summerschool

The starting point of these lectures are certain long-standing conjectures in the character theory of finite groups. Given the classification of all finite simple groups, one can try to attack these problems through reduction theorems and an investigation of the finite simple groups. In recent years, this program has led to substantial advances.

Typically, the simple groups of Lie type are the hardest case to deal with in this set-up. These groups arise as fixed point sets of (connected, reductive) algebraic groups under Frobenius maps and, following the work of Deligne and Lusztig, the character theory of these groups can be approached by geometric methods. We introduce these geometric methods and survey the current state of knowledge.

• Lecture 1: Problems on characters of finite groups
• Lecture 2: The virtual characters of Deligne and Lusztig
• Lecture 3: Lusztig's classification of irreducible characters
• Lecture 4: Examples, outlook, open questions

Full Syllabus in pdf

• Hour 1: Our main example and its properties. Schubert varieties. Bott-Samelson manifolds. The Bruhat stratification of an opposite Bruhat cell. Gröbner degeneration and subword complexes. Theorem: intersections of strata are reduced. Sub-example: matrix Schubert varieties and their defining equations.
• Hour 2: Theorems. Subword complexes.
  A) The stratification is "generated by" the divisor.
  B) The strata are normal, and their boundaries are anticanonical.
  C) The Gr\"obner degeneration commutes with taking limits and intersections.
  D) The initial schemes are Stanley-Reisner schemes of subword complexes.
  E) Schubert varieties are normal and Cohen-Macaulay.
  Main proof technique: Frobenius splitting. Anticanonical sections from T-Poisson structures.
• Hour 3: Bruhat atlases [He-Knutson-Lu].
  Definition: covering a stratified manifold with Bruhat cell charts.
  First example [Snider]: Grassmannians.
  Second example: general $G/P$.
  Third example: wonderful compactifications of groups.
  Fourth example: $P$-orbits on $G/Q$ (for $P$ of finite type).
• Hour 4: Triage presumably; and directions forward.
  Classification? Elek's results in dimension 2. Poisson rank 1? Other examples? Other wonderful compactifications. Quiver variety cores? Conormal varieties of strata?

In these lectures we will discuss various questions related to orbits of the adjoint action of a semisimple algebraic group on its Lie algebra.

A preliminary plan is as follows:

• Nilpotent orbits in semisimple Lie algebras. Singular symplectic varieties.
• $\mathbb{Q}$-factorial terminalizations of singular symplectic varieties. Lusztig-Spaltenstein induction for adjoint orbits.
• Deformation of singular symplectic varieties. Sheets and birational sheets in semisimple Lie algebras.
• Primitive ideals in universal enveloping algebras and orbit method.

Full Syllabus in pdf Problem sheet in pdf

Lecture notes in pdf: Lecture 1 Lecture 2 Lecture 3 Lecture 4

The goal of this series of lectures is to introduce the theory of vertex algebras, with emphasis on their geometrical aspects.

Roughly speaking, a vertex algebra is a vector space $V$, endowed with a distinguished vector, the vacuum vector, and the vertex operator map from $V$ to the space of formal Laurent series with linear operators on $V$ as coefficients. These data satisfy a number of axioms. Although the definition is purely algebraic, these axioms have deep geometric meaning. They reflect the fact that vertex algebras give an algebraic framework of the two-dimensional conformal field theory.

To each vertex algebra $V$ one can naturally attach a certain Poisson variety $X_V$ called the associated variety of $V$. A vertex algebra $V$ is called lisse if $\dim X_V=0$. Lisse vertex algebras are natural generalizations of finite-dimensional algebras and possess remarkable properties. For instance, the characters of simple $V$-modules form vector valued modular functions. More generally, vertex algebras whose associated variety has only finitely symplectic leaves, are also of great interest for several reasons that will be addressed in the lectures.

It is only quite recently that the study of associated varieties of vertex algebras and their arc spaces, has been more intensively developed. In this mini-course I wish to highlight this aspect of the theory of vertex algebras which seems to be very promising. In particular, I will include open problems on associated varieties in the setting of affine vertex algebras (vertex algebras associated with Kac-Moody algebras) and W-algebras (they are certain vertex algebras attached with nilpotent elements of a simple Lie algebra)raised by my recent works with Tomoyuki Arakawa.

The aim of these lectures is to introduce one of the big theories of complex algebraic geometry, the Minimal Model Program (MMP), in the case of $G$-varieties, ie algebraic varieties endowed with an action of an algebraic group $G$.

In the first talk, I will introduce birational geometry and the MMP, and I will explain what happens when an algebraic group acts on the varieties. In particular, I will introduce the notion of contraction, which are morphisms that contracts curves.

In the second talk, I will explain how to construct contractions from a projective variety by using divisors (or line bundles). Especially, we will see that this construction, by using $G$-linearized line bundles over $G$-varieties, naturally reveals a family of $G$-varieties: spherical varieties.

In the third talk, I will introduce two particular subfamily of spherical varieties: toric and horospherical varieties. And I will explain how to study the geometry of these varieties by using the theory of algebraic groups and convex geometry. In the last talk, I will describe the MMP for horospherical varieties via one-parameter families of polytopes

Suggested reading:

• Daniel Perrin; Algebraic Geometry (especially sections about projective varieties)
• David Cox, John Little, Henry Schenck; Toric varieties.