Chi Hin Chan (National Chiao Tung University) The Holder Regularity of Solutions to Hamilton-Jacobi Equations
The set Quo(A) of compatible quasiorders (reflexive and transitive relations) of an algebra A forms a lattice under inclusion, and the lattice Con(A) of congruences of A is a sublattice of Quo(A). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice M3. We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.
In this talk, I will discuss a handful of old and new applications of the geometry of involutions to the study of groups of small Morley rank. I will also present some early meditations on the role that generically defined geometries may play in analyzing simple groups of finite Morley rank with a small Prufer 2-rank.
This talk is dedicated to the proof of the Holder continuity regularization effects of Hamilton- Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent of the regularity of the Hamiltonian.