Nimish A. SHAH
Professor, Department of Mathematics
The Ohio State University, Columbus, Ohio 43210.
Email:
shah@math.osu.edu
1. An upper bound of the Hausdorff
dimension of singular vectors on affine subspaces. (With Pengyu Yang). 18 pages.DOI:10.48550/arXiv.2311.05834
2. Equidistribution of
non-uniformly stretching translates of shrinking smooth curves and weighted
Dirichlet approximation. (With Pengyu Yang). 50 pages. DOI: 10.48550/arXiv.2204.03194
3.
Equidistribution
of expanding degenerate manifolds in the space of lattices. (With Pengyu Yang). 35
pages. DOI:10.48550/arXiv.2112.13952
4. Equidistribution
in the space of 3-lattices and Dirichlet-improvable vectors on planar lines. (With Dmitry Kleinbock, Nicolas de Saxcé, and Pengyu Yang). Annali della Scuola normale superiore di Pisa, Classe di scienze, 2022 Forthcoming Articles, 35 pages. DOI:10.48550/arXiv.2106.08860
5.
Quantitative
behavior of unipotent flows and an effective avoidance principle. (With
Elon Lindenstrauss, Gregory Margulis and Amir Mohammadi).
Journal d'Analyse Mathématique
(2023), 61 pages. DOI:10.1007/s11854-023-0309-9.
6.
Expanding translates of shrinking
submanifolds in homogeneous spaces and Diophantine approximation.(With Pengyu Yang). Journal of Modern Dynamics, 19
(2023), 947-965. DOI:10.3934/jmd.2023027 arXiv.1809.05570
7.
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:10.1017/etds.2020.15. arXiv:1901.03078
8.
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
9.
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
10.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
11.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.
12.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
13.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
14.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
15.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.
16.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.
17.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.
18.Expanding translates
of curves and Dirichlet-Minkowski theorem on linear
forms. J. Amer. Math. Soc. (JAMS), 23
(2010), 563-589.
19.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.
20.Equidistribution of
expanding translates of curves and Dirichlet's theorem on Diophantine
approximation. Inventiones Math., 177 (2009),
509-532.
21.Limiting
distributions of evolution of smooth curves under geodesic flow on hyperbolic
manifolds. Duke Math. J., 148
(2009), no. 2, 281-304.
22.Limiting distributions
of curves under geodesic flow on hyperbolic manifold. Duke Math J.,
148
(2009), 251-279.
23.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.
24.Integral points on
symmetric varieties and Satake compactifications. (With H. Oh and A. Gorodnik). Amer. J. Math., 131
(2009), no. 1, 1-57.
25.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.
26.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.
27.Counting integral
matrices with a given characteristic polynomial. Sankhya (Ser. A), 62 (2000), no. 3, 386--412.
28.On actions of
epimorphic subgroups on homogeneous spaces. (With Barak Weiss). Ergodic
Theory Dynam. Systems, 20
(2000), no. 2, 567-59.
29.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.
30.Correction to
"Unipotent flows and counting lattice points on homogeneous
varieties".(With A. Eskin and S. Mozes). 2 pages, May 1998.
31.Unipotent
flows and counting lattice points on homogeneous varieties. Annals of
Math., 143 (1996), no. 2, 253--299.
32.Non-divergence of
translates of certain algebraic measures. (With A. Eskin
and S. Mozes). Geom. Funct.
Anal. (GAFA), 7 (1997), no. 1,
48-80.
33.Limit distributions
of expanding translate of certain orbits on homogeneous spaces. Proc. Indian
Acad. Sci. Math. Sci., 106 (1996), no. 2,
105--125.
34.On the space of
ergodic invariant measures of unipotent flows. (With S. Mozes). Ergodic
Theory Dynam. Systems, 15
(1995), no. 1, 149--159.
35.Limit distributions of
polynomial trajectories on homogeneous spaces. Duke Math. J.,
75 (1994), no. 3,
711--732.
36.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.
37.Uniformly distributed
orbits of certain flows on homogeneous spaces. Math. Annalen, 289 (1991), no. 2,
315--334.
DISSERTATIONS:
38.Unipotent
flows on homogeneous spaces. Ph.D. Thesis, Tata Institute of
Fundamental Research, Mumbai, 1994. 69 Pages
39.Unipotent flows on
homogeneous spaces of SL(2,C). M.Sc. Thesis,
Tata Institute of Fundamental Research, Mumbai, 1992. 112 pages