Abstract Let XY L , T consist of all countable -structures M that satisfy the axioms and in which homomorphisms type X (these could be plain homomorphisms, monomorphisms, or isomorphisms) between finite substructures are restrictions an endomorphism Y (for example, automorphism a surjective endomorphism). Lockett Truss introduced 18 such classes for relational structures. For given pair however...

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, series of distinguished classes in symplectic cohomology complement $X = M \backslash \mathbf{D}$. Under further "topological" pair, these can organized into Log(arithmic) PSS morph...

Fix a morphism of schemes f : X → S which is flat, proper, and “fiber-by-fiber semi-stable”. Let IV LS be the functor on the category of log schemes over S which to any T associates the isomorphism classes of pairs (MX , f), where MX is a log structure on X×S T and f : pr∗ 2MT →MX is a morphism of log structures making (X×S T ,MX)→ T a log smooth, integral, and vertical morphism. The main resul...

We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N . Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the alge...

The Dual Post Correspondence Problem asks, for a given word α, if there exists a non-periodic morphism g and an arbitrary morphism h such that g(α) = h(α). Thus α satisfies the Dual PCP if and only if it belongs to a non-trivial equality set. Words which do not satisfy the Dual PCP are called periodicity forcing, and are important to the study of word equations, equality sets and ambiguity of m...

Given a complete nonsingular algebraic variety X and a divisor D with normal crossings, we say that X is log homogeneous with boundary D if the logarithmic tangent bundle T X (− log D) is generated by its global sections. We then show that the Albanese morphism α is a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of D are n...

We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is always flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. From de Jong’s theorem, we see that for every variety over an...

Given a complete nonsingular algebraic variety X and a divisor D with normal crossings, we say that X is log homogeneous with boundary D if the logarithmic tangent bundle TX(− log D) is generated by its global sections. Then the Albanese morphism α turns out to be a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of D are non...

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow motives over S, and we show how to associate a motive to any S-variety. We give a geometric proof of relative quantifier elimination for pseudo-finite fields, ...

Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding efficiently, for a word w and a literal anti-/morphism f , whether w contains an instance of a given pattern involving a variab...