Research
My research has two main threads:
mathematical aspects of materials science
and
infinite-dimensional geometry and shape analysis.
Mathematical aspects of materials science:
Many solid bodies in nature are “frustrated” — their internal geometry is incompatible with the surrounding Euclidean space. This happens, for example, in leaves that grow faster near their edges, or in crystalline materials with lattice defects.
Their study — often known as non-Euclidean elasticity — seeks to understand how such incompatibilities determine the shape and mechanical behavior of solids.
While rooted in questions from physics and materials science, this area leads to deep problems in differential geometry and analysis, for example concerning stability of isometric immersions and limits of variational (minimization) problems.
My work ranges from these core mathematical questions to rigorous explanations of physically observed behavior, and to collaborations with theoretical and experimental physicists.
Publications and preprints:
- Curvature Potential Formulation for Thin Elastic Sheets, with Yael Cohen, Animesh Pandey, Yafei Zhang and Michael Moshe, preprint.
- The Willmore energy and curvature concentration, with Raz Kupferman and David Padilla-Garza, preprint.
- Rigorous analysis of shape transitions in frustrated elastic ribbons, with Maria Giovanna Mora, preprint.
- On material-uniform elastic bodies with disclinations and their homogenization, Mathematics and Mechanics of Solids, 30(9), 2043–2053, 2025.
- A continuum geometric approach for inverse design of origami structures, with Alon Sardas and Michael Moshe, Journal of the Mechanics and Physics of Solids, 196 (2025), 106003.
- Linearization in incompatible elasticity for general ambient spaces (previously titled "Elasticity between manifolds: the weakly-incompatible limit"), with Raz Kupferman, SIAM Journal on Mathematical Analysis, 57(5):5598—5627, 2025.
- Stability of isometric immersions of hypersurfaces, with Itai Alpern and Raz Kupferman, Forum of Mathematics, Sigma, 12:e43, 2024.
- From Volterra dislocations to strain-gradient plasticity, with Raz Kupferman, Calculus of Variations and PDEs, 65:102, 2026.
- Hierarchy of Geometrical Frustration in Elastic Ribbons: shape-transitions and energy scaling obtained from a general asymptotic theory, with Ido Levin, Emmanuel Siéfert and Eran Sharon, Journal of the Mechanics and Physics of Solids, 156 (2021), 104579.
- Asymptotic rigidity for shells in non-Euclidean elasticity, with Itai Alpern and Raz Kupferman, Journal of Functional Analysis, 283(6) (2022), 109575.
- Reference configurations versus optimal rotations: a derivation of linear elasticity from finite elasticity for all traction forces, with Maria Giovanna Mora, Journal of Nonlinear Science, 31:62, 2021.
- Limits of distributed dislocations in geometric and constitutive paradigms, with Marcelo Epstein and Raz Kupferman, in Geometric Continuum Mechanics (R. Segev and M. Epstein, ed.), Birkhäuser, 2020.
- A simple example of the weak discontinuity of f↦∫det∇f, a short note (2018).
- On the role of curvature in the elastic energy of non-Euclidean thin bodies, with Asaf Shachar, Journal of Elasticity, 134(2) (2019), 149–173.
- Variational Convergence of Discrete Geometrically-Incompatible Elastic Models, with Raz Kupferman, Calculus of Variations and PDEs, 57:39 (2018).
- Reshetnyak rigidity for Riemannian manifolds (previously titled "Asymptotoc rigidity of Riemannian manifolds"), with Raz Kupferman and Asaf Shachar, Archive for Rational Mechanics and Analysis, 231 (2019), 367–408.
- Limits of elastic models of converging Riemannian manifolds, with Raz Kupferman, Calculus of Variations and PDEs, 55:40 (2016).
- Non-metricity in the continuum limit of randomly-distributed point defects, with Raz Kupferman and Ron Rosenthal, Israel Journal of mathematics 223(1) (2018), 75–139.
- Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities, with Raz Kupferman, Proc. of the Royal Society of Edinburgh, 146A (2016), 741–768.
- The emergence of torsion in the continuum limit of distributed edge-dislocations, with Raz Kupferman, Journal of Geometric Mechanics, 7(3) (2015), 361-387.
Erratum
- A Riemannian approach to the membrane limit in non-Euclidean elasticity, with Raz Kupferman, Communications in Contemporary Mathematics 16 (2014), 1350052.
Infinite-dimensional geometry and shape analysis:
Many problems in imaging, hydrodynamics, and data analysis can be viewed geometrically: one studies spaces of shapes, maps, or deformations, and asks what their natural geometry looks like.
For example, many equations from fluid mechanics can be interpreted as geodesic equations on diffeomorphism groups, while in imaging one may want to deform one shape or image into another along an optimal path.
This leads to infinite-dimensional versions of Riemannian geometry, where familiar notions such as geodesics and distance can behave in unexpected ways (for example, the geodesic distance between two points may be zero!).
My research seeks to understand these geometries; although mostly theoretical, it has connections to algorithms in graphics, imaging, and geometric data science.
Publications and preprints:
- Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces, with Martin Bauer and Benedikt Wirth, preprint.
- A Riemannian viewpoint on the Amari–Čencov α-connections and Proudman–Johnson equations, with Martin Bauer and Alice Le Brigant, preprint.
- Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves, with Martin Bauer and Patrick Heslin, Journal of Geometric Analysis, 34:214, 2024.
- The Lp-Fisher-Rao metric and Amari-Cencov α-connections, with Martin Bauer, Alice Le Brigant and Yuxiu Lu, Calculus of Variations and PDEs, 63:56, 2024.
- A geometric view on the generalized Proudman-Johnson and r-Hunter-Saxton equations, with Martin Bauer and Yuxiu Lu, Journal of Nonlinear Science, 32:17, 2022.
- Sobolev metrics on spaces of manifold valued curves, with Martin Bauer and Peter Michor, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), Vol. XXIV, 1895–1948, 2023.
- Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics, with Martin Bauer, Calculus of Variations and PDEs, 60:54, 2021.
- Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents, with Robert Jerrard, Annals of Global Analysis and Geometry, 56(2) (2019), 351–360.
- Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups, with Robert Jerrard, Annals of Global Analysis and Geometry, 55(4) (2019), 631–656.
My Master's thesis was in game theory:
Co-authors
Itai Alpern,
Martin Bauer,
Yael Cohen,
Marcelo Epstein,
Patrick Heslin,
Robert Jerrard,
Raz Kupferman,
Alice Le Brigant,
Ido Levin,
Yuxiu Lu,
Peter Michor,
Maria Giovanna Mora,
Michael Moshe,
David Padilla-Garza,
Animesh Pandey,
Ron Rosenthal,
Alon Sardas,
Emmanuel Siéfert,
Asaf Shachar,
Eran Sharon,
Eilon Solan,
Benedikt Wirth,
Yafei Zhang.