Nimish A. SHAH

Professor, Department of Mathematics

The Ohio State University, Columbus, Ohio 43210.

Email: shah@math.osu.edu

List of Papers:

1. Limit distributions for SO(n,1) action on k-lattices in Rⁿ⁺¹. (With Michael Bersudsky). 33 pages (May 2025). DOI

2. An effective closing lemma for unipotent flows. (With Elon Lindenstrauss, Gregory Margulis, Amir Mohammadi and and Andreas Wieser). 24 pages (October 2024). DOI

3. Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces. (With Michael Bersudsky and Hao Xing).

62 pages (July 2024). DOI To appear in Duke Math. J.

4. An upper bound of the Hausdorff dimension of singular vectors on affine subspaces. (With Pengyu Yang). Trans. Amer. Math. Soc., Ser. B, Vol. 11 (2024), 1249-1265. DOI

5. Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation. (With Pengyu Yang). Mathematische Zeitschrift, Vol. 308, 60 (2024), 52 pages. DOI

6. Equidistribution of expanding degenerate manifolds in the space of lattices. (With Pengyu Yang). Proc. London Math. Soc., Vol. 129 (4): e12634. 53 pages. DOI

7. Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines. (With Dmitry Kleinbock, Nicolas de Saxcé, and Pengyu Yang). Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XXV (1) (2024), 565-604.  DOI. arXiv

8. Quantitative behavior of unipotent flows and an effective avoidance principle. (With Elon Lindenstrauss, Gregory Margulis and Amir Mohammadi). Journal D’Analyse Mathematique (JAMA), Vol 153 (2024), 1–61. DOI

9. Expanding translates of shrinking submanifolds in homogeneous spaces and Diophantine approximation.(With Pengyu Yang). Journal of Modern Dynamics, 19 (2023), 947-965. DOI arXiv

10. Primitive rational points and higher powers on expanding horocycles in the modular surface.(With Manfred Einsiedler and Manuel Luethi).  Ergodic Theory and Dynamical Systems, 41 (2021), No. 6, 1706-1750. DOI. arXiv

11. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. (With Lei Yang). Discrete and Continuous Dynamical Systems-A, 40 (2020), no. 9, 5247-5287. DOI

12. Equidistribution of dilated curves on nilmanifolds.(With Bryna Kra and Wenbo Sun). J. Lond. Math. Soc. (2) 98 (2018), no. 3, 708--732. DOI

13. Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics.(With Rodolphe Richard) Ergodic Theory Dynamical Systems, 38 (2018), no. 7, 2780--2800. DOI

14. Equidistribution of primitive rational points on expanding horospheres.(With M. Einsiedler, S. Mozes, and U. Shapira) Compositio Mathematica, 152 (2016), 667-692. DOI.

15. Counting visible circles on the sphere and Kleinian groups. (With H. Oh). In: Geometry, topology, and dynamics in negative curvature, London Math. Soc. Lecture Note Ser., 425, (Cambridge: Cambridge Univ. Press, 2016), pp. 272-288. DOI

16. Limits of translates of divergent geodesics and Integral points on one-sheeted hyperboloids. (With H. Oh). Israel J. Math., 199 (2014), no. 2, 915-931. DOI

17. Equidistribution and Counting for orbits of geometrically finite hyperbolic groups. (With H. Oh). J. Amer. Math. Soc. (JAMS), 26 (2013), 511 Р562. DOI

18. The Asymptotic distribution of circles in the orbits of Kleinian groups. (With H. Oh). Inventiones Math., 187 (2012), no. 1, pp. 1-35. DOI

19. Equidistribution of translated curves on homogeneous spaces and Dirichlet's approximation. Proceedings of the International Congress of Mathematicians, Volume 3, 1332--1343, Hindustan Book Agency, New Delhi, 2010.

20. Khinchin theorem for integral points on quadratic varieties. (With A. Gorodnik). Math. Annalen, 350 (2011), no. 2, 357-380. DOI

21. Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms.  J. Amer. Math. Soc. (JAMS), 23 (2010), 563-589. DOI

22. Strong wave front lemma and counting integral points in sectors of symmetric varieties. (With H. Oh and A. Gorodnik).  Israel J. Math., 176 (2010), 419-443. DOI

23. Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation. Inventiones Math., 177 (2009), 509-532. DOI

24. Limiting distributions of evolution of smooth curves under geodesic flow on hyperbolic manifolds.  Duke Math. J., 148 (2009), no. 2, 281-304. DOI

25. Limiting distributions of curves under geodesic flow on hyperbolic manifold.  Duke Math J., 148 (2009), 251-279. DOI

26. Unipotent flows on products of SL(2,K)/'s.  Dynamical systems and Diophantine approximation, 69--104, Séminaires et Congrès, 19, Soc. Math. France, Paris, 2009.

27. Integral points on symmetric varieties and Satake compactifications. (With H. Oh and A. Gorodnik). Amer. J. Math., 131 (2009), no. 1, 1-57. JSTORE

28. Locally compact groups with dense orbits under $Z^d$-actions by automorphisms. (With S.G. Dani and G.A. Willis). Ergodic Theory Dynam. Systems, 26 (2006), no. 5, 1443--1465. DOI

29. Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory.(With D. Kleinbock and A. Starkov).  Handbook of dynamical systems (ed. A. Katok and B. Hasselblatt), Elsevier Science, Volume 1, Part A, 2002. DOI

30. Counting integral matrices with a given characteristic polynomial.  Sankhya (Ser. A), 62 (2000), no. 3, 386--412. JSTORE

31. On actions of epimorphic subgroups on homogeneous spaces. (With Barak Weiss). Ergodic Theory Dynam. Systems, 20 (2000), no. 2, 567-592. DOI

32. Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements. Lie groups and ergodic theory. Proceedings of the International Colloquium held in Mumbai, Jan 4-12, 1996. Tata Inst. Fund.Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998. 

33. Unipotent flows and counting lattice points on homogeneous varieties. Annals of Math., 143 (1996), no. 2, 253--299. DOI

Correction to "Unipotent flows and counting lattice points on homogeneous varieties".(With A. Eskin and S. Mozes). 2 pages, May 1998. 

34. Non-divergence of translates of certain algebraic measures. (With A. Eskin and S. Mozes). Geom. Funct. Anal. (GAFA), 7 (1997), no. 1, 48-80. DOI

35. Limit distributions of expanding translate of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci., 106 (1996), no. 2, 105--125. DOI

36. On the space of ergodic invariant measures of unipotent flows. (With S. Mozes). Ergodic Theory Dynam. Systems, 15 (1995), no. 1, 149--159. DOI

37. Limit distributions of polynomial trajectories on homogeneous spaces.  Duke Math. J., 75 (1994), no. 3, 711--732. DOI

38. Closures of totally geodesic immersions in manifolds of constant negative curvature. Group theory from a geometrical viewpoint (Trieste, 1990), 718--732, World Sci. Publ., River Edge, NJ, 1991. DOI

39. Uniformly distributed orbits of certain flows on homogeneous spaces.  Math. Annalen, 289 (1991), no. 2, 315--334. DOI

DISSERTATIONS:

40. Unipotent flows on homogeneous spaces.  Ph.D. Thesis, Tata Institute of Fundamental Research, Mumbai, 1994. 69 Pages

41. Unipotent flows on homogeneous spaces of SL(2,C).  M.Sc. Thesis, Tata Institute of Fundamental Research, Mumbai, 1992. 112 pages