Research

My main research interest is convex geometry, and anything in differential, integral or metric geometry that has a convex feel or a convex history. I am also interested in spectral, symplectic and contact geometries, and their interaction with convex geometry.

Most recently, I worked in valuation theory (which bridges convex, differential and integral geometries), often in the presence of a non-compact Lie group of symmetries. I also have works related to Finsler geometry, asymptotic geometry, classical analysis, and one very old work in analytic number theory.

Publications

• 1. The Fourier transform on valuations is the Fourier transform, joint with T. Wannerer, submitted, 2023.

  2. Volume growth of Funk geometry and the flags of polytopes, joint with C. Vernicos and C. Walsh, submitted, 2023.

  3. Convex valuations, from Whitney to Nash, joint with G. Hofstaetter, submitted, 2023.

  4. Quasianalyticity, uncertainty, and integral transforms on higher grassmannians, to appear in Advances in Mathematics 2023+.

  5. A Funk perspective on billiards, projective geometry and Mahler volume, to appear in Journal of Differential Geometry 2022+.

  6. Crofton formulas in pseudo-Riemannian space forms, joint with A. Bernig and G. Solanes, to appear in Compos. Math. 158 (2022), no. 10, 1935–1979. 

  7. Curvature measures of pseudo-Riemannian manifolds, joint with A. Bernig and G. Solanes, J. Reine Angew. Math. 788 (2022), 77–127. 

  8. Uniqueness of curvature measures in pseudo-Riemannian geometry, joint with A. Bernig and G. Solanes, J. Geom. Anal. 31 (2021), no. 12, 11819–11848. 

  9. The Weyl principle on the Finsler frontier, joint with T. Wannerer, Selecta Math. (N.S.) 27 (2021), no. 2, Paper No. 27, 30 pp.

  10. Contact integral geometry and the Heisenberg algebra, Geom. Topol. 23(2019) no. 6, 3041-3110.

  11. Crofton Formulas and Indefinite Signature, with an appendix joint with T. Wannerer, Geom. Funct. Anal. 27 (2017), no. 3, 489–540.

  12. Submultiplicative Operators on C^k spaces, joint with H. König and V.D. Milman.  Tribute to Victor Havin. 50 years with Hardy spaces, eds. A. Baranov, S. Kisliakov, N. Nikolski, Operator Theory: Advances and Applications, Birkhäuser Verlag 2017.

  13. Valuation Theory of Indefinite Orthogonal Groups, joint with A. Bernig, J. Funct. Anal. 273(2017) no.6, 2167–2247.

  14.  Generalized translation invariant valuations and the polytope algebra, joint with A. Bernig. Adv. Math. 290 (2016), 36–72.

  15.  Lipschitz functions on the infinite-dimensional torus, joint with B. Klartag.  Commun. Contemp. Math. 18 (2016), no. 1, 1550029, 9 pp.

  16. On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus, joint with B. Klartag and V. Milman. Geometric aspects of functional analysis, 123–131, Lecture Notes in Math., 2116, Springer, 2014. 

  17. Convex valuations invariant under the Lorentz group, joint with S. Alesker, J. Differential Geom., Volume 98, Number 2 (2014), 183-236.

  18. On a stability property of the spherical Radon transform with respect to measure perturbations, Asymptotic geometric analysis, Fields Communication Series, Volume 68, 2013, 55-73.

  19. An extension of Schäffer's dual girth conjecture to Grassmannians, J. Differential Geom., Volume 92, Number 1 (2012), 201-220.

  20. On multiplicative maps of continuous and smooth functions, joint with S. Artstein-Avidan and V. Milman, Geometric Aspects of Functional Analysis, Springer Lecture Notes, 2012, 35-59.

  21. A family of unitary operators satisfying a Poisson-type summation formula, Geometric Aspects of Functional Analysis, Springer Lecture Notes, 2012, 191-204.

  22.  Appendix to Characterizing the derivative and the entropy function by the Leibniz rule, by H. König, V. Milman, J. Funct. Anal. 261(2011), 1325-1344.

  23. A characterization of product preserving maps with applications to a characterization of the Fourier transform, joint with S. Alesker, S. Artstein-Avidan, V. Milman, Illinois J. Math. 54 (2010), no. 3, 1115–1132 (2012).

  24. A characterization of Fourier transform by Poisson summation formula. C. R. Math. Acad. Sci. Paris 348 (2010), no. 7-8, 407–410.

  25. Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field, joint with Z. Rudnick. Compos. Math. 146 (2010), no. 1, 81–101.