Courses

Course 1:   Algebraic Aspects of Arrangements and Matroids

Instructor:  Graham DENHAM, Western University, Canada

Abstract:  Matroids and hyperplane arrangements are basic combinatorial objects whose study over the recent decades has involved a mixture of commutative algebra, geometry, and combinatorics.  This course is intended to be complementary to the course of Prof. Delucchi and will explore some algebraic aspects of the theory, including interactions with toric and tropical geometry.  Topics will include hyperplane arrangement compactifications and the Chow ring of a matroid. The nature of the topic leads to examples and exercises that can be carried out by hand or with help from software like Macaulay2.  

Course 2:  Combinatorics and Topology of Arrangements 

Instructor:  Emanuele DELUCCHI, University of applied arts and sciences of Southern Switzerland, Switzerland

Abstract:  Arrangements of hyperplanes are geometric objects with a rich combinatorial structure, e.g., as related to matroid theory, as well as connections to algebra, for example in the study of Coxeter and Artin groups. Over the last decades all (and more) of these aspects have been developed into a lively theory. Recently, the focus has broadened to include arrangements of hypersurfaces in tori and beyond. This was originally motivated by work of De Concini, Procesi and Vergne on partition functions and box splines, which gave a fresh impulse toward generalizing all aspects of the theory to the non-linear case. In this course I will outline some of the basics of arrangements of hyperplanes and sketch some current developments and open questions in the setting of hypersurface arrangements. I will focus mainly on the combinatorics and on the extension of matroid theory to the hypersurface setting. I will also mention some topological results that have been obtained by combinatorial techniques, since they provide some of the motivation for the constructions and may allow to highlight some connections to Prof. Welker’s lecture.

Course 3:  Extremal ideals

Instructor:  Sara FARIDI, Dalhousie University, Canada

Abstract:   For a positive integer q, the q-extremal ideal is a square-free monomial ideal constructed combinatorially using variables indexed by subsets of the set of integers {1,...,q}. We show that for any square-free monomial ideal I with q generators, there is a ring homomorphism relating I to the q-extremal ideal, and as a result for any r>0, the algebraic behavior of the power ideal I^r can be determined via that of the r-th power of the q-extremal ideal. This course will be an overview of extremal ideals and their known and unknown properties.

Course 4:   Lattice Polytopes, Ehrhart Theory, and Toric Ideals

Instructor:  Martina JUHNKE, University of Osnabrück, Germany

Abstract:  A lattice polytope is a convex polytope whose vertices have integer coordinates. Such objects sit at a rich crossroads of combinatorics, commutative algebra, and algebraic geometry, and their study has seen remarkable growth over the past few decades.

These lectures offer an introduction to three interrelated topics. We begin with the combinatorics of lattice polytopes: their face lattices and we will in particular look at 2-dimensional lattice polygons. We then develop Ehrhart theory in any dimension, which studies how the number of lattice points in dilations of a polytope varies — a story encoded beautifully in the Ehrhart polynomial and its generating series,  as well as the Ehrhart h*-vector. Along the way we encounter fundamental results such as the reciprocity theorem and Stanley's nonnegativity theorems. In the final part we turn to toric ideals: binomial prime ideals arising from integer matrices, encoding the algebraic relations among monomials associated to the lattice points of a polytope. We discuss initial ideals of toric ideals and their connection to triangulations of the polytope, with a special focus on the role of regular and unimodular triangulations.

Exercise sessions will give students the opportunity to work hands-on with small examples, computing Ehrhart polynomials, h*-vectors, and toric ideals explicitly, and to deepen their understanding of the interplay between the geometry, combinatorics, and algebra developed in the lectures.

Course 5:  Binomial Ideals Associated with Graphs Arising from Generic Hankel Matrices  

Instructor:  Sara Saeedi MADANI, Amirkabir University of Technology, Iran

Abstract:  During these lectures we will have an overview on some binomial ideals associated with graphs which arise from a certain matrix of variables. We start with one of the most classical ones which arise from a generic matrix, called the binomial edge ideal of a graph. We will also look at some variations and generalizations of this type. One of the variations is based on a generic Hankel matrix which will be our main topics of discussion. We call such an ideal the Hankel edge ideal of a graph. Regarding these ideals, we mainly focus on specific graphs like labeled Hamiltonian and semi-Hamiltonian graphs as well as certain labeled trees. We discuss the minimal prime ideals of the Hankel edge ideal of the aforementioned classes of graphs, and moreover we investigate certain algebraic properties like radicality, being a complete intersection, almost complete intersection and set-theoretic complete intersection. We may also discuss the initial ideals of such ideals with respect to a certain monomial order.

Course 6:   An Introduction to Matroids and Hyperplane Arrangements

Instructor:  Shaheen NAZIR, LUMS, Pakistan

Abstract: This course develops the combinatorial foundations of hyperplane arrangements and matroid theory, providing essential background for the advanced lectures on their algebraic, geometric, and topological aspects. Beginning with concrete examples of hyperplane arrangements, we examine their intersection structures, intersection posets, and regions. The basic concepts of matroid theory—including rank, circuits, bases, flats, and representability—are developed through linear algebra and graph theory. Particular attention is given to the correspondence between hyperplane arrangements and matroids, illustrating how combinatorial data capture important geometric properties. Characteristic polynomials and their connection with the number of regions of a real arrangement provide a first glimpse of the interplay between combinatorics and geometry. The course concludes with an overview of connections to algebra and topology, including the Orlik–Solomon algebra, compactifications, and the Chow ring of a matroid, together with a brief discussion of hypersurface arrangements. Emphasis is placed on concrete examples and hands-on calculations, establishing a common foundation for the advanced courses.

Course 7:   Ideals of 2-minors

Instructor:  Ayesha Asloob QURESHI, Sabancı Üniversitesi, Turkey 

Abstract:  Ideals generated by 2-minors of generic matrices form a classical class of binomial ideals with important applications in commutative algebra, algebraic geometry, and algebraic statistics. More recently, special families arising from combinatorial configurations, such as graphs, grids, and collections of cells, have been studied from a combinatorial viewpoint. This course provides an introduction to ideals of 2-minors from both algebraic and combinatorial perspectives. We begin with basic definitions and examples of determinantal ideals generated by 2-minors, and discuss fundamental algebraic properties such as primality, radicality, height, and primary decompositions. We then focus on binomial ideals defined by 2-minors associated with combinatorial objects, emphasizing how the underlying combinatorics influences their algebraic behaviour. Topics include Gröbner bases for ideals of 2-minors and connections with toric and lattice ideals. Throughout the course, concrete examples will be used to illustrate general phenomena, and several open problems and current directions of research will be highlighted.

Course 8:  Topological methods in Algebraic Combinatorics

Instructor:  Volkmar  WELKER, Philipps-Universität Marburg, Germany

Abstract:  The use of topological methods for the study of combinatorial questions goes back to the solution of Kneser’s conjecture by Lov´asz and work of Folkman providing homological interpretations of Rota’s cross-cut theorems. The last 40 years have seen a continuous growth of methods, ranging from shellability, over discrete Morse theory to combinatorial adaptations of the theory of homotopy colimits and equivariant methods. In this course we will introduce the students to the basic tools of this theory. We will assume only a very modest knowledge of point set topology and introduce simplicial complexes, their homology and homotopy type at the beginning of the course. We will exhibit their relation to purely combinatorial invariants such a M¨obius number or Euler characteristic. Then we will focus on the purely combinatorial tools for calculating homology and homotopy type, such as shellability, collapsibility and discrete Morse theory. Throughout the course, the theoretical concepts will be accompanied by examples from combinatorics and algebra. For some of the examples the presented methods will lead to complete solutions, which partly will be covered by the problem sets in the first week of the class. Other examples lead to open questions, which then will be addressed by the students in the second week of the course.