Combinatorial Algebraic Topology & Applications III

16 Sept 2025, 08:40 → 19 Sept 2025, 17:10 Europe/Rome

Aula Dini (Palazzo del Castelletto)

Luigi Caputi (Università di Bologna), Carlo Collari (Università di Pisa), Celeste Damiani (Camelot Biomedical Systems), Giovanni Framba (Università di Pisa)

This is the third edition of the workshop “Combinatorial algebraic topology and applications” previously held in Pisa at Centro di Ricerca Matematica “E. De Giorgi”. The aim of this workshop series is to bring together researchers with common interest in combinatorial and algebraic topology, and in their applications, such as topological data analysis and applied topology.

 

This year, the conference will focus on the following main topics: (applied) algebraic and combinatorial topology, and low dimensional/geometric topology.

The talks will cover both theoretical and applied aspects of these subjects. To foster discussions and connections, the workshop will include contributed talks and a free discussion session. Participants are welcome to bring questions, ideas, or problems to share or explore with others.

 

Registration is free, although mandatory, due to restrictions on the number of participants. At the moment, the maximum number of participants has been reached. So the requests from now on will be placed on a reserve list. The deadline for registration is the 30th ofJune 2025. Registration will be confirmed after expiration.

 

LIST OF CONFIRMED SPEAKERS

Elena Bogliolo (Università di Pisa)

Rachael Boyd (University of Glasgow)

Sabino Di Trani (Sapienza Università di Roma)

Ulderico Fugacci (CNR - Genova)

Daisuke Kishimoto (Kyushu University)

Maria Antonietta Pascali (CNR - Pisa)

Mark Powell (University of Glasgow)

Henri Riihimäki (Nordita / Stockholm University)

Baylee Schutte (University of Aberdeen)

Anne-Laure Thiel (Université de Bourgogne)

Special Session: Ran Levi (University of Aberdeen)

Interested participants can apply for a contributed talk.

Limited funding might be available to early registered participants, priority will be given to junior participants and contributed speakers.

The deadline to apply for funding is the 15th JUNE 2025.

 

More information on the application procedure at the REGISTRATION PAGE.

 

Funded by:

FINANZIAMENTO MUR DIPARTIMENTI DI ECCELLENZA 2023-2027 - ATTIVITA' DI ELEVATA QUALIFICAZIONE - CUP I57G22000700001

 

PRIN: PROGETTI DI RICERCA DI RILEVANTE INTERESSE NAZIONALE – Bando 2022 Prot. 2022NMPLT8 "Geometry and topology of manifolds"

LOGHI ppt.jpg

Tuesday, 16 September

08:45 → 09:30 Registration

09:30 → 10:30 Classification of 4-manifolds with infinite dihedral fundamental group

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.

Speaker: Prof. Mark Powell (University of Glasgow)

10:30 → 11:00 Coffee break

11:00 → 12:00 Groups acting on trees with APLA and their bounded cohomology

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.

Speaker: Elena Bogliolo (Università di Pisa)

12:00 → 14:30 Lunch break

14:30 → 15:30 On faithful representations of the braid group

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.

Speaker: Prof. Anne-Laure Thiel (Université de Bourgogne)

15:30 → 16:00 Coffee break

16:00 → 18:00 Contributed Talks

Conveners: Biplab Basak (Indian Institute of Technology Delhi), Ayushi Trivedi (Indian Institute of Technology Delhi), Julian Brüggemann (Ayushi), Tommaso Rossi (Università di Bologna)

16:00 Characterizations of Normal $3$-Pseudomanifolds with Small $g_2$

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, it can be obtained from certain boundary complexes of $4$-simplices by a sequence of operations, including connected sums, edge expansions, and edge folding. Furthermore, we establish that such a $3$-pseudomanifold $K$ is a triangulation of the suspension of $\mathbb{RP}^2$. Additionally, by building upon the results of Walkup, we provide a reframed characterization of normal $3$-pseudomanifolds with no singular vertices for $g_2(K) \leq 9$

Speaker: Biplab Basak (Indian Institute of Technology Delhi)

16:30 Minimal Pseudo-triangulation of the Hopf Map and Its Uniqueness.

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 uniqueness. Additionally, I will show that the pseudo-triangulation corresponding to the $3$-sphere is susceptible to a $4$-coloring.

Speaker: Ayushi Trivedi (Indian Institute of Technology Delhi)

17:00 TBA

Speaker: Julian Brüggemann (Dioscuri Dentre in TDA/IMPAN)

17:30 On the homology on the braid group modulo its center

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 C_n(R^2) is the unordered configuration space of point in the plane. Combining the results of F. Cohen with techniques from equivariant cohomology we can do the computation. This talk is based on https://arxiv.org/abs/2404.10639.

Speaker: Tommaso Rossi (Università di Bologna)

Wednesday, 17 September

09:30 → 10:30 Tverberg’s theorem for cell complexes

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.

Speaker: Prof. Daisuke Kishimoto (Kyushu University)

10:30 → 11:00 Coffee break

11:00 → 12:00 Realisability of fusion systems by discrete groups

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.

Speaker: Prof. Ran Levi (University of Aberdeen)

12:00 → 14:00 Lunch break

14:00 → 15:30 Contributed Talks

Conveners: Mariam Pirashvili (University of Plymouth), Shuchita Goyal (BITS Pilani, India), Andrés Carnero Bravo (Centro de Ciencias Matemáticas, UNAM)

14:00 An isometry theorem for persistent homology of circle-valued functions

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 circle-valued maps in the context of persistent homology. Zig-zag persistence generalises to circle-valued functions, leading to persistence modules which are representations of a zig-zag cyclic quiver of type $\tilde{A_n}$. This approach was first introduced in the work of Burghelea and Dey, who classified the resulting indecomposable represenatitons of the $\tilde{A_n}$ quiver as barcodes and Jordan blocks and proposed an algorithm for computing these. The stability of the numerical invariants of persistent homology with respect to the interleaving distance is the fundamental result in this area that gives this method its strong theoretical foundation. Over the years, this distance has been generalised to the zig-zag setting and to general poset representations, using tools from representation theory. Most notably, the involvement of the Auslander-Reiten translate in the definition of the interleaving distance has meant that the robust machinery of representation theory could be employed to derive algebraic stability theorems in more general settings. Our main result is defining an interleaving distance on circle-valued persistence modules using the Auslander-Reiten translate. Moreover, we propose a novel, computer-friendly way to encode the invariants of circle-valued functions via the so-called geometric model, a relatively new tool from representation theory. We also propose a matching distance based on the geometric model, and show that this matching metric coincides with the interleaving distance.

Speaker: Mariam Pirashvili (University of Plymouth)

14:30 Bousfield localisation on posets

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 category is subjected to right/left Bousfield localization, a technique used to extend the class of weak equivalences. This is a joint work with Andrés Carnero Bravo et al.

Speaker: Shuchita Goyal (BITS Pilani, India)

15:00 Homotopy type of independence complexes of graph products

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 lexicographic products it is known the homotopy of type. We particularly focus in the lexicographic product which is related to the spaces known as polyhedral joins and we will give a homotopy decomposition for the suspension of these spaces under some conditions.

Speaker: Andrés Carnero Bravo (Centro de Ciencias Matemáticas, UNAM)

15:30 → 16:00 Coffee break

16:00 → 18:00 Free Discussion

Thursday, 18 September

09:30 → 10:30 Topology vs. Learning: A Preliminary Comparative Study

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.

Speaker: Dr Ulderico Fugacci (CNR - Genova)

10:30 → 11:00 Coffee break

11:00 → 12:00 Avalanche complex and topology of the sandpile dynamics on digraphs

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.

Speaker: Dr Henri Riihimäki (Nordita / Stockholm University)

12:00 → 14:30 Lunch break

14:30 → 15:30 Real and complex line fields on manifolds

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.

Speaker: Baylee Schutte (University of Aberdeen)

complex_line_fields.pdf

pspan_Wall_manifolds.pdf

15:30 → 16:00 Coffee break

16:00 → 17:00 GKM Theory and Applications to the Study of Quiver Grassmannians

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.

Speaker: Dr Sabino Di Trani (Sapienza Università di Roma))

17:00 → 18:00 Contributed Talks: 3

Conveners: Pratiksha Chauhan (Indian Institute of Technology Mandi), Giulia Maria Menara (Università degli Studi di Milano-Bicocca)

17:00 Shellability of cut complexes of powered cycle graphs

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 master thesis of Denker and were further studied by Bayer et al. in [Topology of cut complexes of graphs, SIAM Journal on Discrete Mathematics, 38(2):1630-1675, 2024]. One of the main motivations behind cut complexes was a famous theorem of Ralf Fr¨oberg, which states that a Stanley–Reisner ideal I∆ generated by quadratic square-free monomials has a 2-linear resolution if and only if ∆ is the clique complex of a chordal graph. This theorem connects commutative algebra and graph theory through topology. While Fr¨oberg’s theorem focuses on ideals generated by degree-two monomials, considering higher-degree monomials leads to the concept of cut complexes. For a positive integer k, the k-cut complex of a graph G, denoted as ∆k(G), is the simplicial complex whose facets (maximal simplices) are the (|V (G)| − k)-subsets σ of the vertex set V (G) of G such that the induced subgraph G[V (G) \ σ] is disconnected. For p ≥ 1 and n ≥ 3, the p-th powered cycle graph Cp n on n vertices is a graph where the set of vertices V (Cp n) = {0, 1, 2, . . . , n − 1} and the set of edges E(Cp n) = {{i, i + j (mod n)} : 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ p}. Bayer et al. conjectured that for k ≥ 3, the k-cut complexes of squared cycle graphs are shellable. Moreover, they also conjectured about the Betti numbers of these complexes when k = 3. We have proven these conjectures for k = 3, showing that the 3-cut complex ∆3(C2 n) is shellable and homotopy equivalent to a wedge of n−4 2 ? − 9 spheres of dimension n − 4. We extend these results for k = 3 to powered cycle graphs proving that ∆3(Cp n) is shellable and homotopy equivalent to a wedge of n−2p 2 ? − (2p2 + p − 1) spheres of dimension n − 4. Our work includes the construction of an explicit shelling order and the characterization of spanning facets, which together determine the homotopy type of these complexes. Further, we investigated the k-cut complexes ∆k(Cp n) for powered cycle graphs Cp n (for small values of k, p and n). Using SageMath, we compute the homology groups of ∆k(Cp n) (with coefficients in Z) and reveal intriguing patterns. These observations suggest new conjectures and open questions about the shellability, homotopy types, and homology of these complexes. The talk will cover the computational techniques used and highlight the key results obtained.

Speaker: Pratiksha Chauhan (Indian Institute of Technology Mandi)

17:30 TBA

Speaker: Giulia Maria Menara (Università degli Studi di Milano-Bicocca)

20:00 → 22:00 Social dinner

Friday, 19 September

09:30 → 10:30 Diffeomorphisms of reducible 3-manifolds

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.

Speaker: Dr Rachael Boyd (University of Glasgow)

10:30 → 11:00 Coffee break

11:00 → 12:00 Topological machine learning and its applications to Raman spectroscopy.

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.

Speaker: Dr Maria Antonietta Pascali (CNR – Pisa)

12:00 → 12:30 Greetings