Topology and Combinatorics Uncovering Structures of Vietoris-Rips Complexes

• Jelena Grbic (University of Southampton)

Room: Aula Dini I will explore the homotopy theory of Vietris-Rips complexes of hypercubes, focusing on the deep interplay between their topology and combinatorial structures. The goal is to understand key structural properties, such as higher connectivity, co-connectivity and the realisation of homotopy types, by introducing a new combinatorial-topological framework. This is joint work with Martin Bendersky.

Khovanov-like categorifications of polynomials in matroid theory

• So Yamagata (Fukuoka University)

Room: Aula Dini Khovanov introduced a bigraded cohomology theory for links whose graded Euler characteristic recovers the Jones polynomial. Analogous Khovanov-like (co)homology theories have since been developed beyond knot theory, including chromatic cohomology for graphs and characteristic homology for hyperplane arrangements. A matroid is a combinatorial structure that captures abstract notions of dependence, encompassing cycles in graphs and linear dependencies of vectors. In particular, matroids arise naturally from both graphs and hyperplane arrangements. In this talk, we introduce (co)homology groups associated with certain polynomials of matroids. This is joint work with Takuya Saito.

Operadic Localization and Applications

• Francesca Pratali (Universiteit Utrecht)

Room: Aula Dini Introduced by May and Boardman–Vogt to study iterated loop spaces, operads are combinatorial gadgets governing homotopy coherent algebraic structures on spaces —and in any other symmetric monoidal category. In homotopy theory, where spaces and objects are considered up to weak equivalence, algebraic structures must be encoded in a homotopy-invariant way. This is achieved using the more flexible formalism of “∞-operads”, which can be thought of as operads up to homotopy. In the first part of this talk, I will introduce the theory of ∞-operads and their algebras, focusing on how the models of Lurie and Moerdijk–Weiss relate to partition posets and trees. In the second part, I will discuss operadic localization, the process of freely inverting a class of morphisms in an operad. I will present criteria that allow one to prove that a given operadic construction realizes a localization, and I will describe applications to factorization algebras and quantum field theory.

The topological evasiveness conjecture

• Bruno Benedetti (University of Miami)

Room: Aula Dini The notion of evasiveness lies at the intersection of combinatorics, topology, and theoretical computer science. In 1984 Kahn, Saks, and Sturtevant conjectured that the simplex is the only vertex-transitive simplicial complex that is non-evasive. In this talk, I will review the history and motivation behind this conjecture and present recent joint work providing a counterexample.

Diagrammatic Hochschild Cohomology, Incidence Algebras, and Filtrations

• Francesco Vaccarino (Politecnico di Torino)
• Luigi Caputi (University of Bologna)

Room: Aula Dini The Hochschild cohomology of a diagram of algebras, as introduced by Gerstenhaber and Schack, provides a natural framework for studying the cohomological properties of presheaves of algebras indexed by a small category. In this talk, we revisit and develop the connection between the Gerstenhaber--Schack complex and the Baues--Wirsching cohomology of categories, showing that a spectral sequence converging to diagrammatic Hochschild cohomology has its second page described by higher limits over the twisted arrow category. A key simplification arises when restricting to diagrams of homological epimorphisms: the Baues--Wirsching cohomology reduces to classical functor cohomology over the indexing category, and the presence of a terminal object forces the spectral sequence to collapse. As a main application, we consider filtrations of finite simplicial complexes and the associated diagrams of incidence algebras. Exploiting the fact that injective simplicial maps induce surjective homological epimorphisms of incidence algebras, and the classical result that simplicial cohomology is Hochschild cohomology, we show that the diagrammatic Hochschild cohomology of such a filtration recovers the simplicial cohomology of the final complex. The spectral sequence itself, however, carries richer information: its first page contains the classical persistent module of the filtration. At the same time, higher columns encode additional data whose geometric meaning remains an open question.

When welded knot theory becomes useful for topology

• Jean- Baptiste Meilhan (Université Grenoble Alpes)

Room: Aula Dini Welded knot theory is a combinatorial and diagrammatic extension of classical knot theory. It arises naturally as a quotient of virtual knot theory, introduced in the early 2000s by Kauffman and by Goussarov–Polyak–Viro. The aim of this talk is to present several results showing that welded knot theory turns out to be a relevant and effective tool for topology — not only in knot theory, but also in the study of knotted surfaces in 4-space. This talk assumes no specialized background in topology.

Quantum groups meet graphs

• Pegah Pournajafi (Collège de France)

Room: Aula Dini Quantum groups and graph theory may seem like distant areas, yet intriguing connections emerge when they intersect. After an introduction to the notion of quantum automorphism groups of finite graphs, we will focus on 0-hyperbolic graphs and a computation of their quantum automorphism group. If time permits, we will also show how their quantum symmetries can be fully understood through their classical properties, due to their structural constraints. This talk is based on joint work with Amaury Freslon and Paul Meunier.

Representing 4-manifolds via branched coverings

• Valentina Bais (SISSA)

Room: Aula Dini Branched coverings can be seen a way to represent a ''complicated manifolds'' M in terms of - a ''simpler'' manifold N (the target of the branched coverig), -a codimension two subcomplex K in N (the branch set), -a representation of the fundamental group of the complement of K into a permutation group (the monodromy). By a classical result of Alexander, every piecewise linear manifold admits a branched covering onto the sphere. On the other hand, given an arbitraty manifold N, its topology might restrict the set of manifolds arising as its branched coverings. I will talk about a recent joint work with Riccardo Piergallini and Daniele Zuddas, where we prove that, given a closed connected 4-manifold N with no 1- and 3-handles, there is a simple d-fold branched covering from M to N if and only if d times the intersection lattice of N isometrically embeds into the intersection lattice of M. We also give conditions on the degree and on the regularity of the branch set.

Moduli Spaces of Spatial Polygons

• Dmitrii Korshunov (Institut de mathématiques de Jussieu – Paris Rive Gauche)

Room: Aula Dini Consider all closed polygonal paths in three-dimensional Euclidean space consisting of n edges of prescribed lengths. We identify those that are related by an isometry of R^3. This moduli space carries a Kähler structure (Deligne–Mostow, Klyachko, Kapovich–Millson). I will discuss the relation between the symplectic geometry of this moduli space, flexible polyhedra, and a solution of Kenyon’s problem on triangulated domes. If time permits, we will discuss the heuristic similarity of this space to Teichmüller space and the moduli space of flat surfaces, together with the probabilistic questions it suggests about the expected shape of random polygonal paths.

Braid groups and related structures

• Anne-Laure Thiel (Université Bourgogne Europe)

Room: Aula Dini The central object of this talk is the braid group. I will recall some combinatorial structures revolving around this group. First two of its generalizations : on one hand virtual braid groups and on the other Artin groups. I will give a construction combining them, namely virtual Artin groups, and review some of its properties. Then I will turn to the Hecke algebra and one of its categorical incarnation Soergel bimodules. I will present a gentle overview of the category of Soergel bimodules and of its importance in (higher) representation theory and knot theory. I will sketch the construction of a similar category but set in a slightly larger scope. In a very special case, I will give a complete description of this category and show how it then gives rise to an algebra related to the affine Hecke algebra. If time permits explain how, in the general case, one could hope to get a better understanding of this category through virtual Hecke-like algebras.