Nimish A. SHAH 

Professor, Department of Mathematics

The Ohio State University, Columbus, Ohio 43210.

Email: shah AT math.osu.edu

List of Papers:

Quantitative behavior of unipotent flows and an effective avoidance principle. (With Elon Lindenstrauss, Amir Mohammadi, Gregory Margulis). 52 pages. arXiv:1904.00290.

Primitive rational points and higher powers on expanding horocycles in the modular surface.(With Manfred Einsiedler and Manuel Luethi).

 Ergodic Theory and Dynamical Systems, 1-45. doi:10.1017/etds.2020.15arXiv:1901.03078

Expanding translates of shrinking submanifolds in homogeneous spaces and Diophantine approximation.(With Pengyu Yang). 16 pages. arXiv:1809.05570

Equidistribution of dilated curves on nilmanifolds.(With Bryna Kra and Wenbo Sun).

J. Lond. Math. Soc. (2) 98 (2018), no. 3, 708--732. DOI:10.1112/jlms.12156

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: 10.3934/dcds.2020227   arXiv:1606.00152

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:10.1017/etds.2016.138

Equidistribution of primitive rational points on expanding horospheres.(With M. Einsiedler, S. Mozes, and U. Shapira) Compositio Mathematica, 152 (2016), 667 - 692. DOI:10.1112/S0010437X15007605.

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 Univ. Press, Cambridge, 2016. DOI:10.1017/CBO9781316275849.009 

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: 10.1007/s11856-013-0063-2

Equidistribution and Counting for orbits of geometrically finite hyperbolic groups. (With H. Oh). 

J. Amer. Math. Soc. (JAMS), 26 (2013), 511 Р562. DOI: 10.1090/S0894-0347-2012-00749-8

The Asymptotic distribution of circles in the orbits of Kleinian groups. (With H. Oh).

Inventiones Math., 187 (2012), no. 1, 1-35. DOI: 10.1007/s00222-011-0326-7.

Equidistribution of translated curves on homogeneous spaces and Dirichlet's approximation

Proceedings of the International Congress of Mathematicians, Volume III,  1332--1343, Hindustan Book Agency, New Delhi,  2010.

Khinchin theorem for integral points on quadratic varieties. (With A. Gorodnik). 

Math. Annalen, 350 (2011), no. 2, 357-380. DOI: 10.1007/s00208-010-0561-z.

Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms. 

J. Amer. Math. Soc. (JAMS), 23 (2010), 563-589

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.

Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation. 

Inventiones Math., 177 (2009), 509-532

Limiting distributions of evolution of smooth curves under geodesic flow on hyperbolic manifolds. 

Duke Math. J., 148 (2009), no. 2, 281-304.

Limiting distributions of curves under geodesic flow on hyperbolic manifold. 

Duke Math J., 148 (2009), 251-279

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.

Integral points on symmetric varieties and Satake compactifications. (With H. Oh and A. Gorodnik). 

Amer. J. Math., 131 (2009), no. 1, 1-57.

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

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), Vol. 1A, Chapter 11, 813--930, North-Holland, Amsterdam, 2002. 

Counting integral matrices with a given characteristic polynomial. 

Sankhya (Ser. A), 62 (2000), no. 3, 386--412.

On actions of epimorphic subgroups on homogeneous spaces. (With Barak Weiss). 

Ergodic Theory Dynam. Systems, 20 (2000), no. 2, 567-59.

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. 

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

Unipotent flows and counting lattice points on homogeneous varieties. 

Annals of Math., 143 (1996), no. 2, 253--299.

Non-divergence of translates of certain algebraic measures. (With A. Eskin and S. Mozes). 

Geom. Funct. Anal. (GAFA), 7 (1997), no. 1, 48-80.

Limit distributions of expanding translate of certain orbits on homogeneous spaces. 

Proc. Indian Acad. Sci. Math. Sci., 106 (1996), no. 2, 105--125

On the space of ergodic invariant measures of unipotent flows. (With S. Mozes). 

Ergodic Theory Dynam. Systems, 15 (1995), no. 1, 149--159.

Limit distributions of polynomial trajectories on homogeneous spaces. 

Duke Math. J., 75 (1994), no. 3, 711--732

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.

Uniformly distributed orbits of certain flows on homogeneous spaces. 

Math. Annalen, 289 (1991), no. 2, 315--334

DISSERTATIONS:

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

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