The Stasheff Associahedra are a family of polytopes that seem to be ubiquitous, but in particular are strongly linked to associativity. Here are the first few: The vertices of the -dimensional associahedron are in bijection with bracketings of the word , or equivalently, with the set of planar binary trees with leaves. For example, there Continue reading How to build the Stasheff Associahedron out…
What is the volume of a ball of dimension and radius ? This ball is the following subset of : and we measure the volume using the standard Euclidean measure. If you only need the answer for even dimensions, here s an easy way to remember: define to be the sum of the volumes of balls Continue reading The Volume of an Even Dimensional Ball
In my last post, I told you how to encode the zero locus of a polynomial function in terms of an -algebra structure on , where lies in degree and lies in degree . Namely, we simply defined the ary bracket on to be the Taylor coefficient of This gave us one of the simplest Continue reading How does a Lie algebra encode a space? (Part 2)
A fundamental principle of derived deformation theory is that a formal space (i.e. formal moduli problem) can be encoded by a differential graded Lie algebra, or more generally, an -algebra. This is usually attributed to a number of famous mathematicians, such as Quillen, Deligne, Drinfeld, Goldman, Millson, Feigin, Manetti, and Kontsevich. More recently, a formalization Continue reading How does…
The character varieties are a remarkable family of spaces that lie at the center of many different active strands of research of the past 30 or 40 years. The basic definition is actually quite simple. We start with a group , and consider the space of representations . This space is an algebraic variety, possibly Continue reading Characters, Brackets, and Skeins
Let be a holomorphic line bundle over a complex manifold . The total space of this bundle is a complex manifold in its own right, and it contains the smooth hypersurface , which is embedded as the set of zero vectors. Let denote the complement. Let be a complex Lie group. In this post we Continue reading A counterexample to Deligne s construction
An absolutely central concept/tool in modern mathematics is that of (co)homology. In one of its earliest incarnations, the singular (co)homology of a topological space, it consists of a sequence of groups that measure the number of ‘holes’ in a space of various dimensions. This can take an inscrutable geometric shape, and boil it down to Continue reading Gugenheim s Theorem
In this post I will discuss a construction, due to Deligne, for extending a flat vector bundle to a flat logarithmic vector bundle. The setting of the construction is as follows: let be a smooth complex manifold, and let be a divisor (i.e. a codimension 1 subvariety). Then given a branch of the logarithm Continue reading Deligne s construction for extending connections.
Consider the exponential map for matrices , which can be defined by the formula where is a square matrix. A matrix logarithm is defined to be a right-inverse to the exponential; that is to say, it is a function such that . It is a standard result of Lie theory that the exponential is a Continue reading Matrix Logarithms