About

Gabriel Paternain is associated with the Math AI Lab at the University of Washington, which is a research and education organization focused on using AI for mathematics. The lab is directed by Vasily Ilin and Jarod Alper and has been involved in numerous projects, including formalization projects using Lean, and teaching courses on Lean. The lab's activities include research on semantic search over mathematical theorems and learning to repair Lean proofs from compiler feedback, among other topics. The Math AI Lab was formerly known as the eXperimental Lean Lab (XLL) and is part of the Department of Mathematics at the University of Washington.

Research topics

• Mathematical analysis
• Mathematics
• Applied mathematics
• Mathematical optimization
• Algorithm
• Geometry
• Pure mathematics
• Statistics

Selected publications

• Quasi‐Fuchsian flows and the coupled vortex equations

  Bulletin of the London Mathematical Society · 2026-03-01

  articleOpen accessSenior author

  Abstract We provide an alternative construction of the quasi‐Fuchsian flows introduced by Ghys. Our approach is based on the coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of the Blaschke metric, uniquely determined by a conformal class and a holomorphic quadratic differential. We also give formulas for the marked length spectrum of a quasi‐Fuchsian flow in the thermostat parametrization.

  PublisherDOI

• Local and Global Blow Downs of Transport Twistor Space

  Symmetry Integrability and Geometry Methods and Applications · 2026-05-05

  preprintOpen accessSenior author

  Transport twistor spaces are degenerate complex $2$-dimensional manifolds $Z$ that complexify transport problems on Riemannian surfaces, appearing, e.g., in geometric inverse problems. This article considers maps $\beta\colon Z\to \mathbb{C}^2$ with a holomorphic blow-down structure that resolve the degeneracy of the complex structure and allow to gain insight into the complex geometry of $Z$. The main theorems provide global $\beta$-maps for constant curvature metrics and their perturbations and local $\beta$-maps for arbitrary metrics, thereby proving a version of the classical Newlander-Nirenberg theorem for degenerate complex structures.

  PublisherOA PDFDOI

• Quasi-Fuchsian flows and the coupled vortex equations

  arXiv (Cornell University) · 2025-01-17

  preprintOpen accessSenior author

  We provide an alternative construction of the quasi-Fuchsian flows introduced by Ghys in \cite{Ghys-92}. Our approach is based on the coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of the Blaschke metric uniquely determined by a conformal class and a holomorphic quadratic differential. We also give formulas for the marked length spectrum of a quasi-Fuchsian flow in the thermostat parametrization.

  PublisherOA PDFDOI

• An inverse problem for the Standard Model of particle physics

  ArXiv.org · 2025-05-30

  preprintOpen accessSenior author

  We pose and solve an inverse problem for the classical field equations that arise in the Standard Model of particle physics. Our main result describes natural conditions on the representations, so that it is possible to recover all the fields from measurements in a small set within a causal domain in Minkowski space. These conditions are satisfied for the representations arising in the Standard Model.

  PublisherOA PDFDOI

• On the interplay between the light ray and the magnetic X-ray transforms

  ArXiv.org · 2025-02-06

  preprintOpen access

  We study the light ray transform acting on tensors on a stationary Lorentzian manifold. Our main result is injectivity up to the natural obstruction as long as the associated magnetic vector field satisfies a finite degree property with respect to the vertical Fourier decomposition on the unit tangent bundle. This is based on an explicit relationship between the geodesic vector field of the Lorentzian manifold and the magnetic vector field.

  PublisherOA PDFDOI

• Resonant forms at zero for dissipative Anosov flows

  Geometry & Topology · 2025-10-10 · 1 citations

  articleOpen accessSenior author

  We study resonant differential forms at zero for transitive Anosov flows on 3-manifolds.We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure.Such flows have two distinguished Sinai-Ruelle-Bowen 3-forms, RB , and the cohomology classes OE X RB (where X is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant 1-forms.When both classes vanish we associate to the flow a helicity that naturally extends the classical notion associated with null-homologous volume-preserving flows.We provide a general theory that includes horocyclic invariance of resonant 1-forms and SRB-measures as well as the local geometry of the maps X 7 !OE X RB near a null-homologous volume-preserving flow.Next, we study several relevant classes of examples.Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys (1992).For these flows we compute explicitly all resonant 1-forms at zero, we show that OE X RB D 0 and give an explicit formula for the helicity.In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is .M /, the same as in the geodesic flow case.In contrast, we show that if .M; g/ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic 1-form has a Ruelle zeta function whose order of vanishing at zero is .M / 1.

  PublisherOA PDFDOI

• On the Interplay between the Light Ray and the Magnetic X-Ray Transforms

  SIAM Journal on Mathematical Analysis · 2025-11-14 · 1 citations

  article
  PublisherDOI

• Marked length spectrum rigidity for Anosov surfaces

  Duke Mathematical Journal · 2025-01-13 · 2 citations

  articleOpen accessSenior author

  Let Σ be a smooth closed oriented surface of genus ≥2. We prove that two metrics on Σ with the same marked length spectrum and Anosov geodesic flow are isometric via an isometry isotopic to the identity. The proof combines microlocal tools with the geometry of complex curves.

  PublisherOA PDFDOI

• Biholomorphism rigidity for transport twistor spaces

  Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2025-04-01

  articleSenior author

  We prove that biholomorphisms between the transport twistor spaces of simple or Anosov surfaces exhibit rigidity: they must be, up to constant rescaling and the antipodal map, the lift of an orientation-preserving isometry.

  PublisherDOI

• Carleman estimates for geodesic X-ray transforms

  Annales Scientifiques de l École Normale Supérieure · 2024-01-22 · 10 citations

  articleOpen access1st authorCorresponding

  I will describe a new energy estimate for the geodesic vector field of a manifold of negative curvature. The estimate has several\napplications including injectivity of non-abelian X-ray transforms.\n\nThis is joint work with Mikko Salo.

  PublisherOA PDFDOI

Frequent coauthors

• K� Falconer

  Kinokuniya

  234 shared

• J Woodhouse

  Kinokuniya

  234 shared

• J Luk

  University College London

  207 shared

• Martin R. Bridson

  University of Oxford

  207 shared

• Nathanaël Berestycki

  207 shared

• Benny Sudakov

  ETH Zurich

  198 shared

• Henry Wilton

  198 shared

• Le Gross

  Cambridge University Press

  198 shared

Labs

• Math AI Lab, University of WashingtonPI