Abstract: Theories of shadowing and specification, originating with the works of Anosov and Bowen have been developing parallel with the theory of hyperbolic systems. In some crude sense, one may say that these notions are similar. The common goal is to find a true trajectory near an approximate one, but they differ in understanding what constitutes the approximate trajectory. In shadowing one traces a pseudo-orbit, while in specification arbitrarily assembled finite pieces of orbits are supposed to be followed by a true orbit.

Abstract: This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connected coverings and subgroups of the fundamental group. Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire's algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illustrate by numerous examples. As an application we obtain a simple formula for the second (co)homology group of a quandle Q. It has long been known that H_1(Q) = H^1(Q) = \Z[\pi_0(Q)], and we construct natural isomorphisms H_2(Q) = \pi_1(Q,q)_{ab} and H^2(Q,A) = Ext(Q,A) = Hom(\pi_1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever the fundamental group is known, (co)homology calculations in degree 2 become very easy.

Abstract: We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).

Abstract: We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.

Abstract: We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.

Abstract: We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group. 1. Introduction. A dimer is an edge in a bipartite graph, and a dimer covering is a perfect matching for that graph. The study of dimer coverings started in the 1960's with the work of Kasteleyn (Kas63) and Temperley{ Fisher (TF61) who used it as a tool for studying statistical physics. Kasteleyn showed that the partition function on weighted bipartite planar graphs can be expressed as the determinant of a suitable matrix. The last ten years have seen a resurgence of the study of dimers and the application of this theory to many other areas of mathematics. Our interest is in exploring the opposite direction. We have a given ma- trix, and we want to

Abstract: Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computacion; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria; Argentina

Abstract: A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that in $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set.

Abstract: The Rothberger number b(I) of a definable ideal I on w is the least cardinal such that there exists a Rothberger gap of type (w, k) in the quotient algebra P(w)/I. We investigate b(I) for a class of F-sigma ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is N-1, while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.

Abstract: We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on ω2 using finite conditions. In previous work [3] we introduced the idea of an adequate set of models and showed how to use adequate sets as side conditions in forcing with finite conditions. We gave several examples of forcing with adequate sets, including forcing posets for adding a generic function on ω2, adding a nonreflecting stationary subset of ω2, adding a Kurepa tree on ω1, and in [4] adding a club to a fat stationary subset of ω2. The main result of the present paper is to define a forcing poset using adequate sets which adds a ω1-sequence. The idea of using models as side conditions in forcing goes back to Todorcevic [6], where the method was applied to add generic objects of size ω1 with finite approximations. In the original context of applications of PFA, the preservation of ω2 was not necessary. To preserve ω2, Todorcevic introduced the requirement of a system of isomorphisms on the models in a condition. In the present paper we introduce the idea of a coherent adequate set of models. A coherent adequate set is essentially an adequate set in the sense of [3] which also satisfies the existence of a system of isomorphisms in the sense of Todorcevic. Combining these two ideas turns out to provide a powerful method for forcing with side conditions. As an application we define a forcing poset which adds a square sequence on ω2 with finite conditions. We assume that the reader is familiar with the basic theory of adequate sets as described in Sections 1–3 of [3]. Our treatment of coherent adequate sets owes a lot to the presentation of nicely arranged families given by Abraham and Cummings [1]. Forcing a square sequence with finite conditions was first achieved by Dolinar and Dzamonja [2] using the Mitchell style of models as side conditions [5]. An important difference is that the clubs which appear in the square sequence added by their forcing poset belong to the ground model, whereas for us the clubs are themselves generically approximated by finite fragments.

Abstract: Given a projection $f$ of a product of real analytic manifolds onto one factor, let us say, $S$, and a subanalytic sheaf $\mathcal{F}$ on the associated subanalytic site, we give a natural construction of the (subanalytic) relative sheaf $\mathcal{F}^S$. Applying our construction to the subanalytic sheaves of tempered distributions, holomorphic functions and Whitney $\mathcal{C}^{\infty}$-functions we obtain their relative versions and study their properties.

Abstract: On the one hand, the ideals of a well quasi-order (wqo) naturally form a compact topological space into which the wqo embeds. On the other hand, Nash-Williams’ barriers are given a uniform structure by embedding them into the Cantor space. We prove that every map from a barrier into a wqo restricts on a barrier to a uniformly continuous map, and therefore extends to a continuous map from a countable closed subset of the Cantor space into the space of ideals of the wqo. We then prove that, by shrinking further, any such continuous map admits a canonical form with regard to the points whose image is not isolated. As a consequence, we obtain a simple proof of a result on better quasi-orders (bqo); namely, a wqo whose set of non principal ideals is bqo is actually bqo.

Abstract: We show that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod ym property does not nec- essarily contain a bilipschitz image of a thick family of geodesics. This is done by showing that any thick family of geodesics is not Markov convex, and comparing this result with results of Cheeger-Kleiner, Lee-Naor, and Li. The result contrasts with the earlier result of the author that any Banach space without the Radon-Nikod ym property contains a bilipschitz image of a thick family of geodesics.

Abstract: We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on X (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of X, and we show that the resulting hierarchies of degrees are much more complicated than the classical Wadge hierarchy; in particular, they always contain large infinite antichains, and in most cases also infinite descending chains.

Abstract: In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hern\'andez-Guti\'errez and Hru\v{s}\'ak. The method of the proof also allows us to obtain a metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.

Abstract: The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open subgroup of a free Markov topological group is a free Markov topological group if and only if it is disconnected.

Abstract: Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but that there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on H_g which is quadratic in g. For the proofs we develop a calculus for finite group-actions on H_g, by codifying such actions by handle-orbifolds and finite graphs of finite groups.

Abstract: The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary. As a corollary, we show that the homotopy fiber is weakly equivalent to a space on which the framed little disks operad acts possibly nontrivially, and hence its rational homology is a (higher) BV-algebra in a stable range of dimensions.

Abstract: We characterize which automorphisms of an arbitrary complete bipartite graph Kn;m can be induced by a homeomorphism of some embedding of the graph in S 3 . 1. Introduction. Knowing the symmetries of a molecule helps to pre- dict its chemical behavior. Chemists use the point group as a way to rep- resent the rigid symmetries of a molecule. However, molecules which are large enough to be exible (such as long polymers) or have pieces which

Abstract: In this note we prove that every finitely presented subgroup of a systolic group is itself systolic.

Abstract: The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this paper that the one-term homology of a finite spindle can have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, we show that if a shelf satisfies a certain, rather general, condition then the one-term homology is trivial.

Abstract: It is convenient to express the conclusion of Silver’s theorem in terms of the continuous reducibility of equivalence relations. Let id denote the equivalence relation of equality on Cantor space 2ω. If E,F are equivalence relations on Polish spaces then we say that E is continuously reducible to F (written E ≤c F ) iff there is a continuous function f such that E(x, y) iff F (f(x), f(y)). Then Silver’s theorem says that if E is a Borel equivalence relation on the reals with uncountably many classes then id is continuously reducible to E. A more generous notion is Borel reducibility, where the “reduction” f is allowed to be Borel (we then write E ≤B F ).

Abstract: We prove a game theoretic dichotomy for Gδσ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis’ proof of Σ3 determinacy.

Abstract: We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ?-sequence or a non-trivial converging ?1-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ?1-sequences is first-countable and, in addition, has many ?1-sized Lindelof subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.