A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs hav...

In this paper, we describe the spaces of stability conditions for the triangulated categories associated to three dimensional Calabi-Yau fibrations. We deal with two cases, the flat elliptic fibrations and smooth K3 (Abelian) fibrations. In the first case, we will see there exist chamber structures similar to those of the movable cone used in birational geometry. In the second case, we will com...

We consider the homotopy type of the space Mσ(Σ, Γ) of maps between symplectic surfaces (Σ, σΣ) and (Γ, σΓ) whose graphs form symplectic submanifolds of the product. We give a purely topological model for this space in terms of maps with constrained numbers of pre-images. We use this to show that the dependence of the homotopy type on the forms σΣ and σΓ is quantizedit changes only when the par...

It is proved that the generalized T-homotopy equivalences preserve the underlying homotopy type of a flow.

Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. This is applied to classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant η-equalities and consequently do not admit dependent eliminators. To recover η and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then ex...

We present an ansatz which enables us to construct heterotic/M-theory dual pairs in four dimensions. It is checked that this ansatz reproduces previous results and that the massless spectra of the proposed new dual pairs agree. The new dual pairs consist of M-theory compactifications on Joyce manifolds of G 2 holonomy and Calabi-Yau compactifications of heterotic strings. These results are furt...

There is an on-going connection between type theory and homotopy theory, based on the similarity between types and the notion of homotopy types for topological spaces. This idea has been made precise by exhibiting the category cSet of cubical sets as a model of homotopy type theory. It is natural to wonder, conversely, to what extend this model can be reflected in a type theory. The homotopy st...

We consider the homotopy type of maps between symplectic surfaces (Σ, σΣ) and (Γ, σΓ) whose graphs form symplectic submanifolds of the product. We give a purely topological model for this space in terms of maps with constrained numbers of pre-images. We use this to show that the dependence of the homotopy type on the forms σΣ and σΓ is quantizedit changes only when the parameters pass certain d...

Dependent type theory associates to any elements x, y of a type A an identity type x =A y, the type of proofs of equality of these elements. In PER Martin-Löf’s extensional type theory, the identity type is a subsingleton inhabited precisely when x and y are judgmentally equal. Semantically, types are thus sets equipped with equivalence relations given by their identity types. Groupoid Martin-L...