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Introductory Course: Representation Theory and Symmetric Polynomials
Instructor: Shaheen Nazir
Description: This course comprehensively introduces representation theory and its significant connections with symmetric functions, emphasizing their crucial role in algebraic combinatorics. It begins with an overview of representation theory, covering basic definitions of groups, rings, modules, and algebras, along with examples such as symmetric and cyclic groups. The course explores finite groups and their representations, including reducible and irreducible representations, and delves into character theory, highlighting the significance of characters in algebraic combinatorics. Students will examine Young tableaux and their role in the representation theory of symmetric groups, as well as the connections between Young tableaux and various combinatorial structures. The course also introduces symmetric functions, including elementary, complete, power sum, and Schur functions, discussing their importance and applications in combinatorial problems. In particular, Schur functions are studied through their definitions via Young tableaux and their combinatorial interpretations, including Pieri's rule and the Littlewood-Richardson rule. Furthermore, the course investigates the relationship between representation theory and symmetric functions, focusing on concepts such as the Frobenius characteristic map and Schur-Weyl duality.
Course 1: Computational Algebraic Geometry
Instructor: Henry Schenck
Description: This course will be an introduction to the syzygy-based computational tools for the analysis and study of geometric objects. It will be divided into three main chapters. Lecture 1 will cover graded rings and modules, Hilbert function, Hilbert polynomials, Hilbert series, as well as the central topic of finite free resolutions and how Gröbner basis are used to compute them. Lecture 2 will introduce some basics of homological algebra, including Ext and Tor functors, and their computation, Hilbert syzygy theorem and the use of Ext to stratify associated primes. Lecture 3 will provide an introduction to some more advanced topics in combinatorial algebraic geometry, mainly illustrating the concepts above with Stanley-Reisner ideals, Alexander duality, Eagon-Reiner theorem and a little taste of toric geometry.
Course 2: Graph ideals and their properties and invariants
Instructor: Sara Madani
Description: 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 3: (Symbolic) Powers of Monomial Ideals
Instructor: Somayeh Moradi
Description: The study of powers and symbolic powers of ideals is an active area of research with numerous open questions and conjectures. Symbolic powers have a long history in commutative algebra, and find applications in algebraic geometry, due to their connection to certain geometric properties of varieties. A compelling question in the study of powers is to investigate relations between the symbolic powers and the ordinary powers of an ideal and their homological invariants. In this course, we will focus on the study of symbolic powers of monomial ideals, which due to their combinatorial nature, serve as a rich testing ground for conjectures and theorems in commutative algebra. We pay special attention to squarefree monomial ideals which can be associated to graphs and simplicial complexes. After providing the necessary algebraic background, we present an overview of recent advances on some algebraic and combinatorial properties of (symbolic) powers of such ideals. Also we will highlight some open problems in this area.
Course 4: Some Combinatorics of Coxeter Groups
Instructor: Susanna Fishel
Description: The first lecture will be the basics of Coxeter groups, with special attention paid to how they generalize the symmetric group. We will also emphasize the geometric representation and provide many examples. I will assign a problem or two on Coxeter groups of other types, including affine versions. Lecture two will be on partial orders associated with Coxeter groups. We will discuss lattice properties, maximal chains and reduced words, and inversions. I will assign a problem or two on using sagemath for poset questions. The third lecture will be on the Shi arrangement, a hyperplane arrangement related to Coxeter groups. We'll start with its original definition, then show how it was extended and generalized (affine case only). Again, the combinatorics will be stressed. I will assign a problem on drawing the Shi arrangement. Lecture four will be much more technical. We'll go into the details about root systems and Coxeter groups we'll need for our project. Homework will be in sagemath and on posets.
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