Description-Periods and restrictions of eigenfunctions on locally symmetric spaces: A brief introduction to Hecke theory for SL(2,Z), and describe the method of arithmetic amplification, which uses Hecke operators to give improved bounds for periods and L^p restrictions.

Description-Harish-Chandra’s tempered representations and geometry:
In the series of lecture, I plan to explain some recent topics on quantitative estimate on proper actions with emphasis on their relation to representation theory. I begin with some geometric problems of group actions including properness criterion for reductive homogeneous spaces. In turn, I introduce a ''quantification” of proper actions and bring geometric ideas to analytic representation theory such as Harish-Chandra’s temperedness criterion. Basic notion will be illustrated by examples.

Description-Representation theory, Harmonic analysis, and intrinsic Diophantine approximation on homogeneous varieties. The spectral approach to Diophantine approximation averaging operators in dynamical systems, unitary representations, spectral estimates, and operator norm bounds. A brief introduction to some aspects of the representation theory of semisimple algebraic groups and the integrability of their matrix coefficients. The spectral transfer principle and effective ergodic theorems. The automorphic representation associated with a lattice subgroup, and effective counting of lattice points. Best possible spectral estimates for subgroup actions. Lattice actions on homogeneous spaces and fast equidistribution of dense lattice orbits. Intrinsic Diophantine approximation on homogeneous algebraic varieties, and best possible exponent for tempered subgroups. The effective duality principle on homogeneous spaces.

Description-Plancherel theory for real spherical spaces: Harish-Chandra's Plancherel formula for real reductive groups G. Harish-Chandra's Plancherel formula up to the formal degrees of the discrete series as well as the theorem of Delorme and van den Ban-Schlichtkrull for symmetric spaces.

Description-The Fourier transform on semisimple Lie Algebras and Lie groups. Ramifications and Applications.