Can one hear the shape of a drum (isospectral domains)
Probabilistic representation of heat kernels
Heat content
Wave equation methods
Shape optimization of spectral invariants (where to place an obstacle so as to maximize a given spectral invariant)
Nodal sets
Spectral computation

List of Publications

Spectral Theory

[S1] (with Quan Zheng, Xiaohua Yao, Dashan Fan) The spectrum of differential operators in H^p spaces, Illinois J. Math. 49 (2005) 45–62.

[S2] (with Lan Tang) Some upper bounds for sums of eigenvalues of the Neumann Laplacian, Proc. Amer. Math. Soc. 134 (2006) 3301–3307.

[S3] On the second eigenvalue of the Laplacian in an annulus, Illinois J. Math. 51 (2007) 913–925.

[S4] Nonexistence of local minima of supersolutions for the polyharmonic problems, Math. Nachr. 281 (2008) 710–714.

[S5] (with Lan Tang) On a new form of the ergodic theorem for the unit sphere with application to spectral theory, Math. Nachr. 281 (2008) 715–720.

[S6] (with Alexander Strohmaier) The local counting function of operators of Dirac and Laplace type, J. Geom. Phys. 104 (2016) 204–228.

[S7] (with Alexander Strohmaier) Heat kernel estimates for general boundary problems, J. Spectral Theory 6 (2016) 903–919.

Additive Combinatorics

[A1] Collinear triples in permutations, Innov. Incidence Geom. 8 (2008) 171–173.

[A2] (with Jian Shen) A sum-division estimate of reals, Proc. Amer. Math. Soc. 138 (2010) 101–104.

[A3] Slightly improved sum-product estimates in fields of prime order, Acta Arith. 147 (2011) 153–160.

[A4] (with Oliver Roche-Newton) An improved sum-product estimate for general finite fields, SIAM J. Discrete Math. 25 (2011) 1285–1296.

[A5] (with Oliver Roche-Newton) Convexity and a sum-product type estimate, Acta Arith. 156 (2012) 247–255.

[A6] (with Derrick Hart, Chun-Yen Shen) Fourier analysis and expanding phenomena in finite fields, Proc. Amer. Math. Soc. 141 (2013) 461–473.

Metric number theory

[M1] Zero-one laws in simultaneous and multiplicative Diophantine approximation, Mathematika 59 (2013) 321–332.

[M2] A note on the Duffin-Schaeffer conjecture, Uniform Distribution Theory 8 (2013) 151–156.

[M3] The Duffin-Schaeffer-type conjectures in various local fields, Mathematika 62 (2016) 753–800.

Probabilistic inequalities

[P1] (with Chunrong Feng, Jian Shen) On the Borel-Cantelli lemma and its generalization, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1313–1316.

[P2] (with Chunrong Feng, Jian Shen) Some inequalities in functional analysis, combinatorics, and probability theory, Elec. J. Combinatorics 17 (2010) R58.

[P3] (with Chunrong Feng) On the Mori-Szekely conjectures for the Borel-Cantelli lemma, Studia Scientiarum Mathematicarum Hungarica 50 (2013) 280–285.

Classical Analysis