We give an introduction to cellular algebras. These are finite dimensional algebras defined by Graham and Lehrer in 1996 that allow for a complete description of their simple modules. In 2012 this idea was generalized by Koenig and Xi to affine cellular algebras, including infinite dimensional algebras. We illustrate these definitions with the classical and the affine Temperley-Lieb algebra, and we also discuss how the affine nilTemperley-Lieb algebra fails to inherit this affine cellular structure.

We give an introduction to combinatorially defined particle configurations and explain how they are related to crystals of (affine) type A. On fermionic particle configurations, it is possible to define a faithful representation of the (affine) nilTemperley-Lieb algebra. We explore this algebra and give descriptions of bases, its center and its simple modules. Similar results can be obtained for bosonic particle configurations. We will mainly present results from joint work with G. Benkart and the speaker's PhD thesis, but also talk about some ongoing work.

Divided difference operators are discrete analogs of differential operators. The goal of the talk is to show how they appear naturally in the representation theory for the nil-Coxeter algebra, and — as recently discovered — the general linear Lie algebra.

We shall give an introduction to the discriminant of a reflection group. No prior knowledge of reflection groups will be assumed. Let \(W\) be a finite (complex) reflection group acting on a (complex) vector space \(V\). Let \(H\) be the union of the mirrors (fixed points of reflections in \(W\)). By Chevalley theorem \(V/W\) is an affine space. The image of \(H\) inside the \(V/W\) is a singular affine hypersurface called the discriminant of the reflection group \(W\). This generalized the discriminant of a quadratic or cubic polynomial. There are many interesting things to say about the discriminant. For example, the fundamental group of the complement \((V/W) - H\) generalizes the braid group.

We shall give an introduction to the discriminant of a reflection group. No prior knowledge of reflection groups will be assumed. Let \(W\) be a finite (complex) reflection group acting on a (complex) vector space \(V\). Let \(H\) be the union of the mirrors (fixed points of reflections in \(W\)). By Chevalley theorem \(V/W\) is an affine space. The image of \(H\) inside the \(V/W\) is a singular affine hypersurface called the discriminant of the reflection group \(W\). This generalized the discriminant of a quadratic or cubic polynomial. There are many interesting things to say about the discriminant. For example, the fundamental group of the complement \((V/W) - H\) generalizes the braid group.

The ideal-variety correspondence is classical and connects commutative algebra and algebraic geometry.  With the advent of computer algebra systems, like Macaulay2, and Groebner basis algorithms, we can efficiently compute many complicated invariants of ideals and varieties such as dimension, multiplicity, free resolutions, etc.  I will give a brief summary of the theory, including Hilbert's big three theorems (Basis Theorem, Syzygy Theorem, Nullstellensatz) and show what the corresponding Macaulay2 computations like.

We will give an outline of the work due to Shimura and Deligne which associates a 2-dimensional Galois representation to some special types of modular forms. This construction has had far reaching applications, including in finite Galois Theory. In the first of two talks, we will develop some prerequisites including talking about (elliptic) curves, Jacobians and their Tate modules.

Perfectoid spaces have revolutionized arithmetic geometry and commutative algebra in the last ten years.  In this talk, I'll describe the most basic perfectoid algebra corresponding to a point — a perfectoid field.  I'm assuming nothing other than basic field theory.

I’ll give a gentle introduction to certain surfaces called Kleinian singularities. They are important objects in algebraic geometry and are related to group theory, invariant theory, root systems, and representation theory. I will show some pictures of what they look like, and go over their classification in terms of ADE Dynkin diagrams. (This talk can be viewed as a prequel to my talk on March 10, 2017 which dealt with analogs of these objects in noncommutative geometry.)

Let \(G\) be a group of linear maps on a vector space \(V\). The invariant ring of \(G\) is the ring of polynomial functions on V that are invariant under \(G\) action. We will discuss the invariant rings of finite complex reflection groups. We shall illustrate how properties of the reflection group are encoded in the ring of invariants and co-invariants.

We will start with a gentle introduction to Galois Representations. We will then discuss how questions in Galois Theory can be answered using techniques from Galois Representations. If there is time, I will discuss a project that I have been pursuing with Yeansu Kim to realise spin groups over \(\mathbb{Q}\).

A crystal is a type of directed graph that encodes essentially all the combinatorial information about of a Lie algebra, its associated quantum group, and their representations. In this talk we will study the structure of a crystal with respect to an automorphism of the underlying Dynkin diagram, via a process known as folding.

The folding procedure introduced here extends a well-known Lie theoretic construction based on fixed-points. The latter was used by Lusztig in his celebrated construction of the canonical basis for a quantized enveloping algebra of multiply-laced type. Folding does not preserve the property of being highest-weight. This leads to a new type of crystal that corresponds to a new type of representation of a quantized enveloping algebra. We call these crystals multi-highest-weight.

In certain cases the multi-highest-weight crystal obtained from folding \(B(\infty)\) — the crystal for the lower triangular half of a quantized enveloping algebra — can be combinatorially generated using subsets of the Weyl group known as balanced parabolic quotients. I will explain how this generation can be accomplished both intrinsically and using the semigroup structure of Nakashima and Zelevinsky's polyhedral realization.

We will introduce the class of finite complex reflection groups and give many examples. We will describe their Coxeter-like diagrams and braid groups and talk about some of the mysteries surrounding them.

We present classification theorems for maximal subalgebras of finite-dimensional algebras over a field \(k\). This is done by first classifying maximal subalgebras of semisimple algebras, and then lifting to the  general case. When \(k\) is nice (ex. algebraically closed), the classification can be understood in terms of the ideal structure of the Jacobson radical. For bound quiver algebras, this gives us nice presentations for subalgebras. Trivial extensions and separable extensions feature prominently in the classification, and allow us to relate representation-theoretic properties of an algebra to those of its subalgebras via induction and restriction. If time permits, we discuss related problems and applications, ex. determining isomorphism classes of subalgebras, and finding minimal generating sets of algebras.

In this talk, we discuss the results of investigations that began with a solution to an open problem posed by Schwabhäuser and Szczerba regarding definability (without parameters) in the three dimensional Euclidean geometry of lines, asking whether intersection was definable from perpendicularity (two lines intersecting at a right angle). It is not. The result is a “new” 3-dimensional geometry of lines, which we call perpendicular geometry, since it can be formalized from perpendicularity. Further investigations produce a rather complete classification of possible geometries arising from elementary Euclidean binary relations between lines in \(\mathbb{R}^3\), modulo the determination of (metrical) projective geometries formalized by binary relations between points. The classification shows that in a sense made precise, perpendicular geometry is the only new geometry that can arise from binary geometric relations, except for possible new projective plane geometries, which we conjecture do not exist. Generalizing to geometries of s-flats in n-dimensional Euclidean geometry, we state a theorem which provides the essential first step towards a similar classification for the general case. The theorem states that parallel is definable in such a geometry no matter what binary geometric relation that one might chose to formalize geometry (with an enumerated list of trivial exceptions). To what extent this is true for ternary or higher order geometric relations is open, even for lines in \(\mathbb{R}^3\). We conjecture that it remains true, that is, that parallel is definable from anything “except for the exceptions”. Finally, we note that perpendicular geometry, whose automorphism group is connected with derivations, sheds some rather curious light on the relationship between the product rule for derivations over a ring, and the sum rule. For example, it is a direct consequence of perpendicular geometry that the product rule for the cross product of 3-dimensional vectors implies the sum rule. It is conjectured that this is also true of all finite dimensional semi-simple Lie algebras over the complex numbers.

The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a reductive Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type \(C\) (Symplectic group). To make the symplectic case look more like the Type \(A\) case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux and obtain an exotic Robinson-Schensted correspondence. This is joint work with Vinoth Nandakumar and Neil Saunders.

Galois rings and orders, defined by Futorny and Ovsienko in 2010, form a class of algebras which include many important algebras in representation theory, such as the (quantized) enveloping algebra of \(\mathfrak{gl}_n\), type \(A\) restricted Yangians and finite W-algebras. I will present a new criterion for determining when a Galois ring is a Galois order. This can be applied in particular to \(U_q(\mathfrak{gl}_n)\) which also proves the quantum Gelfand-Zeitlin subalgebra is maximal commutative. This establishes several conjectures including bounds on fibers of irreducible Gelfand-Zeitlin characters. The method in fact applies to all of the mentioned examples and give unified new proofs.

In this talk we will discuss the structure of non standard maximal parabolics of twisted affine Lie algebras, global Weyl modules and the associated commutative associative algebras. Since the global Weyl modules associated with the standard maximal parabolics have found many applications the hope is that these non-standard maximal parabolics will lead to different, but equally interesting applications.

An introduction to the current algebra of \(\mathfrak{sl}_2\) will be given. This Lie algebra is infinite-dimensional and has non-semisimple finite-dimensional modules (i.e. Weyl's theorem fails). Some classes of modules will be discussed, and open problems in the area stated.