String Theory and the Fuzzy Torus

We outline a brief description of non commutative geometry and present some applications in string theory. We use the fuzzy torus as our guiding example.Comment: Invited review for IJMPA rev1: an imprecision corrected and a reference adde

Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications

The moduli space of matroids

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.Comment: 83 page

LINEAR TO NON-LINEAR TOPOLOGY VIA γ-OPEN SETS IN THE ENVIRONMENT OF BITOPOLOGICAL SPACES

Generalizations of open sets always gives a linear structure in an ordinary topological space. This paper proposes that there exists a non-linear structure in a given bitopological space via $\gamma$-open sets of the context. The new structure is also studied in the light of hyperconnectedness to show that it is completely independent with the original one. Also, the relationships between extremally disconnectedness, connectedness and hyperconnectedness are presented in the same environment by means of $\gamma$-open set. Moreover, the idea of maximal $\gamma$-hyperconnectedness is initiated in this work and some important results related to filter, ultrafilter, door space are established. Finally, some functions concerned with $(1, 2) \gamma$-open sets are introduced and interrelationships among them are produced. Some suitable examples and counter examples are properly placed to make the paper self sufficient