Geometry of the Trace Formula

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The goal of this proposal is to pursue the study of geometry of the trace formula, which commenced with PI's work on the fundamental lemma. A particular emphasis will be put on the limiting form of the trace formula carried out in a previous his joint work with Frenkel and Langlands. The PI also intends to work on related problems in number theory and topology including: Hausel's conjecture of the topology of Hitchin system, Deligne-Flicker's conjecture on the number of l-adic local systems and the moduli space related to Bhargava-Shankar?s work on the average rank of rational elliptic curve. These problem, in addition to their intrinsic interest, should provide insight PI's main project on the geometry of the trace formula.With algebraic geometry of moduli space and automorphic representation as basic area of the proposed activity, it should have impact on related mathematical domain as number theory and mathematical physics. The PI will train young mathematician and give them way to some of the most exciting problems in current mathematical research. He will give popular talks on mathematics to school children in order to help them acquiring taste for mathematics and fundamental sciences.