I will discuss what is known about homotopy and homeomorphism classifications of closed 4-manifolds. Then I will report on joint work with Hillman, Kasprowski, and Ray, in which we classify 4-manifolds whose fundamental group is that of a 3-manifold, up to homotopy equivalence. I will then specialise to the case of the infinite dihedral group, to obtain a homeomorphism classification. In...
We present a family of groups, introduced by Le Boudec, consisting of automorphisms of a regular tree that have almost prescribed local action (APLA) on the edges around the vertices.
In this talk, we prove a condition for the vanishing of their continuous bounded cohomology, which is a functional analytic version of group cohomology. Moreover, we show that when this condition is not...
The famous Burau representation of the braid group is known to be unfaithful for braids with at least five strands. In the early 2000s two constructions were provided to fix faithfulness: the first being the Lawrence-Krammer-Bigelow linear representation, hence proving linearity of braid groups, and the second being the Khovanov-Seidel categorical representation. In this talk, based on joint...
In this talk, we characterize normal $3$-pseudomanifolds $K$ with $g_2(K) \leq 4$. It is known that if a normal $3$-pseudomanifold $K$ with $g_2(K) \leq 4$ has no singular vertices, then it is a triangulated $3$-sphere. We first prove that a normal $3$-pseudomanifold $K$ with $g_2(K) \leq 4$ has at most two singular vertices. Subsequently, we show that if $K$ is not a triangulated $3$-sphere,...
The Hopf map is a continuous map from the $3$-sphere to the $2$-sphere, exhibiting a many-to-one relationship, where each unique point on the $2$-sphere originates from a distinct great circle on the $3$-sphere. This mapping is instrumental in generating the third homotopy group of the $2$-sphere. In this talk, I will present a minimal pseudo-triangulation of the Hopf map and establish its...
Let B_n be the braid group with n-strands and Z(B_n) its center. The (integral) homology of B_n was computed in the seventies by F. Cohen. In this talk we will see how to compute the homology of H_*(B_n/Z(B_n); F_p) for any n natural number and p prime. The approach will be topological, since the classifying space of B_n/Z(B_n) can be realized as the homotopy quotient C_n(R^2)//S^1, where...
The advent of machine and deep learning has driven major advances in computer vision and data analysis, enabling a shift from handcrafted features to automatic extraction of meaningful features through representation learning. At the same time, topological invariants provide computable shape descriptors well-suited for distinguishing complex structures, though when applied to real-world data,...
For a prime p, fusion systems over discrete p-toral groups are categories that model and generalise the p-local structure of Lie groups and certain other infinite groups in the same way that fusion systems over finite p-groups model and generalise the p-local structure of finite groups. In the finite case, it is natural to say that a fusion system F is realizable if it is isomorphic to the...
This talk is based on the preprint arXiv:2506.02999. Circle-valued functions provide a natural extension of real-valued functions, where instead of measuring values along a linear scale the values lie on a circle. This opens up new possibilities for analysing data in settings where the underlying structure is periodic or has a direction associated to it. There has been significant work on...
A transfer system on a poset P is a wide subcategory of P closed under pullbacks. Since the data of a model structure can be entirely determined by its classes of weak equivalences (W) and acyclic fibrations (AF) on a lattice, the model category information is given by the class W and a transfer system, AF contained in W. This talk focuses on how this transfer system changes when a model...
The independence complex of a graph is the simplicial complex with independent sets as simplices. This complex is one of the most studied graph complexes, but to determine its homotopy type is not an easy task even for highly symmetric graphs. In this talk we will focus in the independence complexes of graph products, we will talk about for which families of categorical, strong and...
A commonly adopted quote in TDA is that of Gunnar Carlsson: “Data has shape and shape has meaning”. But is that really the case? Starting from this provocative question, we will overview different application domains to test the validity of such an assumption and we exploit this trigger for setting up a debate and exchanging ideas.
Sandpile model, or chip firing, is a discrete dynamical system on (di)graphs with connections to combinatorics, algebra, and statistical physics, and which poses open questions related to certain fractal type patterns emerging from the dynamics. Configurations of this system are natural number valued vertex functions, also called chip configurations. Vertex whose value exceeds its (out-)degree...
In this talk, I will explore algebraic invariants that govern certain geometric properties of manifolds. The prototypical example is Hopf’s theorem, which states that a smooth manifold admits a non-vanishing vector field if and only if the Euler characteristic vanishes. To begin, I will define and motivate the projective span of a smooth manifold, which is the maximal number of pointwise...
GKM theory provides a powerful tool to describe the (equivariant) cohomology of certain T-varieties, where T is an algebraic torus. In particular, the cohomology of such varieties can be determined by studying the combinatorics of a graph—the moment graph—constructed from the geometry of the 0- and 1-dimensional T-orbits. In the first part of the talk, we will recall the main results of GKM...
The study of graph complexes, which are simplicial complexes associated with graphs, has led to deep con- nections between topology and combinatorics. Some well-known examples are neighborhood complexes, clique complexes, independence complexes, and matching complexes. Recently, a new family of graph complexes, called cut complexes, has been introduced. These complexes first appeared in the...
I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher...
Tverberg’s theorem states that given any (d+1)(r-1)+1 points in the d-dimensional Euclidean space, there are pairwise r subsets whose convex hulls have a point in common. This can be restated in terms of an affine map from a (d+1)(r-1)-simplex to the d-dimensional Euclidean space, and the topological Tverberg’s theorem generalizes it to a continuous map. I will further generalize it to a...
