(ii) (Lazard’s Theorem) L is the polynomial algebra Z[x1, x2, ...]. We proved Lazard’s theorem last semester. We also proved that MU∗ is isomorphic to Z[x1, x2, ...], so MU∗ and L are both polynomial algebras with generators in even degrees (positive even degrees in MU∗, and negative even degrees in L). The usual complex orientation on MU gives rise to a formal group law, and thus we have a hom...

For an orbifold X and α ∈ H(X,Z), we introduce the twisted cohomologyH c (X, α) and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups K α (X)⊗C and twisted cohomologyH c (X, α). This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an is...

A graph G is r-ramsey-minimal with respect to Kk if every rcolouring of the edges of G yields a monochromatic copy of Kk, but the same is not true for any proper subgraph of G. In this paper we show that for any integer k ≥ 3 and r ≥ 2, there exists a constant c > 1 such that for large enough n, there exist at least c 2 non-isomorphic graphs on at most n vertices, each of which is r-ramsey-mini...

The ultraproduct construction is generalized to $p$-ultramean constructions ($1\leqslant p<\infty$) by replacing ultrafilters with finitely additive measures. These correspond the linear fragments $\mathscr L^p$ of continuous logic. A powermean variant Keisler-Shelah isomorphism theorem proved for L^p$. It then that L^p$-sentences (and their approximations) are exactly those sentences logic whi...

An L{structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in L are internal. A nonstandard universe is said to satisfy the {isomorphism property if for any two internally presented L{structures A and B, where L has less than many symbols, A is elementarily equivalent to B implies that A is isomorphic to B. In this paper we prove t...

A problem which is appealing to the intuition in view of the relative frequency interpretation of probability is to define a measure on a countable space which assigns to each point the measure 0. Such a measure of course becomes trivial if it is countably additive. Finitely additive measures of this type have been discussed by R. C. Buck [l] and by E. F. Buck and R. C. Buck [2]. In a discussio...

Let G and K be groupoids. We present the notion of a (Gα,Kβ)-set we prove duality theorem in this context, which extends for graded algebras by groups. For A unital G-graded algebra X finite split G-set, show that there is an isomorphism between category X-graded left A-modules A#αGX-modules. As application isomorphism, construct Morita context.