IFS 13/11/20).
7. Euler and Pontryagin classes (SAS 29/10/20).
8. Constructions with vector bundles (SAS 28/08/20).
9. Introduction to vector bundles (SAS 21/08/20).
10. What is modular representation theory? (WiSe 20/08/20, exercises).
11. Combinatorial actions of Soergel bimodules II (IFS 01/11/19).
12. Combinatorial actions of Soergel bimodules I (IFS 25/10/19).
13. Tensor products of finite and infinite dimensional representations of semisimple Lie algebras II (IFS 07/06/19).
14. Tensor products of finite and infinite dimensional representations of semisimple Lie algebras I (IFS 31/05/19).
15. Perverse sheaves and the weak Lefschetz theorem (IFS 09/11/18).