Restriction of free arrangements
Speaker: Takuro Abe
Affiliation: Rikkyo University, Japan
Abstract: The classical Orlik’s conjecture asserted that the restriction of free arrangement was free, but a counter example was found by Edelman and Reiner. So this is no more conjecture, but in many free arrangements, their restrictions are free, though they may not satisfy Terao’s addition-deletion theorem. So still it is hard to characterize the freeness of the restriction of an free arrangement without using Terao’s restriction theorem.
In this talk, we consider this problem and give some criterions for the freeness of the restrictions. Also, we give some connections to tame arrangement theory.
Monodromy of supersolvable toric arrangements
Speaker: Dan Cohen
Affiliation: Louisiana State University
Abstract: TBA (joint with Bibby and Delucchi)
The ribbon conjecture in even Artin groups
Speaker: Maria Cumplido
Affiliation: University of Seville
Abstract: TBA
TBA
Speaker: Corrado De Concini
Affiliation: University of Rome "La Sapienza"
Abstract: TBA
Distances in trees and inequalities for matroids
Speaker: Graham Denham
Affiliation: University of Western Ontario
Abstract: The distance matrix of a tree appears in a range of contexts, from phylogenetics to physical chemistry. I will describe a new result about the spectrum of this matrix, one that gives affirmative answers to two questions about matroid positivity properties. These are both strengthenings of Mason’s conjectures about the log-concavity of sequence of numbers of independent sets of a matroid, proposed by Igor Pak and by Giansiracusa, Rincón, Schleis, Ulirsch, respectively.
Combinatorics and Topology of Conditional Oriented Matroids
Speaker: Galen Dorpalen-Barry
Affiliation: Texas A&M
Abstract: Oriented matroids are combinatorial objects that capture much of the topology of (central) real arrangements. A well-know theorem of Salvetti, for example, describes the homotopy type of the complexitied complement of a real arrangement using only the data of its oriented matroid. A conditional oriented matroid plays the role of an oriented matroid when one has a convex body cut by hyperplanes in a real vector space. These arise, for example, in the study of Coxeter arrangements, convex polytopes, and affine arrangements. In this talk, we will give an overview of what’s known about conditional oriented matroids and share new results about their combinatorics and topology. This is joint work with Nick Proudfoot.
Orlik-Solomon algebras and 2-isomorphisms
Speaker: Michael Falk
Affiliation: Northern Arizona University
Abstract: I’ll talk about a perspective on Orlik-Solomon algebras that allows the application of the 2-isomorphism theorem of Vertigan and Whittle to study isomorphisms and automorphisms.
A New Approach to Biparameter Persistence Based on Varying the Associated Group-Equivariant Non-Expansive Operator
Speaker: Patrizio Frosini
Affiliation: University of Pisa
Abstract: As is already known, one way to approach certain concepts in TDA is to interpret them as Group-Equivariant Non-Expansive Operators (GENEOs), which transform topological filtrations into other, simpler types of data. In particular, in topological biparameter persistence, the concept of matching distance can be introduced via a family of GENEOs F(a,b), each of which transforms every filtering function f with values in ℝ2 into the persistence diagram of a real-valued function f(a,b), associated with a fixed straight line r(a,b) in the plane, having positive slope.
This approach to biparameter persistence has yielded many theoretical and practical results, but it presents two limitations: 1) The set of GENEOs used depends on two parameters (a,b), resulting in a relatively high computational cost; 2) These GENEOs are not smooth operators, preventing their direct application through methods derived from differential calculus.
It is therefore natural to ask whether replacing the traditional GENEOs F(a,b) with alternative, simpler types of GENEOs could address these two issues. In this talk, we will demonstrate how the theory can indeed be simplified through the use of a smooth family of operators Ft, depending on a single parameter t. Specifically, we will show how this approach enables the definition of a new distance between vector-valued filtering functions, which can be proven to be stable. Moreover, we will illustrate how—quite surprisingly—the concept of the extended Pareto grid, originally developed to study the classical matching distance, can also be directly applied to this new metric, allowing for the localization of points in the persistence diagrams of the filtering functions Ft(f).
Canonical reduction systems for braid groups and Artin groups
Speaker: Juan Gonzalez Meneses
Affiliation: University of Seville
Abstract: Joint work with María Cumplido and Davide Perego. Recently, some geometric and topological objects, procedures and properties of braid groups are being translated to algebraic terms, so that they can be generalized to other Artin groups of spherical type. One important example is the complex of curves, which can be described as a complex of irreducible parabolic subgroups. The canonical reduction system of a braid is a distinguished set of curves, invariant under the action of the braid, along which one performs Thurston’s decomposition. We give a group-theoretical algorithm to compute this set of curves, using the Garside structure of braid groups.
Combinatorial Topology and its applications in Distributed Computing
Speaker: Dmitry Kozlov
Affiliation: University of Bremen
Abstract: In this talk we will discuss applications of combinatorial topology, including studying the computational complexity of distributed protocols for standard tasks in theoretical distributed computing. In particular, we shall state an open problem in combinatorial topology which is related to the complexity of the Weak Symmetry Breaking distributed task.
Alcoved Polytopes and Arrangements
Speaker: Lukas Kühne
Affiliation: University of Bielefeld
Abstract: Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots ei − ej. This fundamental class of polytopes appears in several applications such as optimization, tropical geometry or physics. Symmetric alcoved polytopes are dual to the fundamental polytopes attached to finite metric spaces.
This talk focuses on the type fan of alcoved polytopes which is the subdivision of the metric cone by combinatorial types of alcoved polytopes. The type fan governs when the Minkowski sum of alcoved polytopes is again alcoved. For symmetric alcoved polytopes, this fan is the intersection of the metric cone with the Wasserstein arrangement.
I will discuss both theoretical and computational results for the symmetric and asymmetric type fan of alcoved polytopes.
This talk is based on joint works with Emanuele Delucchi, Nick Early, Leonid Monin, and Leonie Mühlherr.
Parametrized chain complexes in Topological Data Analysis
Speaker: Claudia Landi
Affiliation: University of Modena and Reggio Emilia
Abstract: Functors indexed by posets with values in the category of non-negative chain complexes are a case of generalized persistence modules convenient for encoding homotopical and homological properties of objects in topological data analysis. We will start discussing some circumstances in which the category of such functors admits both an Abelian and a model structure, highlighting the case of posets of dimension 1. Next, we will consider structure theorems in this category. We will see that the structure theorem for filtered (i.e. with internal maps that are monomorphisms) tame functors indexed by non-negative reals gives insights into the usual persistent homology barcoding algorithm. In the case of factored (i.e. with internal maps that are epimorphisms) tame functors, again indexed by non-negative reals, the structure theorem yields the construction of barcodes for Morse-Smale vector fields. Generally, when functors need not be filtered or factored, the family of indecomposables is wild. However, if the indexing poset is of dimension 1, any functor admits a cofibrant replacement for which a structure theorem will be presented.
Complex braid groups, their parabolic subgroups and Hecke algebras
Speaker: Ivan Marin
Affiliation: University of Amiens
Abstract: TBA
Topological Machine Learning: Applications in Raman Spectroscopy and the Challenge of Explainable AI
Speaker: Davide Moroni
Affiliation: CNR, Pisa
Abstract: In this talk, based on joint work with Francesco Conti and Maria Antonietta Pascali, I will survey current trends in Topological Machine Learning (TML). We will begin by exploring the synergy between persistent homology and machine learning, which allows for the precise processing of complex data structures. We will then present sample applications in the health domain, focusing specifically on Raman spectrography, where our TML approach has yielded promising results. Finally, we will address a transversal requirement of modern Artificial Intelligence: the need for explainability (XAI). We will present a novel approach that leverages the principles of TML to respond to this critical need in Raman spectra analysis, demonstrating how topological methods can contribute to creating more transparent and interpretable AI systems.
Chow rings with building sets: an application to 0,n
Speaker: Roberto Pagaria
Affiliation: University of Bologna
Abstract: We introduce and study the Chow ring of a lattice equipped with a building set. We derives formulas for computing their Hilbert series. As a key application, we consider partition lattices endowed with the minimal building sets and we provide a new formula for the Poincaré polynomial of the moduli space 0,n.
Factoring isometries into reflections
Speaker: Giovanni Paolini
Affiliation: University of Bologna
Abstract: Any element of the orthogonal group O(n) can be expressed as a product of at most n reflections. Less well known is the structure of the poset formed by the minimal such factorizations, or its analogue for isometries of arbitrary quadratic spaces. In this talk, I will survey these questions at the crossroads of linear algebra and combinatorics, and illustrate how the resulting insights help us study Coxeter and Artin groups.
Trickle Groups
Speaker: Luis Paris
Affiliation: University of Burgundy
Abstract: We present a new family of groups, called trickle groups, which generalize the right angled Artin and Coxeter groups, as well as the Cactus groups. These are defined by relations of the form xy = yz and xμ = 1, which depend on some combinatorial data called trickle graph. The aim of the talk is to provide the definition of a trickle group, to show several important examples, such as the Thomson group F, and to give some combinatorial results, such as a solution to the word problem.
On arrangements stemming from hyperpolygon spaces
Speaker: Gerhard Röhrle
Affiliation: University of Bochum
Abstract: TBA
Semi-infinite Hodge structure and primitive forms for hyperbolic root systems of rank 2
Speaker: Kyoji Saito
Affiliation: RIMS
Abstract: This is a test study of hyperbolic period maps, on which hyperbolic Artin groups act. The hyperbolic period domain is identified with certain Bridgland’s stability condition space and the Jacobi-inversion problem is solved in a renewed style. More precisely, we construct semi-infinite Hodge structure equipped with primitive forms for hyperbolic root systems of rank 2, to which, we associate the flat structure and the Frobenius manifold structure. Using the data, we get descriptions of the period maps for the virtual period integrals of the primitive forms, on which hyperbolic Artin groups acts. Since hyperbolic root systems are the first case of root systems which do not have geometric origin as vanishing cycles, we calculate down to earth, since we need to renew definition of primitive derivation (=the unit tangent vector) and the wrapping quotient space (=the domain for the virtual” period map).
Algebraic statistics and particle physics meet hypersurface arrangements
Speaker: Hal Schenck
Affiliation: Auburn University
Abstract: An arrangement of hypersurfaces in projective space is SNC if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomous prime ideal of this variety.
(joint with T. Kahle, B. Sturmfels, M. Wiesmann)
Twisted characteristic varieties
Speaker: Alex Suciu
Affiliation: Northeastern University
Abstract: The characteristic varieties of a space are defined as the jump loci for cohomology with coefficients in rank one local systems. As such, they generalize the classical Alexander polynomial from knot theory. In this talk, I will discuss a twisted version of these varieties, which generalize the twisted Alexander polynomials of knots. I will present some applications of these notions, including sharper tropical bounds for the Bieri-Neumann-Strebel-Renz invariants. Based on work in progress with Yongqiang Liu.
TBA
Speaker: Ulrike Tillmann
Affiliation: University of Oxford
Abstract: TBA
Magnitude of finite metric spaces and poset topology
Speaker: Masahiko Yoshinaga
Affiliation: Osaka University
Abstract: Magnitude is an invariant of metric spaces introduced by Leinster. Later, Hepworth, Willerton, Leinster and Shulman introduced the notion of magnitude homology group. In this talk, we will report a poset-topological construction of the space "magnitude homotopy type" whose homology group is isomorphic to magnitude homology (joint work with Y. Tajima). If time permits, we will also discuss continuity and discontinuity of the magnitude on the Gromov-Hausdorff space of finite metric spaces (joint work with E. Roff and H. Katsumasa.)
