The order of finite algebraic fundamental groups of surfaces with K^2<=3\chi-2
We study the structure of the algebraic fundamental group for minimal surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic fundamental group of S has order at most 5, and equality only occurs if S is a Godeaux surface. We also show that if K_S^2<= 3\chi-3 and the algebraic fundamental group of S is not trivial, then it is Z_2, or Z_2^2 or Z_3. Furthermore in this last case one has: 2<=\chi<=4, K^2=3\chi-3 and these possibilities do occur.
Gorenstein stable Godeaux surfaces
We classify Gorenstein stable numerical Godeaux surfaces with worse than canonical singularities and compute their fundamental groups.
Educating the Educators: Rethinking Subject Matter and Methods
gate? In addition to general requirements in their professional education such as psychological and social foundations and student teaching, what specifically do social studies teachers need to know and be able to do? What, in particular, do they need to study in the subject matters of the social sciences (defined here, for purposes of brevity, to include history and geography), and what should they learn about its effective direction to desired results, that is, method? Usually answers to the first question require reference to the liberal arts component of teacher education and the second to methods in the professional education component. As we shall see, this separation of subject matter and method in the education of social studies teachers, although conventional, poses problems.