Jayadev Athreya
University of Washington
August 8, 2024
Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori: This is joint work with Jean Lagacé, Martin Möller, and Martin Raum. Generalizing the construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. We give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows us to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections, and new phenomena for the spectrum appear. No prior knowledge of Eisenstein series, the Langlands program, or Siegel-Veech transforms is required.
University of Washington
August 8, 2024
Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori: This is joint work with Jean Lagacé, Martin Möller, and Martin Raum. Generalizing the construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. We give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows us to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections, and new phenomena for the spectrum appear. No prior knowledge of Eisenstein series, the Langlands program, or Siegel-Veech transforms is required.