The n-Category Café

Here are some of the other talks:

• Minicourse by Nicholas Proudfoot: Categorification via geometry.

Lecture 1: A geometric realization of the regular representation of the symmetric group

We construct two commuting actions of the symmetric group on the cohomology of the flag variety, and show that these are isomorphic to the left and right actions on the regular representation.

Lecture 2: Categorification of the regular representation of the symmetric group

Next we promote this action to a pair of braid group actions on a certain category of D-modules on the flag variety (Bernstein-Gelfand- Gelfand category O), such that when we pass to the Grothendieck group, we recover the regular representation of Lecture 1.

Lecture 3: Geometric category O and symplectic duality

All of the beautiful structure on BGG category O, including the two commuting braid group actions, come from the geometry of the cotangent bundle of the flag variety. In this lecture we show that, given any sufficiently nice symplectic variety (of which the cotangent bundle of the flag variety is a special case), we can construct a category that shares many of the properties of BGG category O.

All of the material in Lectures 1 and 2 are due to other people. All of the material in Lecture 3 is joint work with Braden, Licata, and Webster.

• Minicourse by Dorette Pronk: Equivariant cohomology with local coefficients for representable orbifolds.

Orbifolds are spaces that can locally be described as the quotient of an open subset of Euclidean space by a smooth action of a finite group. To obtain a notion of morphism between orbifolds that is appropriate to study orbifold homotopy theory, orbifolds can be represented by proper foliation groupoids (i.e., Lie groupoids with a proper diagonal and discrete isotropy groups). Two such groupoids represent the same orbifold precisely when they are Morita equivalent. More specifically, the category of orbifolds can be viewed as the bicategory of fractions of the 2-category of Lie groupoids with respect to the essential/Morita equivalences. An orbifold is representable when it can be obtained as the quotient of a manifold by the action of a compact Lie group. The subcategory of representable orbifolds is equivalent to the bicategory of fractions of the 2-category of translation groupoids with equivariant maps with respect to equivariant Morita equivalences. This prepares the way to generalize invariants from equivariant Bredon cohomology to orbifolds. We construct an equivariant fundamental groupoid and describe Bredon cohomology with twisted/local coefficients for orbifolds. This is joint work with Laura Scull.

• Talk by Melchior Grutzmann: Cohomology theories of H-twisted Lie and Courant algebroids

In the first part I introduce the structure of H-twisted Lie and Courant algebroids together with examples, some of which are motivated by supermanifolds. In brief we ask for the usual axioms of Lie (resp. Courant) algebroids, but twist the Jacobi identity by a 3-form H with values in the kernel of the anchor map. In addition we have a closedness condition for this H. Already the twist over a point (the algebra) gives new structures.

In the second part I introduce three cohomology theories for subclasses of H-twisted Lie algebroids. The first occuring from the naive idea of cutting down cochains until the naive differential from the Cartan formula (that works for Lie algebroids) squares to 0. Also examples will be given. The second one is for splittable H-twisted Lie algebroids which are in 1:1-correspondence to symplectic dg-manifolds of maximal degree 3. This structure is also known under the name Lie algebra with representation up to homotopy in the literature. The third one finally works for regular anchor maps where we can promote its kernel to a degree 2 vector bundle and the underlying vector bundle of the algebroid to degree 1. Then it is possible to introduce another dg-structure that defines a third kind of cohomology.

• Talk by Rongmin Lu: Higher algebraic structures in parabolic geometry

Let G be a semi-simple Lie group, with Lie algebra g, and P be a parabolic subgroup with Lie algebra p. A parabolic geometry on a manifold M is then given by a principal P-bundle over M equipped with a principal g-valued connection satisfying certain conditions. It is known that a parabolic geometry carries a Lie algebroid structure, but recent progress made by various researchers have uncovered more general algebroid structures. In this talk, we give a survey of these developments.

• Talk by Shusen Ding: Norm Inequalities for Differential Forms and Related Operators

Differential forms are extensions of functions, and have become invaluable tools for many fields of sciences and engineering, including theoretical physics, general relativity, potential theory and electromagnetism. Differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. For instance, some kinds of differential forms are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals, and certain geometric invariance. The theory about operators applied to functions has been very well studied. However, the study about operators, such as Green’s operator G, the Laplace-Beltrami operator $\Delta$ , the maximal operator M, and the homotopy operator T, applied to differential forms just began. As we know, the operator $\Delta$ is widely used to define the different versions of the harmonic equations. Green’s operator G is often applied to study the solutions of various differential equations and to define Poisson’s equation for differential forms. These operators are critical tools to establish existence and regularity for solutions to PDEs, and to control oscillatory behavior on a manifold. In many situations, the process to solve a PDE or a system of PDEs involves integration or integral estimate (for numerical solutions) and the operators are often used to represent solutions. Hence, we have to investigate the involved operators and their compositions. The main purpose of this presentation is to introduce some recent results about differential forms, including inequalities with different norms, such the $L^p$-norms, BMO norms, $L^{p,q}$-norms and Orlicz norms. We will also discuss inequalities for the related operators and their compositions.