Overview Days for nonspecialists
Tuesday 8th January 2013 to Thursday 10th January 2013
09:00 to 09:55  Registration  
09:55 to 10:00  Welcome from INI Deputy Director, Christie Marr  INI 2  
10:00 to 11:00  Deformation theory  INI 2  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Operads
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 nfold loop spaces in algebraic topology (70's). It enjoyed a renaissance in algebra and geometry with the Koszul duality theory and the deformationquantization of Poisson manifolds respectively (90's). Recently, it was proved to be explicitely connected to GrothendieckTeichmüller theory (2010's).

INI 2  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
P Lochak (Institut de Mathématiques de Jussieu) GrothendieckTeichmüller theory
In this talk I am planning to review both the first inputs and some of the main sources and currents of GrothendieckTeichmueller 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.

INI 2  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
Multiple Zeta Values
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 renaissanceafter Euler's seminal work which had been mostly forgottenabout 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.

INI 2  
17:30 to 18:30  Welcome Wine Reception 
10:00 to 11:00 
Braids and the GrothendieckTeichmuller Group
I will explain what are associators (and why are they useful and natural) and what is the GrothendieckTeichmüller group, and why it is completely obvious that the GrothendieckTeichmuller 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 boundeddegree 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 GrothendieckTeichmuller 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/Newton1301/ .

INI 2  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
H Nakamura (Okayama University) Arithmetic and topological problems in universal monodromy representation of GaloisTeichmueller groups 
INI 2  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
padic multiple zeta values
I will introduce a construction of padic multiple zeta values and explain their basic properties including a relationship with GT.

INI 2  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
The BoardmanVogt resolution and algebras uptohomotopy
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 Palgebra structure is in general not invariant under homotopy: a space which is homotopy equivalent to one carrying a Palgebra structure only has a "Palgebra structure uptohomotopy". We will address the questions whether these Palgebra structures uptohomotopy can be controlled by another operad, and whether they can be "strictified" to true Palgebra 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
categories.

INI 2  
19:30 to 22:00  Conference Dinner at Emmanuel College 
10:00 to 11:00 
Periods
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.

INI 2  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
V Dotsenko (Trinity College Dublin) Shuffle operads
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.

INI 2  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
K Hess (EPFL  Ecole Polytechnique Fédérale de Lausanne) Operads and their (bi)modules in topology
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.

INI 2  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
A line in the plane and the GrothendieckTeichmueller group
The GrothendieckTeichmueller 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 BatalinVilkovisky formalism are discussed.

INI 2 