Name: Combinatorial Algebraic Topology & Applications III
Start: 2025-09-16T08:40:00+02:00
End: 2025-09-19T17:10:00+02:00
Location: Palazzo del Castelletto

Combinatorial Algebraic Topology & Applications III

from Tuesday, 16 September 2025 (08:40) to Friday, 19 September 2025 (17:10)

Monday, 15 September 2025
Tuesday, 16 September 2025

08:45 Registration
Registration
08:45 - 09:30
Room: Aula Dini
09:30 Classification of 4-manifolds with infinite dihedral fundamental group - Mark Powell (University of Glasgow)
Classification of 4-manifolds with infinite dihedral fundamental group
- Mark Powell (University of Glasgow)
09:30 - 10:30
Room: Aula Dini 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 particular I will explain what invariants one needs to compute to deduce that two such 4-manifolds coincide, with explicit examples.
10:30 Coffee break
Coffee break
10:30 - 11:00
Room: Aula Dini
11:00 Groups acting on trees with APLA and their bounded cohomology - Elena Bogliolo (Università di Pisa)
Groups acting on trees with APLA and their bounded cohomology
- Elena Bogliolo (Università di Pisa)
11:00 - 12:00
Room: Aula Dini 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 satisfied, the continuous bounded cohomology in degree two is infinite-dimensional.
12:00 Lunch break
Lunch break
12:00 - 14:30
Room: Aula Dini
14:30 On faithful representations of the braid group - Anne-Laure Thiel (Université de Bourgogne)
On faithful representations of the braid group
- Anne-Laure Thiel (Université de Bourgogne)
14:30 - 15:30
Room: Aula Dini 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 work in progress with Licata, Queffelec and Wagner, I will investigate the interplay between these two representations.
15:30 Coffee break
Coffee break
15:30 - 16:00
Room: Aula Dini
16:00
16:00 - 18:00
Room: Aula Dini
Contributions

16:00 Characterizations of Normal $3$-Pseudomanifolds with Small $g_2$ - Biplab Basak (Indian Institute of Technology Delhi)

16:30 Minimal Pseudo-triangulation of the Hopf Map and Its Uniqueness. - Ayushi Trivedi (Indian Institute of Technology Delhi)

17:00 TBA - Julian Brüggemann (Dioscuri Dentre in TDA/IMPAN)

17:30 On the homology on the braid group modulo its center - Tommaso Rossi (Università di Bologna)
Wednesday, 17 September 2025
09:30 Tverberg’s theorem for cell complexes - Daisuke Kishimoto (Kyushu University)
Tverberg’s theorem for cell complexes
- Daisuke Kishimoto (Kyushu University)
09:30 - 10:30
Room: Aula Dini 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 continuous map out of a certain CW complex, e.g. a simplicial ((d+1)(r-1)-1)-sphere. This is joint work with Sho Hasui. Masahiro Takeda, and Mitsunobu Tsutaya.
10:30 Coffee break
Coffee break
10:30 - 11:00
Room: Aula Dini
11:00 Realisability of fusion systems by discrete groups - Ran Levi (University of Aberdeen)
Realisability of fusion systems by discrete groups
- Ran Levi (University of Aberdeen)
11:00 - 12:00
Room: Aula Dini 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 fusion system of a finite group, but it is less clear what realizability should mean in the discrete p-toral case. In this talk I will discuss recent joint work with Carles Broto and Bob Oliver. We look at some of the different types of realizability for fusion systems over discrete p-toral groups, including realisability by linear torsion groups and sequential realisability, of which the latter is the most general. We show in particular that fusion systems of compact Lie groups are always realised by linear torsion groups (hence sequentially realisable). We also present two large families of examples of fusion systems that are not sequentially realisable. I will proceed by comparing four different types of realisability for saturated fusion systems over discrete p-toral groups. For example, when G is a locally finite group all of whose p-subgroups are artinian (hence discrete p-toral), we show that it has “weakly Sylow” p- subgroups and give explicit constructions of saturated fusion systems and associated linking systems associated to G. We also show that a fusion system over a discrete p-toral group S is saturated if its set of morphisms is closed under a certain topology and the finite subgroups of S satisfy the saturation axioms, and present a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.
12:00 Lunch break
Lunch break
12:00 - 14:00
Room: Aula Dini
14:00
14:00 - 15:30
Room: Aula Dini
Contributions

14:00 An isometry theorem for persistent homology of circle-valued functions - Mariam Pirashvili (University of Plymouth)

14:30 Bousfield localisation on posets - Shuchita Goyal (BITS Pilani, India)

15:00 Homotopy type of independence complexes of graph products - Andrés Carnero Bravo (Centro de Ciencias Matemáticas, UNAM)
15:30 Coffee break
Coffee break
15:30 - 16:00
Room: Aula Dini

16:00

16:00 - 18:00
Room: Aula Dini
Thursday, 18 September 2025
09:30 Topology vs. Learning: A Preliminary Comparative Study - Ulderico Fugacci (CNR - Genova)
Topology vs. Learning: A Preliminary Comparative Study
- Ulderico Fugacci (CNR - Genova)
09:30 - 10:30
Room: Aula Dini Adopting the perspective of novice users, we aim at assessing which class of methods may be more advantageous among learning techniques and topological data analysis (TDA). In particular, we compare persistent homology-based approaches with traditional machine learning and deep learning techniques in the context of label-efficient classification. We evaluate simple topological methods - such as persistence thresholding and Bottleneck distance classification - alongside conventional learning algorithms and hybrid strategies on two binary classification tasks: surface crack detection and malaria cell identification.
10:30 Coffee break
Coffee break
10:30 - 11:00
Room: Aula Dini
11:00 Avalanche complex and topology of the sandpile dynamics on digraphs - Henri Riihimäki (Nordita / Stockholm University)
Avalanche complex and topology of the sandpile dynamics on digraphs
- Henri Riihimäki (Nordita / Stockholm University)
11:00 - 12:00
Room: Aula Dini 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 is unstable and fires by sending one chip to each of its (out-)neighbours, thus taking the system to a new configuration. A variant of the vertex firing is the avalanche, in which all unstable vertices in a chip configuration fire simultaneously. To study sandpile dynamics topologically, we introduce a simplicial complex, called the avalanche complex, generated by the avalanching sets of vertices. Our results concern the topology of the avalanche complex for certain classes of graphs, particularly cycles, and decomposition of the configuration space into domains by Betti numbers, a kind of hyperplane arrangement. This is a joint ongoing work with Jason Smith.
12:00 Lunch break
Lunch break
12:00 - 14:30
Room: Aula Dini
14:30 Real and complex line fields on manifolds - Baylee Schutte (University of Aberdeen)
Real and complex line fields on manifolds
- Baylee Schutte (University of Aberdeen)
14:30 - 15:30
Room: Aula Dini 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 linearly independent line fields. The determination of the projective span of a given smooth manifold (or family of manifolds) is referred to as the line field problem. Along the way, I will mention joint work with Mark Grant [Bol. Soc. Mat. Mex. 30(3):75, 2024] in which we solve the line field problem for all of the Wall manifolds. Chiefly, from work joint with Nikola Sadovek [arXiv:2411.14161], I will identify complete obstructions to the existence of 1, 2, or 3 linearly independent line fields on certain classes of almost-complex manifolds, thereby improving the tractability of the solving the line field problem for such manifolds. Finally, I will comment on an application of these results to complex geometry.
15:30 Coffee break
Coffee break
15:30 - 16:00
Room: Aula Dini
16:00 GKM Theory and Applications to the Study of Quiver Grassmannians - Sabino Di Trani (Sapienza Università di Roma))
GKM Theory and Applications to the Study of Quiver Grassmannians
- Sabino Di Trani (Sapienza Università di Roma))
16:00 - 17:00
Room: Aula Dini 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 theory through a number of examples. Afterwards, we will show how the combinatorics of the moment graph can be applied to investigate geometric properties of certain quiver Grassmannians for equioriented quivers of type A.
17:00 3
3
17:00 - 18:00
Room: Aula Dini
Contributions

17:00 Shellability of cut complexes of powered cycle graphs - Pratiksha Chauhan (Indian Institute of Technology Mandi)

17:30 TBA - Giulia Maria Menara (Università degli Studi di Milano-Bicocca)
20:00 Social dinner
Social dinner
20:00 - 22:00
Friday, 19 September 2025
09:30 Diffeomorphisms of reducible 3-manifolds - Rachael Boyd (University of Glasgow)
Diffeomorphisms of reducible 3-manifolds
- Rachael Boyd (University of Glasgow)
09:30 - 10:30
Room: Aula Dini 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 and McCullough. The theory we develop to prove this theorem has other applications, and I’ll provide an overview of these.
10:30 Coffee break
Coffee break
10:30 - 11:00
Room: Aula Dini
11:00 Topological machine learning and its applications to Raman spectroscopy. - Maria Antonietta Pascali (CNR – Pisa)
Topological machine learning and its applications to Raman spectroscopy.
- Maria Antonietta Pascali (CNR – Pisa)
11:00 - 12:00
Room: Aula Dini 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, these descriptors might seem too rigid. Persistent homology (PH), overcomes this limitation by enabling intrinsically multiscale analysis. In joint work with Davide Moroni e and Francesco Conti, we combined PH and machine learning to develop a Topological Machine Learning (TML) pipeline, which has demonstrated promising results in challenging classification tasks. This talk will describe the synergy between persistent homology and machine learning, and survey recent trends in TML. We will present applications to Raman spectroscopy data, with a focus on their potential impact in the medical domain.
12:00 Greetings
Greetings
12:00 - 12:30
Room: Aula Dini