ABSTRACTS

• Tattwamasi Amrutam, IM PAN, On amenable subalgebras of the group von Neumann algebra.

Given a discrete group Γ, we will associate with it the von Neumann group algebra L(Γ). We will study the subalgebras of L(Γ) with respect to the Effros-Marechal topology. We shall also put an invariant probability measure on the collection of such subalgebras and discuss the amenability of such subalgebras. No knowledge of von Neumann algebras shall be assumed in this talk. This is a joint work with Yair Hartman and Hanna Oppelmayer.

• Fredric D. Ancel, University of Wisconsin - Milwaukee,  Rolfsen’s Conjecture and wild knots that pierce wild disks.

Rolfsen’s Conjecture (1974): Every knot (tamely or wildly embedded S1) in S3 is non-ambiently isotopic to an unknot. We will discuss the status of this unresolved conjecture. We will show that every knot in S3 that pierces a possibly wild disk is non-ambiently isotopic to an unknot.  Also, we will exhibit a wild knot in S3 that pierces a wild disk but pierces no tame disk.

• Hector Barge, Universidad Politecnica de Madrid,  Non-connected compacta and attractors of dynamical systems.

It is well known that if K = K1 ⊔ . . . ⊔ Kr ⊂ Rn, where Ki is compact for each i, is an attractor for a dynamical system (discrete or continuous) in

Rn then Ki is an attractor for each i. In this talk we shall study whether the converse statement holds: 

(S): If each Ki is an attractor for a dynamical system so is K.

We shall see that (S) holds true in the case of continuous dynamical systems. However, the discrete case is more involved. In particular, we will see that there exist compacta in R3 comprised of two connected components that can be realized as attractors separately but whose union cannot be so realized. Finally, we will introduce a sufficient condition for (S) to hold true in the discrete case and we shall see that this condition is also sufficient in some suitable situations.

These results have been obtained in collaboration with J.J. Sanchez-Gabites.

• Alexander Dranishnikov, University of Florida,   On symmetric product of surfaces.

We will discuss some topology and geometry questions about symmetric products of orientable surfaces SP^n(M_g). We show how topological results there can answer some geometry questions like the existence of Riemannian metric with different curvature conditions.

• Jerzy Dydak, University of Tennessee,    Boundaries for geodesic spaces.

For every proper geodesic space X we introduce its quasi-geometric boundary ∂_QG X with the following properties:

1. Every geodesic ray g in X converges to a point of the boundary  ∂_QG X and for every point p in   ∂_QG X  there is a geodesic ray in X converging to p,

2. The boundary ∂_QG X    is compact metric,

3. The boundary   ∂_QG X   is an invariant under quasi-isometric equivalences,

4. A quasi-isometric embedding induces a continuous map of quasi-geodesic boundaries, 

5. If X is Gromov hyperbolic, then ∂_QG X   is the Gromov boundary of X.

6. If X is a Croke-Kleiner space, then   ∂_QG X a point.

Joint work with my former PhD student, Hussain Rashed. 

• Craig Guilbault, University of Wisconsin - Milwaukee, On the theory of Z-boundaries of groups.

Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups are useful tools in geometric group theory and in geometric and algebraic topology. By defining the general notion of a Z-boundary, Bestvina extended  the applicability of group boundaries to a much broader class of groups. Variations on his original definition have been developed by Dranishnikov, Farrell, Lafont, and others. The key ingredient in all versions of the definition is that of a Z-compactification. One begins with a proper cocompact action of G on a “nice” contractible space X then looks to compactify X in such a way that the boundary captures as much information about X (and thereby G) as possible.  In this talk, we will explain why a Z-compactification of X is an ideal choice. We will analyze the definition(s) and describe the information captured by a Z-boundary and the extent to which it is well-defined for a given group G. Along the way, we will review new developments in this area—some very recent. Among the foundational concepts to be employed are the theory of ANRs and the theory of shapes, both due to Borsuk.

• Michael Levine, IM PAN, Ben Gurion University,  Equivariant maps to cubical shifts and mean dimension.

The mean dimension of a dynamical system is a topological invariant introduced by M. Gromov. We will discuss possible generalizations and limitations of the celebrated theorem of E. Lindensrauss saying that a minimal Z-action on a compact metric space with finite mean dimension is equivariantly embeddable into a cubical shift.

• Piotr W. Nowak, IM PAN,  Higher dimensional expanders and Gromov hyperbolicity.

Coboundary expanders are sequences of finite simplicial complexes, whose simplicial coboundary maps satisfy certain spectral gap type condition in a particular dimension. I will discuss the relationship between coboundary expanders and Gromov hyperbolicity. This is joint work with Dawid Kielak.

• Mark Pengitore, IM PAN,   Growth Functions and linearity of automorphism groups of hyperbolic groups.

This talk will introduce various growth functions associated to a finitely generated group which measure the difficulty of separating an element from the identity using epimorphisms to a finite products of nonabelian finite simple groups of Lie type with characteristic kernels as a function of the word length. Using these functions, we provide a characterization of when the automorphism group of a hyperbolic group is linear. As an application, we investigate the question of linearity of the mapping class group.

• Jose M.R. Sanjurjo, Universidad Complutense de Madrid (in collaboration with Hector Barge 

and Jaime Sanchez-Gabites),  Topology and dynamics of isolated invariant compacta.

This talk examines the topological and dynamical structure of attractors and other isolated invariant sets in flows, with a focus on non-saddle sets. These sets are characterized by their asymptotic behavior and the absence of certain unstable dynamics. Using notions like strong influence and isolating blocks, the presentation links dynamical properties to topological shape, often revealing polyhedral or simple homotopy types. The role of dissonant points is discussed in shaping the complexity of the region of influence. On manifolds such as the torus or Euclidean spaces, non-saddle sets exhibit especially rich and structured behavior. The concepts of topological and dynamical robustness are shown to coincide under certain conditions. The presentation also addresses bifurcations, Morse decompositions, and classification results for surfaces based on the dynamics of global non-saddle sets.