Abstract: Let ℕ be the set of non negative integers and S ⊆ ℕd be an affine semigroup, that is a finitely generated submonoid of ℕd. The set cone(S) = Spanℚ≥0(S) denotes the rational cone generated by S, and H(S) = (cone(S) \ S) ∩ ℕd is referred to as the set of gaps of S. The affine semigroup S is called a C-semigroup if H(S) is finite. When d = 1 and S is a C-semigroup, it is known as a numerical semigroup. If cone(S) ∩ ℕd = ℕd, then S is known as a generalized numerical semigroup. The notions of C-semigroup and generalized numerical semigroup, introduced for the first time by [2] and [1] respectively, have been considered in order to study how some known properties in the context of numerical semigroups can be extended to submonoids in ℕd, focusing on the finite-complement property. In this talk we present some results and developments, with a glance both at the characteristic of these objects as monoids and at algebraic properties of the associated semigroup ring (for this one, we refer to [3]).

References

[1] Gioia Failla, Chris Peterson, and Rosanna Utano. Algorithms and basic asymptotics for generalized numerical semigroups in Nd. Semigroup Forum, 92:460–473, 2016.
[2] Juan Ignacio García-García, Daniel Marín-Aragón, and Alberto Vigneron-Tenorio. An extension of Wilf’s conjecture to affine semigroups. Semigroup Forum, 96(2):396–408, 2018.
[3] Om Prakash Bhardwaj, Carmelo Cisto. Buchsbaumness of finite complement simplicial affine semigroups. arXiv:2507.09267, 2025.

Short Bio: Carmelo Cisto has developed his research at the University of Messina during the PhD and postdoc. His interests focus on commutative algebra, regarding mainly numerical semigroups, affine semigroups and commutative rings arising from combinatorial objects like polyominoes.

Abstract: Let R be a commutative ring with unity and let Q(R) be its total ring of fractions. A reflexive (or divisorial) ideal for R is a fractional ideal such that R : (R : I) = I (where the colon is taken on Q(R)). Since Bass, it is classically known that Gorenstein one-dimensional local rings are characterized by the property that every regular fractional ideal is reflexive. Recently, the problem of determining regular reflexive ideals has been addressed in the one-dimensional local, Noetherian case, but no complete result has been proved.
In this talk, I will give some new necessary conditions for an ideal to be reflexive and I will show how other classes of one-dimensional rings, like Arf rings and almost Gorenstein rings, can be characterized by looking at their reflexive ideals.
The content of the talk is a joint work with P. Campochiaro and F. Strazzanti.

Short bio: Marco D'Anna has been working at the University of Catania since 1997, and he is full professor of Algebra. His research mainly focuses on commutative rings and semigroups. He teaches in both master degree course and bachelor degree course in Mathematics. He is also tutor for the Scuola Superiore di Catania and advisor of several PhD students.

Abstract: The r-th squarefree Veronese subring of a polynomial ring over a field has been studied extensively, owing to its close connections with simplicial complexes. In this talk, I will describe a connection between 3-uniform hypergraphs and the third squarefree Veronese subring Sn(3), the subring of K[x1,…,xn] generated by all squarefree monomials of degree 3, where deg(xi)=1. I will introduce a new notion of ridge-chordality and present some related results. This is joint work with Zhongming Tang.

Short bio: Gioia Failla is Associate Professor of Geometry at the University Mediterranea of Reggio Calabria, Italy. Her research focuses on Algebraic Geometry, Commutative Algebra, and Algebraic Combinatorics,. She collaborates with research groups in Europe, the United States and Mexico.

Abstract: I will discuss finite configurations of pairwise skew lines in PK3 and a naturally associated group of projective transformations. Using a matrix model for skew lines, these transformations become explicit Möbius transformations, and the group can be studied as a finite subgroup of PGL2(K).
I will describe recent progress on this realization problem, explaining both how several finite groups can be realized by configurations of skew lines and why some natural candidates cannot occur. I will also discuss extensions of the construction and applications to algebraic geometry. Some of the results presented are joint work with L. Chiantini, L. Farnik, B. Harbourne, J. Migliore, T. Szemberg, and J. Szpond.

Short bio: Giuseppe Favacchio is Associate Professor at the University of Palermo. He obtained his PhD in Mathematics from the University of Catania in 2014. His research interests lie at the intersection of commutative algebra, algebraic geometry, and combinatorics, with a focus on projective configurations, interpolation problems, and Lefschetz properties. 

Abstract: In the present talk, extending significantly the polynomial ring case due to Muta and Terai, we will introduce the concept of Serre depths of a module in either a Noetherian local ring or a standard graded algebra over a field. These invariants measure the deviation of a module from satisfiying Serre’s conditions, in the same way that depth measures deviation from the Cohen-Macaulay property. We develop the basic theory of Serre depth, its invariance under completion, its behaviour under Gröbner degenerations, and its asymptotic behaviour along powers. In the monomial case, we will show that the Serre depth of powers is eventually constant. The proof deeply uses Presburger arithmetic, an unusual tool in Commutative Algebra.

Short bio: Antonino Ficarra is a researcher in commutative algebra and its interactions with combinatorics, with a particular focus on monomial ideals. He obtained his Ph.D. in Mathematics and Computational Sciences under the supervision of Marilena Crupi and Jürgen Herzog. He subsequently held a postdoctoral position at the University of Évora, working under the supervision of Pedro Macias Marques. He is currently an Ikerbasque Juan de la Cierva Research Fellow at BCAM (the Basque Center for Applied Mathematics) in Bilbao, Spain. His research interests lie primarily in commutative algebra, combinatorial aspects of algebraic structures, and the study of monomial ideals and their applications.

Abstract: In this talk we study some properties of the so called Hadamard fat grids (HFG) in P2, which are the result of the Hadamard product of two sets of collinear points with given multiplicites. The most important invariants of Hadamard fat grids, as minimal resolution, Waldschmidt constant and resurgence, are then computed. In particular, after introducing the minimum Hamming distance and the minimum socle degree of a set of fat points in the projective space, we answer a question of Tohaneanu and Van Tuyl which relates these quantities in the case of Hadamard fat grids. Based on joint works with I. Bahmani Jafarloo, C. Bocci, G. Malara, and C. Bocci, M. Sciuto.

Short bio: Elena Guardo is Full Professor of Geometry at the University of Catania. Since October 2024 she has been the Head of the Mathematics Section of the Department of Mathematics and Computer Science. From January 2019 to September 2023 she served as Coordinator of the Bachelor’s Degree Programme in Mathematics, and since November 2023 she has been its Vice‑President. She teaches in the Engineering and Mathematics degree programmes. Her research interests include Algebraic Geometry, Commutative Algebra and Combinatorics.

Abstract: We consider a relevant class of bipartite planar graphs that generalizes those introduced in [1], and investigate their minimal vertex covers and the graded Betti numbers in the minimal graded resolution of their edge ideal. We refer to [2] for more information and details.
Remember that planar graphs are made up of regions of the plane delimited by their edges without any edge crossing and bipartite graphs can be represented by partitioning their vertex set into two disjoint vertex subsets each of which without any edge. Due to their structure, bipartite planar graphs play a crucial role in various practical sectors of algebra and geometry.
The vertex covering of any graph consists in finding a minimal vertex subset such that any edge of the graph is incident with one vertex in that subset. The study of vertex covers of minimum cardinality has significant implications in optimization and algebraic combinatorics. For the examined graphs we describe the minimal vertex covers together with their cardinality and connect to the vertex covers some algebraic quantities such as the height of the corresponding edge ideal, the dimension and the projective dimension of the associated quotient polynomial ring.
Using geometric and combinatorial matters we are also able to study all the graded Betti numbers that appear in the minimal graded resolution of the quotient ring associated with such graphs and give upper bounds for them in terms of the number of regions. As an application we calculate the second graded Betti number in degree 3 in terms of graph theoretical properties.

References

[1] L.R. Doering, and T. Gunston, Algebras arising from bipartite planar graphs, Commun. Algebra 24 (1996), 3589-3598.
[2] M. Imbesi, and M. La Barbiera, Algebraic and geometric aspects of bipartite planar graphs, Ann. Acad. Rom. Sci. Ser. Math. Appl. 17, 3 (2025), 197-210.

Short bio: Maurizio Imbesi is an Associate Professor of Geometry SSD MATH-02/B at the Department MIFT: Mathematical and Computer Sciences, Physical Sciences and Earth Sciences of the University of Messina. He is the author/co-author of about 30 scientific papers published in journals indexed in Scopus and WoS. His research concerns Commutative, Combinatorial and Computational Algebra, Algebraic Graph Theory, Linear and Nonlinear Algebra, and Discrete Variational Analysis. Currently, he deals with:
1) Determination of spanning trees and graceful labelings of bipartite planar graphs.
2) Graded Betti numbers and projective dimension of the quotient ring associated with bipartite graphs. Minimal vertex covering and maximal independent edges.
3) Existence and convergence of solutions of nonlinear equations on locally finite weighted graphs.
4) Geometric and combinatorial aspects of projective varieties and graphs.
5) Term orders that descend from the geometry of polyhedra and combinatorial algebra.

Abstract: In this work, we study the finite generation of the effective monoids of Looijenga pairs with a 4-cycle obtained as blow-ups of Hirzebruch surfaces. To do so, we give a numerical condition that ensures the finite generation of the effective monoids and provides a tool to determine explicitly their minimal generating sets. Also, it turns out that such numerical condition implies that these surfaces are big and as a consequence, we obtain that they are Harbourne-Hirschowitz surfaces and it ensures the finite generation of their Cox rings. The ground field in this work is assumed to be an algebraically closed field of arbitrary characteristic.

Short bio: Mustapha Lahyane obtained his Ph.D. degree in Mathematics on December 3, 1998, from Laboratoire Jean-Alexandre Dieudonné (LJAD, UMR7351) of the University of Nice Sophia-Antipolis, nowadays known as Université Côte d'Azur (Nice, France). His main research topics are Algebraic Geometry, Commutative Algebra, and applications to Coding Theory. Lahyane has supervised three Ph.D. theses and at least ten master and undergraduate theses. Currently, he works at the Faculty of Physics and Mathematics of the Universidad Michoacana de San Nicolás de Hidalgo (Morelia, State of Michoacán de Ocampo, Mexico). Lahyane has carried out research stays at several prestigious institutions, including the Laboratoire Jean-Alexandre Dieudonné (Nice), ICTP (Trieste), McGill University, University of Valladolid, University of Messina, Mediterranean University of Reggio Calabria, University of Jaume I, CIMAT (Guanajuato), University of Nebraska-Lincoln, Universidad Nacional Autónoma de México (Mexico City and Oaxaca), and Galatasaray University.

Abstract: The results presented in this talk are drawn from [2]. Binomial edge ideals provide a rich class of ideals connecting combinatorial properties of graphs with homological invariants of the associated quotient rings. In particular, the sequentially Cohen-Macaulay property plays a central role in understanding their algebraic and combinatorial structure, extending the classical Cohen-Macaulay case. In this talk, we revisit the sequentially Cohen-Macaulay property for binomial edge ideals through three complementary perspectives. First, we discuss a homological technique based on deficiency modules, due to Schenzel, which provides intrinsic criteria for detecting the property. We then present a more recent combinatorial approach based on special ideals, due to Goodarzi, which gives effective characterizations for several classes of binomial edge ideals. Finally, we address the computational aspects of the theory and describe how these methods can be implemented in practice in the Macaulay2 software. In this framework, we present the package SCMAlgebras [1], providing methods for computing deficiency modules, filter ideals, and related invariants, giving effective tools for testing the sequentially Cohen-Macaulay property in practical examples. Our aim is to illustrate how homological, combinatorial, and computational methods interact in the study of sequentially Cohen-Macaulay binomial edge ideals, and how this interplay leads both to a better structural understanding and to practical verification techniques.

References

[1] E. Lax: SCMAlgebras: a Macaulay2 package to check sequential Cohen-Macaulayness. arXiv preprint (2024) [arxiv:2409.15134]. Available at https://github.com/ErnestoLax/SCMAlgebras.
[2] E. Lax, G. Rinaldo, F. Romeo: Sequentially Cohen-Macaulay binomial edge ideals. arXiv preprint (2024) [arxiv:2405.08671].

Short bio: Ernesto Lax received his Ph.D. in Mathematics and Computational Sciences (38th cycle) at the University of Messina in 2026. He currently holds the position of teaching assistant in Discrete Mathematics (SSD: MATH-02/A) at the Department of Mathematics and Computer Science, Physics and Earth Sciences (MIFT) of the University of Messina. His research interests are in Commutative Algebra and Combinatorics, with a particular focus on homological invariants of graded ideals associated with combinatorial structures, especially graphs and simplicial complexes. His work explores the interplay between algebraic properties and the combinatorial features of the underlying structures. He is also active in Computational Algebra, working with Computer Algebra Systems (CAS) such as Macaulay2. More informations can be found at his website.

Abstract: The study of ideals generated by arbitrary set of minors of a generic matrix has deep roots in commutative algebra, algebraic geometry and combinatorics. In this context, the polyomino ideal was introduced in a recent work of A.A. Qureshi as the ideal generated by those sets of 2-minors of a matrix that can be combinatorially characterized as a polyomino, that is, a collection of equally sized squares joined edge to edge, similar to a pruned chessboard.
In this talk we will focus on the theory of the rook polynomial of a polyomino and its connection with the corresponding polyomino ideal. In particular, we will survey several results from the literature and discuss recent developments and open problems in the area.

Short bio: Francesco Navarra is a postdoctoral researcher at Sabanci University, working under the supervision of Ayesha Asloob Qureshi. He received his PhD in Mathematics from the University of Palermo in July 2023, under the supervision of Marilena Crupi and Rosanna Utano. His research lies in Combinatorial Commutative Algebra, with a focus on monomial and binomial ideals associated with hypergraphs, simplicial complexes, and polyominoes. He is also interested in computational methods and software development in Macaulay2.

Abstract: Let I be a squarefree monomial ideal. The k-th squarefree power I[k] of I is the ideal generated by the squarefree monomials among the generators of Ik. The study of squarefree powers of squarefree monomial ideals is closely connected with the classical theory of matchings in hypergraphs. We will discuss the recent results on squarefree powers of facet ideals associated with simplicial trees (equivalently, totally balanced hypergraphs), focusing on the linearity of their minimal free resolutions, Castelnuovo–Mumford regularity and projective dimension.

Short bio: Ayesha Asloob Qureshi is an Assistant Professor of Mathematics at Sabancı University, Istanbul. Her research focuses on Commutative Algebra and Combinatorics, with particular interests in monomial and binomial ideals, Gröbner bases, and toric algebras.

Abstract: Image processing and computer vision techniques have experienced remarkable progress over the last decades, evolving from classical image analysis methods to advanced machine learning and deep learning frameworks. The primary objective of image processing is the extraction of meaningful information from digital images through enhancement, segmentation, classification, detection, and interpretation processes. A wide range of methodologies has been developed to address these tasks, each offering advantages depending on image characteristics and application requirements.

Classical image segmentation approaches include threshold-based methods, edge detection, region growing, watershed transformation, clustering algorithms, and graph-based optimization techniques. Thresholding methods, such as global and adaptive thresholding, separate objects from the background based on pixel intensity values and remain widely used due to their simplicity and computational efficiency. Edge-based techniques identify object boundaries through gradient analysis, while region-growing methods group neighboring pixels with similar properties. Clustering approaches, including K-means and Fuzzy C-means, partition images into homogeneous regions without requiring extensive prior information. More advanced optimization methods, such as Graph Cut and GrabCut, formulate segmentation as an energy minimization problem, enabling accurate object extraction while preserving boundary information.

Recent developments in machine learning and artificial intelligence have substantially improved image analysis performance. Traditional machine learning algorithms, including Support Vector Machines (SVM), Random Forests, and Artificial Neural Networks, utilize handcrafted features for classification and pattern recognition tasks. Deep learning architectures, particularly Convolutional Neural Networks (CNNs), U-Net, Mask R-CNN, and Vision Transformers (ViTs), automatically learn hierarchical image representations and achieve state-of-the-art results in segmentation, detection, and classification. These methods reduce the need for manual feature engineering and demonstrate high robustness when processing large and complex datasets.

The application of these techniques extends across numerous domains. In Synthetic Aperture Radar (SAR) image analysis, image processing methods are employed for target recognition, terrain classification, environmental monitoring, change detection, and disaster assessment. The ability of machine learning algorithms to handle speckle noise and complex backscatter characteristics has significantly enhanced SAR image interpretation. In medical imaging, advanced segmentation and classification methods support automated analysis of MRI, CT, PET, and ultrasound data for tumor detection, organ delineation, disease diagnosis, and treatment planning. Deep learning models have become particularly valuable in assisting clinicians through accurate and efficient image interpretation.

In civil engineering, image processing techniques play an increasingly important role in structural health monitoring and infrastructure inspection. Automated crack detection and characterization in concrete structures, bridges, tunnels, pavements, and buildings are performed using thresholding methods, edge detection, texture analysis, machine learning classifiers, and deep neural networks. These approaches improve inspection efficiency, reduce human subjectivity, and enable early identification of structural deterioration.

The convergence of traditional image processing methods, optimization-based segmentation techniques, and modern artificial intelligence algorithms provides powerful tools for analyzing increasingly complex image data. Future research directions include explainable artificial intelligence, multimodal image fusion, self-supervised learning, real-time processing, and digital twin integration, which are expected to further enhance the reliability and applicability of intelligent image analysis systems across remote sensing, healthcare, and civil engineering applications.

Short bio: Maria Zdimalova is an Associate Professor of Applied Mathematics at the Slovak University of Technology in Bratislava. Her research focuses on graph theory, mathematical modelling, and applying artificial intelligence to image processing. She leads the international project Arch Math, which connects mathematics with architecture, design, fashion, and art. She has published over 60 research papers, with 41 indexed in the Web of Science and Scopus databases. As an invited speaker and researcher, she has worked globally, including in India, Mexico, Italy, and South Africa. She serves on international editorial boards and acts as an external PhD examiner for African universities. At the university, she teaches calculus, logic, and statistics, while actively engaging students in scientific research.