Union Theorem for Cohomological Dimension: A Simple Counterexample

Cohomological Dimension of Markov Compacta

We rephrase Gromov’s definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum X, dimZ(p) X = dimQ X for all but finitely many primes p where Z(p) is the localization of Z at p. We construct Markov compacta of arbitrarily large dimension having dimQ X = 1 as well ...

Cohomological Approach to Asymptotic Dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdimZ X of metric spaces. We show that it agrees with the asymptotic dimension asdimX when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdimX. In pa...

A counterexample to the a-‘theorem’

We exhibit a renormalization group flow for a four-dimensional gauge theory along which the conformal central charge a increases. The flow connects the maximally superconformal point of an N = 2 gauge theory with gauge group SU(N+1) and Nf = 2N flavors in the ultraviolet, to a strongly-coupled superconformal point of the SU(N) gauge theory with Nf = 2N massless flavors in the infrared. Our exam...

A Counterexample to a Theorem of Chase

Birkhoff's Theorem from a geometric perspective: A simple example

‎From Hilbert's theorem of zeroes‎, ‎and from Noether's ideal theory‎, ‎Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original examples of algebraic geometry‎. ‎I will describe a simple example that illustrates some of the aspects of this relationship‎. The dualization from algebra to geometr...