The goal of this introductory talk on operads will be to give several definitions of this notion as well as its main applications discovered so far. An operad is an universal device which encodes multiple inputs operations and all the ways of composing them. This notion was first used to recognize n-fold loop spaces in algebraic topology (70's). It enjoyed a renaissance in algebra and geometry with the Koszul duality theory and the deformation-quantization of Poisson manifolds respectively (90's). Recently, it was proved to be explicitely connected to Grothendieck-Teichmüller theory (2010's).
In this talk I am planning to review both the first inputs and some of the main sources and currents of Grothendieck-Teichmueller theory. I will start by quickly recalling the existence of an action of the arithmetic Galois group on (various versions of) the fundamental group of an algebraic variety (resp. scheme, stack) in general, then single out (as Grothendieck first did) the moduli stacks of curves, which feature the defining objects of GT theory. I will then give some indications about the contents of four `pionneering' papers, by A.Grothendieck, V.Drinfel'd, Y.Ihara and P.Deligne respectively. This will lead in particular to underlining some crucial differences in goals and approaches for the various (at least two) versions of the theory, which of course are still in the making.
Special values of the famous Riemann zeta function at integer points have long been known to be of high arithmetic significance.
They can be regarded as a special (depth 1) case of multiple zeta values whose renaissance--after Euler's seminal work which had been mostly forgotten--about 25 years ago, in particular by Zagier and Goncharov in an arithmetic context and by Broadhurst in particle physics, has triggered a flurry of activity producing lots of results and many more conjectural properties about these numbers. We will try to give some of the basic properties and a glimpse of a few of the many different contexts in which they appear.
I will explain what are associators (and why are they useful and natural) and what is the Grothendieck-Teichmüller group, and why it is completely obvious that the Grothendieck-Teichmuller group acts simply transitively on the set of all associators. Not enough will be said about how this can be used to show that "every bounded-degree associator extends", that "rational associators exist", and that "the pentagon implies the hexagon".
In a nutshell: the filtered tower of braid groups (with bells and whistles attached) is isomorphic to its associated graded, but the isomorphism is neither canonical nor unique - such an isomorphism is precisely the thing called "an associator". But the set of isomorphisms between two isomorphic objects *always* has two groups acting simply transitively on it - the group of automorphisms of the first object acting on the right, and the group of automorphisms of the second object acting on the left. In the case of associators, that first group is what Drinfel'd calls the Grothendieck-Teichmuller group GT, and the second group, isomorphic (though not canonically) to the first, is the "graded version" GRT of GT.
All the references and material for this talk can be found there: http://www.math.toronto.edu/~drorbn/Talks/Newton-1301/ .
In this lecture, we will present some basic properties and constructions of topological operads and their algebras. For an operad P, the property of having a P-algebra structure is in general not invariant under homotopy: a space which is homotopy equivalent to one carrying a P-algebra structure only has a "P-algebra structure up-to-homotopy". We will address the questions whether these P-algebra structures up-to-homotopy can be controlled by another operad, and whether they can be "strictified" to true P-algebra structures. Much of this goes back to Boardman and Vogt's book "Homotopy Invariant Algebraic Structures", but can efficiently be cast in the language of Quillen model
This talk will complement Herbert Gangl's talk. I will give a leisurely introduction to some of the ideas underlying the motivic philosophy of periods, with particular emphasis on the case of the projective line minus 3 points.
Symmetric operads, usually arising in applications, provide a language to work with algebraic properties exhibited by substitutions of operations with many arguments into one another. Because of multiple arguments of operations, one naturally has symmetries (permutations of arguments of an operation) present in all arising questions. Contrary to what they teach us, in many of these questions symmetries rather get in the way than are helpful. A way to make symmetries almost disappear is to move to a large universe of "shuffle operads". Shuffle operads are easier to handle, and answers to many questions on symmetric operads can be derived from the respective answers in the shuffle world. The goal of this talk is to give an introduction to this circle of questions for beginners, and to outline some applications.
Lunch at Wolfson Court
Hess, K (EPFL - Ecole Polytechnique Fédérale de Lausanne)
I will introduce the notions of left and right modules over an operad and explain their significance. In particular I will present important examples of operads and their (bi)modules that arise in algebraic topology.
The Grothendieck-Teichmueller group (GT) appears in many different parts of mathematics: in the theory of moduli spaces of algebraic curves, in number theory, in the theory of motives, in the theory of deformation quantization etc. Using recent breakthrough theorems by Thomas Willwacher, we argue that GT controls the deformation theory of a line in the complex plane when one understands these geometric structures via their associated operads of (compactified) configuration spaces. Applications to Poisson geometry, deformation quantization, and Batalin-Vilkovisky formalism are discussed.