Courses

Course 1:  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 2:   Ideas 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 3:  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 4:  

Instructor:  Graham DENHAM, University of Western, Canada

Abstract:

Course 5:  Graph ideals and their properties and invariants 

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

Abstract:  In this series of talks, we look at certain interesting monomial and binomial ideals in the polynomial ring attached to simple graphs. We give an overview on their algebraic and combinatorial properties and invariants. We will also provide some open problems related to the topic.

Course 6:   An Introduction to Algebraic Methods in Combinatorics 

Instructor:  Shaheen NAZIR, LUMS, Pakistan

Abstract:  Algebraic combinatorics lies at the intersection of algebra and combinatorics, where algebraic structures and techniques are used to understand and solve combinatorial problems. This introductory course will develop some of the fundamental ideas and methods of the subject, with an emphasis on the interplay between algebraic and combinatorial perspectives. The course will begin with basic enumerative techniques, generating functions, permutations, partitions, and Young tableaux. We will then introduce symmetric functions and explore their connections with tableaux and representations of symmetric groups. Other topics will include partially ordered sets and Möbius inversion, graph polynomials and spectral methods, and an introduction to matroids and their associated invariants. Where appropriate, we will also discuss connections with simplicial complexes, commutative algebra, and algebraic geometry. The aim is to provide participants with a broad foundation in algebraic combinatorics while highlighting the connections between different areas. The course will be suitable for graduate students and researchers with a basic background in algebra and combinatorics, and will provide preparation for more advanced topics in algebraic, geometric, and topological combinatorics.

Course 7:  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.