An introduction to Smarandache multi-spaces and mathematical combinatorics

These Smarandache spaces are right theories for objectives by logic. However, the mathematical combinatorics is a combinatorial theory for branches in classical mathematics motivated by a combinatorial speculation. Both of them are unifying theories for sciences and contribute more and more to mathematics in the 21st century. In this paper, I introduce these two subjects and mainly concentrate on myself research works on mathematical combinatorics finished in past three years, such as those of map geometries, pseudo-manifolds of dimensional n, topological or differential structures on smoothly combinatorial manifolds. All of those materials have established the pseudo-manifold geometry and combinatorially Finsler geometry or Riemannian geometry. Other works for applications of Smarandache multi-spaces to algebra and theoretical physics are also partially included in this paper.