**Benjamin Linowitz,**Associate Professor, Oberlin College

**Research Interests:** I am interested in the geometry and topology of arithmetic hyperbolic manifolds and more generally, arithmetic locally symmetric spaces. In particular, I like to consider questions which arise in spectral geometry; that is, the extent to which one can "hear" the geometry of a manifold given knowledge of its Laplace eigenvalue spectrum or geodesic length spectrum.

**Undergraduate research:** If you are a student at Oberlin and are interested in conducting research in algebraic number theory or hyperbolic geometry, please let me know. Preferred prerequisites are number theory and rings and fields for algebraic number theory, and geometry (or topology) and group theory for hyperbolic geometry.

## Publications

- The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems (with E. Lauret), submitted.
- Systoles of arithmetic hyperbolic 2- and 3-manifolds (with R. Bozzai), to appear in Annales de la Faculte des Sciences de Toulouse.
- Constructing multi-cusped hyperbolic manifolds that are isospectral and not isometric to appear in Rocky Mountain Journal of Mathematics.
- Universal systole bounds for arithmetic locally symmetric spaces (with S. Lapan and J. Meyer), Proc. Am. Math. Soc. 150, No. 2, 795-807 (2022).
- Systole inequalities for arithmetic locally symmetric spaces (with S. Lapan and J. Meyer), to appear in Comm. Anal. Geom.
- Areas of totally geodesic surfaces of hyperbolic 3-orbifolds (with D. B. McReynolds and N. Miller), Pure Appl. Math. Q. 17 (2021), no. 1, 1–25.
- Fields of definition of arithmetic hyperbolic reflection groups, Oberwolfach Rep. 16 (2019), no. 2, 1064-1065.
- Commensurability classes of fake quadrics (with M. Stover and J. Voight), Selecta Math. (N.S.) 25 (2019), no. 3, Paper No. 48.
- The Fourier coefficients of Eisenstein series newforms (with L. Thompson), Automorphic Forms and Related Topics, 169-176, Contemp. Math., 732, Amer. Math. Soc., Providence, RI, 2019.
- Brauer equivalent number fields and the geometry of quaternionic Shimura varieties, Q. J. Math. 70 (2019), no. 2, 675-687. Errata
- Counting and effective rigidity in algebra and geometry (with D. B. McReynolds, P. Pollack and L. Thompson), Invent. Math. 213 (2018), no. 2, 697-758.
- Counting isospectral manifolds (with M. Belolipetsky), Counting isospectral manifolds. Adv. Math. 321 (2017), 69–79.
- Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds (with D. B. McReynolds, P. Pollack and L. Thompson), C. R. Math. Acad. Sci. Paris 355 (2017), no. 11, 1121-1126.
- Counting problems for geodesics on arithmetic hyperbolic surfaces, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1347-1361.
- Systoles of arithmetic hyperbolic surfaces and 3-manifolds (with D. B. McReynolds, P. Pollack and L. Thompson), Math. Res. Lett. 24 (2017), no. 5, 1497-1522.
- On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces (with J. Meyer), Geom. Dedicata 188 (2017), 165–169.
- Bounds for arithmetic hyperbolic reflection groups in dimension 2, Transform. Groups 23 (2018), no. 3, 743–753.
- Parameterizing Shimura subvarieties of A
_{1}Shimura varieties and related geometric problems (with M. Stover), Arch. Math. (Basel) 107 (2016), no. 3, 213–226. - Local Selectivity of Orders in Central Simple Algebras (with T. Shemanske), Int. J. Number Theory 13 (2017), no. 4, 853-884.
- Locally equivalent correspondences (with D. B. McReynolds and N. Miller), Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 451–482.
- The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces (with J. Meyer and P. Pollack), New York J. Math. 21 (2015) 955 - 972.
- Systolic surfaces of arithmetic hyperbolic 3-manifolds (with J. Meyer), In the Tradition of Ahlfors–Bers, VII, 215–223, Contemp. Math., 696, Amer. Math. Soc., Providence, RI, 2017.
- Selective orders in central simple algebras and isospectral families of arithmetic manifolds, Manuscripta Math. 147 (2015), no. 3, 399 - 413.
- A non-commutative analogue of the Odlyzko bounds and bounds on performance for space time lattice codes (with M. Satriano and R. Vehkalahti), IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 1971-1984, April 2015.
- Small isospectral and nonisometric orbifolds of dimension 2 and 3 (with J. Voight), Math. Z., vol. 281, no. 1 (2015), pp. 523-569.
- Families of mutually isospectral Riemannian orbifolds, Bull. London Math. Soc. (2015) 47 (1): 47-54.
- The sign changes of Fourier coefficients of Eisenstein series (with L. Thompson), Ramanujan J., 37 (2015), no. 2, 223-241.
- On fields of definition of arithmetic Kleinian reflection groups. II (with M. Belolipetsky), Int. Math. Res. Not. IMRN 2014, no. 9, 2559-2571.
- Characterizing adelic Hilbert modular cusp forms by coefficient size, Kyushu J. Math. 68 (2014) no. 1, 105-111.
- A newform theory for Hilbert Eisenstein series (with T. Atwill), Ramanujan J. 30 (2013), no. 2, 257-278.
- Decomposition theorems for Hilbert modular newforms , Funct. Approx. Comment. Math. Volume 47, Number 2 (2012), 157-172.
- Isospectral towers of Riemannian manifolds, New York J. Math. 18 (2012) 451 - 461.
- Embedding orders in central simple algebras (with T. Shemanske), J. Theor. Nombres Bordeaux 24 (2012), 405-424.
- Selectivity in quaternion algebras, J. Number Theory 132 (2012), pp. 1425-1437.
- Modular Forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations (with L. Fang, J. Hoffman, A. Rupinski and H. Verill), Exp. Math., Volume 19, Number 1 (2010), 1-27.