I work on algebraic and differential geometry. Often the structures I study are related to ideas from mathematical physics.

The analytic Langlands correspondence: conjectured by Teschner and developed by Etingof-Frenkel-Kazhdan, this correspondence shifts the focus from sheaves in the geometric Langlands correspondence to functions which are eigenfunctions of Hecke operators. This calls for an exploration of rather explicit nature. My projects with Teschner involve revisiting and making explicit certain aspects of two early approaches to the geometric Langlands correspondence: the first one by Drinfeld using Radon transform and the second by Beilinson-Drinfeld using conformal field theories techniques.

Moduli of Higgs bundles, flat/projective connections: these moduli display a rich variety of geometric structures. Wobbly bundles, conformal limits, Białynicki-Birula stratification and deformation of symplectic structures are the usual ingredients I work with. Recently, I enjoy thinking about giving a concrete description of branes in the moduli of Higgs bundles and their Fourier-Mukai transforms.

Other research interests include:

• The relation between Donagi-Pantev-Simpson's approach to geometric Langlands, which uses mirror symmetry and non-abelian Hodge theories, to analytic Langlands correspondence and Drinfeld's approach;
• Bridgeland's spaces of stability conditions and his complexification of moduli of Higgs bundles;
• the interaction between random geometry (e.g. Schramm-Loewner evolution), real enumerative geometry and conformal field theories;
• a geometric understanding of Verlinde formula.

• From λ-connections to PSL(2,C)-opers with apparent singularities, (2024) [arXiv].
• Rank-2 wobbly bundles from special divisors on spectral curves, (2024) [arXiv].
• Classical limit of the geometric Langlands correspondence for SL(2,C), with J. Teschner, (2023) [arXiv].
• Stratified description of the moduli spaces of Higgs bundles and connections, PhD Thesis, (2023) [electronic publication].

The following papers are in preparation:

• Bialynicki-Birula stratification and apparent singularities of projective structures, with T. Figiel and J. Teschner.
• Quantum Separation of Variables of the Hitchin system, with J. Teschner.