Present Challenging Problems

Present CMC Fellow’s Challenging Problems

• A structural refinement of the Birch and Swinnerton-Dyer conjecture - Kim, Chan-Ho

  The aim of this research project is to understand the structure of Selmer groups in terms of modular symbols. This can be regarded as a structural refinement of the Birch and Swinnerton-Dyer conjecture.

• Classification of even unimodular lattices – Lee, Chul-hee

  The E8 lattice in dimension 8 and the Leech lattice in dimension 24 are remarkable examples of even unimodular lattices, which mysteriously show up in diverse areas of mathematics such as finite simple groups, sphere packing and string theory. Such lattices can occur only in dimensions divisible by 8, and their classification is known in dimension 8,16 and 24. The classification in higher dimension is an old unsolved problem, and is closely related to the classification of automorphic representations

• Schoen's Conjecture – Shin, Jinwoo

  We attempt to answer the following question posed by R. Schoen : “If we have a metric g∈M_1:={g \ ∫_M▒〖dV_g 〗=1} which realizes the Yamabe invariant σ(M), is g is an Einstein metric?”