Speaker: Zhao Peng, Princeton University Title: Quantum Variance of the Modular Surface Venue: B1044 Time: 14:00 – 15:20, 27 June Abstract: We discuss the quantum variance, which is introduced by Zelditch and describes the fluctuations of a quantum observable, on the phase space of modular surface. We asymptotically evaluate the quantum variance and show that it is equal to the classical variance of the geodesic flow on the phase space after inserting the correction factor of certain L-function's central value on each irreducible subspace. It is also very close to the arithmetic variance associated to closed geodesics on the modular surface studied by Luo, Rudnick and Sarnak. This talk is based on a joint work with Peter Sarnak. Speaker: Li Han, Yale University Title: Oppenheim Conjecture and Markov Spectrum Venue: B1044 Time: 15:30 – 17:00, 27 June Abstract: In the mid 1980's, Margulis completed the proof of the Oppenheim conjecture: If $Q$ is a non-degenerate, indefinite real quadratic form in n-variables (n>2), and is not proportional to a quadratic form of rational coefficients. Then for every (sufficiently small) $a>0$ there is a non-zero integral vector v in $R^n$, such that $|Q(v)| We will show that “every bounded SO(2,1)-orbit in SL(3, R)/ SL(3, Z) is closed” implies the Oppenheim conjecture. We will also discuss a recent progress on the counting problem of the 3-dimensional Markov spectrum, and this is a joint work with Prof. Margulis. .