**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. Perhaps surprisingly, these questions can often be reduced to problems in algebraic and analytic number theory relating to things like orders in central simple algebras, the structure of affine buildings and Mahler measures of algebraic integers.

I am a member of the GEAR network.

**Publications (submitted)**:

- Systole inequalities for arithmetic locally symmetric spaces (with S. Lapan and J. Meyer)
- Areas of totally geodesic surfaces of hyperbolic 3-orbifolds (with D. B. McReynolds and N. Miller)
- The Fourier coefficients of Eisenstein series newforms (with L. Thompson)
- Fake quadrics (with M. Stover and J. Voight)

**Publications (published or accepted)**:

- Counting and effective rigidity in algebra and geometry (with D. B. McReynolds, P. Pollack and L. Thompson), to appear in Invent. Math.
- 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), to appear in Math. Res. Lett.
- 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, to appear in Transformation Groups.
- Parameterizing Shimura subvarieties of A
_{1}Shimura varieties and related geometric problems (with M. Stover), Archiv der Mathematik, 107(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), Mathematische Zeitschrift, vol. 281, no. 1 (2015), pp. 523-569 [Errata]
- 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.

**Lecture Notes:**

**Theses:**

- Selectivity in central simple algebras and isospectrality, Ph.D. thesis, Dartmouth College, 2012.
- An Exposition of the AKS Polynomial Time Primality Testing Algorithm, Master’s Thesis, University of Pennsylvania, 2006.