Separation of Variables and the analytic Langlands correspondence. Conjectured by Teschner and developed by Etingof-Frenkel-Kazhdan, the analytic Langlands 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 involve revisiting and making explicit certain aspects, both classical and quantum, 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 theory techniques.

Results

• With J. Teschner, the geometry underlying the classical limit, i.e. the Separation of Variables map in rank 2 [arXiv].
• An analogue of this map for λ-connections, in which opers with apparent singularities play the central role [arXiv].
• With P. Dimakis and S. Xu, a variant for higher rank: Lagrangian correspondences between moduli spaces of Higgs bundles (respectively, flat connections) and Hilbert schemes of points [arXiv].

Ongoing

• With R. Abedin, F. Ambrosino and J. Teschner: construction of an integral transform that realizes the analytic Langlands correspondence for compact Riemann surfaces of genus g ≥ 2, generalizing the case of punctured Riemann spheres proved by Ambrosino-Teschner.
• With S. Maibach, A. Pajala, E. Peltola and J. Teschner: relation between the Schramm-Loewner evolution and a variant of the analytic Langlands correspondence.

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.

Results

• Wobbly bundles, i.e. stable bundles which admit nonzero nilpotent Higgs fields, form the singular loci of Hecke eigensheaves, and I gave an explicit description of rank-2 wobbly bundles in terms of special divisors on spectral curves [arXiv].
• The conformal limit maps the Hitchin section to the space of opers, and more generally relates the Białynicki-Birula stratification of the moduli of Higgs bundles to Simpson's partial oper stratification of the moduli of flat connections. With T. Figiel and J. Teschner, I related crossing to lower strata in this stratification to isomonodromic bubbling of projective connections [arXiv].

Forthcoming and ongoing

• With R. Donagi, C. Nollau and T. Pantev, I study shaky and wobbly loci of C^*-fixed points in general semi-projective varieties from the point of view of Białynicki-Birula theory. For moduli spaces of Higgs bundles, we proved that the two loci coincide, with the shaky locus defined via a single smooth Hitchin fiber, for stable bundles of rank 2 and for Hodge bundles of type (1, …, 1) in arbitrary rank (paper to appear).
• With D. Arinkin, I am working to make explicit the passage between the de Rham and Dolbeault geometric Langlands correspondences.

Hyperbolic band theory and quantum error correction. This is the part of my research which makes surprising contact with experimental science. In the hyperbolic band theory of Maciejko-Rayan, a crystal whose lattice is given by a Fuchsian group corresponds to a compact Riemann surface, and Kienzle-Rayan observed that Higgs bundles enter this picture naturally, as spectral curves encode crystal lattices and Hitchin fibers the corresponding spaces of abelian crystal momenta.

Results

• With S. Rayan, I studied this dictionary in the presence of a uniform magnetic field. For hyperbolic crystals of genus two, we related the jump of the lowest Landau level at zero momentum to the crossing of Higgs bundles to the Hitchin section, i.e. the lowest Białynicki-Birula stratum in this setting [arXiv].

Ongoing

• We are also working on a precise dictionary of Higgs bundles in the quantum error correction setting.

Other research interests include:

• the relation of Donagi-Pantev-Simpson's approach to geometric Langlands, which uses mirror symmetry and non-abelian Hodge theory, to the 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 the Verlinde formula.