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Seminars (GDO)

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Event When Speaker Title Presentation Material
GDOW01 8th January 2013
10:00 to 11:00
Deformation theory
GDOW01 8th January 2013
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 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).
GDOW01 8th January 2013
14:00 to 15:00
P Lochak Grothendieck-Teichmüller theory
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.
GDOW01 8th January 2013
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 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.
GDOW01 9th January 2013
10:00 to 11:00
Braids and the Grothendieck-Teichmuller Group
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/ .
GDOW01 9th January 2013
11:30 to 12:30
H Nakamura Arithmetic and topological problems in universal monodromy representation of Galois-Teichmueller groups
GDOW01 9th January 2013
14:00 to 15:00
p-adic multiple zeta values
I will introduce a construction of p-adic multiple zeta values and explain their basic properties including a relationship with GT.
GDOW01 9th January 2013
15:30 to 16:30
The Boardman-Vogt resolution and algebras up-to-homotopy
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 categories.
GDOW01 10th January 2013
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.
GDOW01 10th January 2013
11:30 to 12:30
V Dotsenko 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.
GDOW01 10th January 2013
14:00 to 15:00
K Hess 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.
GDOW01 10th January 2013
15:30 to 16:30
A line in the plane and the Grothendieck-Teichmueller group
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.
GDO 15th January 2013
14:00 to 17:00
Grothendieck-Teichmüller groups, graph complexes and deformation quantization (Informal talk)
GDO 16th January 2013
10:30 to 12:00
Operads: definition, examples and first properties
GDO 16th January 2013
14:00 to 15:30
Koszul duality theory for algebras
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 16th January 2013
16:00 to 17:30
Operads: exercise session
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 17th January 2013
14:00 to 16:00
Meta-Groups, Meta-Bicrossed-Products, and the Alexander Polynomial
I will define "meta-groups" and explain how one specific meta-group, which in itself is a "meta-bicrossed-product", gives rise to an "ultimate Alexander invariant" of tangles, that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.

This will be a repeat of a talk I gave in Regina in August 2012 and in a number of other places, and I plan to repeat it a good further number of places. Though here at the Newton Institute I plan to make the talk a bit longer, giving me more time to give some further fun examples of meta-structures, and perhaps I will learn from the audience that these meta-structures should really be called something else. The slides of the talk are availble here: http://www.math.toronto.edu/~drorbn/Talks/Newton-1301/#Talk2.

GDO 22nd January 2013
14:00 to 15:00
Dendroidal sets and infinity-operads I
Dendroidal sets form an extension of simplicial sets, suitable for dealing with nerves of (simplicial) operads rather than just categories. They also give rise to a natural definition of infinity-operad, completely parallel to that of infinity-category. In these lectures, I aim to introduce the category of dendroidal sets together with its monoidal structure and its compatible Quillen model category structure. If time allows, I will also discuss the comparison of the dendroidal theory of infinity-operads to Lurie's approach to infinity-operads.
GDO 22nd January 2013
15:00 to 16:00
Dendroidal sets and infinity-operads II
Dendroidal sets form an extension of simplicial sets, suitable for dealing with nerves of (simplicial) operads rather than just categories. They also give rise to a natural definition of infinity-operad, completely parallel to that of infinity-category. In these lectures, I aim to introduce the category of dendroidal sets together with its monoidal structure and its compatible Quillen model category structure. If time allows, I will also discuss the comparison of the dendroidal theory of infinity-operads to Lurie's approach to infinity-operads.
GDO 23rd January 2013
10:30 to 12:00
Koszul duality theory for operads
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 23rd January 2013
14:00 to 15:30
Methods for Koszul duality
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 23rd January 2013
16:00 to 17:30
Operads: exercise session
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 24th January 2013
14:00 to 15:00
Morita equivalence, Tannakian categories and motivic periods I
GDO 24th January 2013
15:00 to 16:00
Morita equivalence, Tannakian categories and motivic periods II
GDO 29th January 2013
14:00 to 15:00
Dendroidal sets and infinity-operads III
Dendroidal sets form an extension of simplicial sets, suitable for dealing with nerves of (simplicial) operads rather than just categories. They also give rise to a natural definition of infinity-operad, completely parallel to that of infinity-category. In these lectures, I aim to introduce the category of dendroidal sets together with its monoidal structure and its compatible Quillen model category structure. If time allows, I will also discuss the comparison of the dendroidal theory of infinity-operads to Lurie's approach to infinity-operads.
GDO 29th January 2013
15:00 to 16:00
Dendroidal sets and infinity-operads IV
Dendroidal sets form an extension of simplicial sets, suitable for dealing with nerves of (simplicial) operads rather than just categories. They also give rise to a natural definition of infinity-operad, completely parallel to that of infinity-category. In these lectures, I aim to introduce the category of dendroidal sets together with its monoidal structure and its compatible Quillen model category structure. If time allows, I will also discuss the comparison of the dendroidal theory of infinity-operads to Lurie's approach to infinity-operads.
GDO 30th January 2013
10:30 to 12:00
Homotopy algebras
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 30th January 2013
14:00 to 15:30
(Co)homology theories and deformation theory
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 30th January 2013
16:00 to 17:30
Operads: exercise session
An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.

Reference: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012). [Available for free at http://math.unice.fr/~brunov/Operads.pdf]
GDO 31st January 2013
14:00 to 15:00
Tree-indexed series, theory and examples I
To any operad, one can associate a pro-unipotent group much alike the group of formal power series in one variable. This provides a theoretical framework for the use of tree-indexed series in numerical analysis (under the name of Butcher series) by using the PreLie operad. Some examples of tree-indexed series will be presented, with relations to polytopes and Lie idempotents.
GDO 31st January 2013
15:00 to 16:00
Tree-indexed series, theory and examples II
To any operad, one can associate a pro-unipotent group much alike the group of formal power series in one variable. This provides a theoretical framework for the use of tree-indexed series in numerical analysis (under the name of Butcher series) by using the PreLie operad. Some examples of tree-indexed series will be presented, with relations to polytopes and Lie idempotents.
GDO 5th February 2013
10:30 to 12:00
The Hochschild cochain complex, operads, and the Deligne conjecture I
GDO 5th February 2013
14:00 to 15:30
The Hochschild cochain complex, operads, and the Deligne conjecture II
GDO 5th February 2013
16:00 to 17:30
Deformation theory: exercise session
GDO 6th February 2013
10:30 to 12:00
A brief introduction to the etale fundamental group and to the Galois action
GDO 6th February 2013
14:00 to 15:30
Profinite braid groups; first appearance of GT
GDO 6th February 2013
16:00 to 17:30
Special Topic: Belyi Theorem and dessins d'enfants
GDO 7th February 2013
14:00 to 15:00
S Shadrin Hypercommutative algebras, Batalin-Vilkovisky algebras, and Givental group I
I'll make a survey of several different ways to define a so-called Givental group. This group is, in some interpretations, a group of symmetries of the representations of the operad Hycomm, and plays an important role in the correspondence between the operads Hycomm and BV. I'll also try to make a link of this formalism to the topological recursion arising in the matrix models theory.
GDO 7th February 2013
15:00 to 16:00
S Shadrin Hypercommutative algebras, Batalin-Vilkovisky algebras, and Givental group II
GDO 8th February 2013
10:30 to 12:00
Dendroidal sets and infinity-operads III
GDO 12th February 2013
10:30 to 12:00
Formality, obstructions to formality, and a key argument in the proof of the formality conjecture I
GDO 12th February 2013
14:00 to 15:30
B Vallette Formality, obstructions to formality, and a key argument in the proof of the formality conjecture II
GDO 12th February 2013
16:00 to 17:30
Deformation theory: exercise session
GDO 13th February 2013
10:30 to 12:00
Deforming braided categories and a useful dictionary
GDO 13th February 2013
14:00 to 15:30
Infinitesimal braid and the Lie algebra of (the prounipotent version of) GT
GDO 13th February 2013
16:00 to 17:30
Special Topic: Associator relations satisfy double shuffle (H. Furusho)
GDO 14th February 2013
14:00 to 16:00
The classical master equation
We formalize the construction of Batalin and Vilkovisky of a solution of the master equation associated with a polynomial in n variables (or a regular function on a nonsingular affine variety). We show existence and uniqueness up to "stable equivalence" and discuss the associated BRST cohomology (joint work with David Kazhdan).
GDO 19th February 2013
10:30 to 12:00
Etingof - Kazhdan quantization of Lie bialgebras I
GDO 19th February 2013
14:00 to 15:30
Etingof - Kazhdan quantization of Lie bialgebras II
GDO 19th February 2013
16:00 to 17:30
Deformation theory: exercise session
GDO 20th February 2013
10:30 to 12:00
Moduli spaces of curves, curve complexes
GDO 20th February 2013
14:00 to 15:30
The Teichmueller lego
GDO 20th February 2013
16:00 to 17:30
Special Topic: Origamis
GDO 21st February 2013
14:00 to 16:00
Factorization algebras and formality theorems I
In this talk I will explain a conjectural approach to unify many formality results in deformation quantization. I will take this opportunity to give an introduction to factorization algebras.
GDO 21st February 2013
16:00 to 17:00
Factorization algebras and formality theorems II
In this talk I will explain a conjectural approach to unify many formality results in deformation quantization. I will take this opportunity to give an introduction to factorization algebras.
GDO 26th February 2013
10:30 to 12:00
From associators to the Kashiwara-Vergne conjecture I (after Alekseev-Torossian)
GDO 26th February 2013
14:00 to 15:30
From associators to the Kashiwara-Vergne conjecture II (after Alekseev-Enriquez-Torossian)
GDO 26th February 2013
16:00 to 17:30
Deformation theory: exercise session
GDO 27th February 2013
10:30 to 12:00
Lie algebras I
GDO 27th February 2013
14:00 to 15:30
Lie algebras II
GDO 27th February 2013
16:00 to 17:30
Special Topic
GDO 28th February 2013
10:30 to 12:00
Proof of the Deligne conjecture after McClure-Smith
GDO 28th February 2013
14:00 to 16:00
Cohomology of braid groups and mapping class groups I
I will explain how "group completion" techniques lead to a description of the cohomology of braid groups(=spaces of configurations in the plane) and mapping class groups(=moduli spaces of Riemann surfaces). I hope to explain the method completely for braid groups, and indicate what one needs to worry about for mapping class groups. I will then survey explicit calculational information which can be obtained from this method (and others).
GDO 28th February 2013
16:30 to 17:00
Cohomology of braid groups and mapping class groups II
I will explain how "group completion" techniques lead to a description of the cohomology of braid groups(=spaces of configurations in the plane) and mapping class groups(=moduli spaces of Riemann surfaces). I hope to explain the method completely for braid groups, and indicate what one needs to worry about for mapping class groups. I will then survey explicit calculational information which can be obtained from this method (and others).
GDO 5th March 2013
14:00 to 15:15
Introduction to E_n-operads and little discs operads (mini-course)
The little n-discs operads, where n = 1,2,...,infinity, are used to define a hierarchy of homotopy commutative structures, from fully associative but non-commutative (n=1) up to fully associative and commutative (n=infinity).

In the construction of such homotopy commutative structures, we generally have to deal with operads which are just weakly-equivalent to the little n-discs, and the name of an E_n-operad has been introduced to refer to this notion. In this first lecture, I will give an introduction to E_n-operads and their applications. I will notably report on the following significant issues: formality, Koszul duality, and recognition theorems.

General reference:B. Fresse, "Homotopy of operads and Grothendieck-Teichmüller Groups". Book project. First volume available on the web-page "http://math.univ-lille1.fr/%7Efresse/OperadGT-December2012Preprint.pdf"

GDO 5th March 2013
15:45 to 17:00
Homotopy automorphisms of operads in topological spaces (mini-course)
The Grothendieck-Teichmüller group can be defined algebraically, as the automorphism group of an operad in groupoids, the operad of parenthesized braids. I will explain that this operad represents the fundamental groupoid of the little 2-discs operad. The homotopy automorphisms considered in my lecture series represent a topological counterpart of the automorphisms of an algebraic operad. In this lecture, I will explain the precise definition of this notion, and of a rational version of the notion of a homotopy automorphism, where we neglect torsion phenomena.

The result which I aim to establish precisely asserts that, in the case of the little 2-discs operad, the group of rational homotopy automorphisms reduce to homotopy automorphisms that can be detected by their action on fundamental groupoids.

General reference:B. Fresse, "Homotopy of operads and Grothendieck-Teichmüller Groups". Book project. First volume available on the web-page "http://math.univ-lille1.fr/%7Efresse/OperadGT-December2012Preprint.pdf"

GDO 6th March 2013
10:30 to 12:00
Iterated integrals and de Rham fundamental group of P^1 minus 3 points
GDO 6th March 2013
14:00 to 15:30
Malcev completion and Betti fundamental group of P^1 minus 3 points
GDO 6th March 2013
16:00 to 17:30
Discussion Session
GDO 7th March 2013
14:00 to 16:00
Polynomial functors on free groups
GDO 12th March 2013
14:00 to 15:15
The rational homotopy theory of spaces (mini-course)
The rational homotopy theory of spaces is the study of topological spaces modulo torsion. The rational homotopy of a space is determined by two dual algebraic models: the Sullivan model, defined by a commutative dg-algebra structure, and the Quillen model, defined by a Lie dg-algebra.

Furthermore, each space has a minimal Sullivan model, which can be used to prove that the rational homotopy automorphisms of a space form an algebraic group, with a deformation complex of the Sullivan model as Lie algebra. The purpose of this lecture is to provide a survey of these results, after short recollections on the definition of the Sullivan model of a space.

General reference: B. Fresse, "Homotopy of operads and Grothendieck-Teichmüller Groups". Book project. First volume available on the web-page "http://math.univ-lille1.fr/%7Efresse/OperadGT-December2012Preprint.pdf"

GDO 12th March 2013
15:45 to 17:00
The rational homotopy theory of operads (mini-course)
The first purpose of this lecture is to explain the definition of an analogue of the Sullivan model for the rational homotopy of topological operads, and the definition of a rationalization functor on operads.

The Sullivan model of an operad involves both a commutative dg-algebra structure, which encodes the rational homotopy type of the spaces underlying the operad, and a cooperad structure, which models the composition structure attached to our topological operad. If we neglect the commutative algebra part of the structure, then we get a model for the stable rational homotopy of operads, and an operadic version of the Sullivan miminal model can also be defined in this setting. But the construction of minimal models fails when we deal with the combination of commutative algebra and operad structures involved in our model for the rational homotopy of operads in topological spaces. So does the definition of the Quillen model, as well as the classical approach to integrate deformation complexes into rational homotopy automorphism groups.

I will explain methods to bypass these difficulties, and which can be used to establish the main result of this lecture series in the little 2-discs case.

General reference:B. Fresse, "Homotopy of operads and Grothendieck-Teichmüller Groups". Book project. First volume available on the web-page "http://math.univ-lille1.fr/%7Efresse/OperadGT-December2012Preprint.pdf"
GDO 13th March 2013
10:30 to 12:00
The Ihara action on the motivic fundamental group of P^1 minus 3 points
GDO 13th March 2013
14:00 to 15:30
Motivic multiple zeta values and Galois coaction
GDO 13th March 2013
16:00 to 17:30
Discussion Session
GDO 14th March 2013
10:30 to 12:00
R Kaufmann Arcs as an effective model
The arc operad and its variations have proven itself in many instances to be a very clean minimal and explicit tool; for instance in giving models for the little discs, framed little discs, E_n operads, Sullivan PROP, etc on the topological level by basically the same formalism on surface. Furthermore, these models have a very nice cellular structure and hence are naturally suited to solve chain level conjectures. We will present some of these constructions and draw parallels to other activities at the program.
GDO 14th March 2013
14:00 to 15:00
T Terasoma Brown-Zagier relation for associators: I
GDO 14th March 2013
15:00 to 16:00
T Terasoma Brown-Zagier relation for associators: II
GDO 19th March 2013
14:00 to 16:00
M Spitzweck The abelian category of mixed Tate motives with integer coefficients
GDO 20th March 2013
10:30 to 12:00
Relations among Multiple Zeta Values I
GDO 20th March 2013
14:00 to 15:30
Double Zeta Values and Modular Forms
GDO 20th March 2013
16:00 to 17:30
Multiple Zeta Values: discussion session
GDO 21st March 2013
14:00 to 16:00
R Kaufmann Feynman categories
There is a plethora of operad type structures and constructions which arise naturally in classical and quantum contexts such as operations on cochains, string topology or Gromov-Witten invariants.

We give a categorical framework which allows us to handle all these different beasts in one simple fashion. For instance operads, cyclic operads, modular operads, and other new types. In the enriched case, this also covers twisted modular operads, giving a natural description of hyper-operads. In this context, many of the relevant constructions are simply Kan extensions.

We are also able to show how in this framework bar constructions, Feynman transforms, master and BV equations appear naturally

GDO 26th March 2013
10:30 to 12:00
Derivations for multiple zeta values
GDO 27th March 2013
10:30 to 12:00
Relations among multiple zeta values II
GDO 28th March 2013
14:00 to 16:00
Multiple L-functions of modular forms
GDOW02 2nd April 2013
09:30 to 10:30
U Tillmann Homology stability for symmetric diffeomorphisms and their mapping classes
A central object of study in geometry and topology are diffeomorphism groups of compact manifolds. Nevertheless, these remain generally poorly understood objects. For surfaces homology stability has played a crucial role in advancing our understanding (Madsen-Weiss). We will present some homology stability results for 'symmetric' diffeomorphisms for manifolds of arbitrary dimension and put them into context.
GDOW02 2nd April 2013
11:00 to 12:00
Factorization homology of topological manifolds
I'll describe factorization homology and some classes of computations therein which relate Koszul duality, Lie algebra homology, and configuration space models of mapping spaces. Parts of this work are joint with David Ayala, Kevin Costello, and Hiro Lee Tanaka.
GDOW02 2nd April 2013
13:30 to 14:30
A Berglund Rational homotopy theory of automorphisms of highly connected manifolds
I will talk about joint work with Ib Madsen on the rational homotopy type of classifying spaces of various kinds of automorphisms of highly connected manifolds. The cohomology of the classifying space of the automorphisms of a g-fold connected sum of S^d x S^d stabilizes degreewise as g tends to infinity. For diffeomorphisms, the stable cohomology has been calculated by Galatius and Randal-Williams. I will discuss recent results on the stable cohomology for homotopy equivalences and for block diffeomorphisms. Curiously, the calculation in these cases involves certain Lie algebras of symplectic derivations that have appeared before in Kontsevich's work on the cohomology of outer automorphisms of free groups.
GDOW02 2nd April 2013
15:00 to 16:00
D Roytenberg A Dold-Kan-type correspondence for superalgebras of differentiable functions and a "differential graded" approach to derived differential geometry
The commutative algebra appropriate for differential geometry is provided by the algebraic theory of $C^\infty$-algebras -- an enhancement of the theory of commutative algebras which admits all $C^\infty$ functions of $n$ variables (rather than just the polynomials) as its $n$-ary operations. Derived differential geometry requires a homotopy version of these algebras for its underlying commutative algebra. We present a model for the latter based on the notion of a "differential graded structure" on a superalgebra of differentiable functions, understood -- following Severa -- as a (co)action of the monoid of endomorphisms of the odd line. This view of a differential graded structure enables us to construct, in a conceptually transparent way, a Dold-Kan-type correspondence relating our approach with models based on simplicial $C^\infty$-algebras, generalizing a classical result of Quillen for commutative and Lie algebras. It may also shed new light on Dold-Kan-type co rrespondences in other contexts (e.g. operads and algebras over them). A similar differential graded approach exists for every geometry whose ground ring contains the rationals, such as real analytic or holomorphic.

This talk is partly based on joint work with David Carchedi (arXiv:1211.6134 and arXiv:1212.3745).

GDOW02 2nd April 2013
16:30 to 17:30
B Tsygan Operations on Hochschild complexes, revisited
Algebraic structures on Hochschild cochain and chain complexes of algebras are long known to be of considerable interest. On the one hand, some of them arise as standard operations in homological algebra. On the other hand, for the ring of functions on a manifold, Hochschild homology, resp. cohomology, is isomorphic to the space of differential forms, resp. multivector fields, and many algebraic structures from classical calculus can be defined (in a highly nontrivial way) in a more general setting of an arbitrary algebra and its Hochschild complexes. Despite an extensive amount of work on the subject, it appears that large parts of the algebraic structure on the Hochschild complexes of algebras and their tensor products are still unknown. I will review the known results (Kontsevich-Soibelman, Dolgushev, Tamarkin and myself, Keller) and then outline some provisional results and conjectures about new operations. The talk will not assume any prior knowledge beyond the basic facts about complexes and (co)homology.
GDOW02 3rd April 2013
09:30 to 10:30
Classical and quantum Lagrangian field theories on manifolds with boundaries
Classical and quantum field theories may be thought of as appropriate functors from (some version of) the cobordism category. At the quantum level this was proposed by Segal as an axiomatization. Incarnations of this exist for nonperturbative topological field theory (by Witten following Atiyah's version of the axioms for TFTs) as well as in one and two dimensional field theories. This talk (based on joint work with Mnev and Reshetikhin) will give an introduction to the classical version and to the Batalin-Vilkovisky version, which forms the starting point for the perturbative quantization. The possibility of including boundaries of boundaries (and so on) naturally yields to a Lurie-type description. Eventually, one might be able to reconstruct perturbative quantum theories on manifolds by gluing simple pieces together. Work in progress on this will be presented.
GDOW02 3rd April 2013
11:00 to 12:00
P Severa Integration of differential graded manifolds
I will explain why the Sullivan simplicial set given by a differential non-negatively graded manifold is actually a Kan simplicial manifold. The basic tool is an integral transformation that linearizes the corresponding PDE. By imposing a gauge condition, we can then locally find a finite-dimensional Kan submanifold, which can be seen as a local Lie n-groupoid integrating the dg manifold. These local n-groupoids are (non-uniquely) isomorphic on the overlaps. I will mention many open problems. Based on a joint work in progress with Michal Siran.
GDOW02 3rd April 2013
13:30 to 14:30
G Segal Bott periodicity in Algebra
Topological K-theory was based on the Bott periodicity theorem, which made it possible to define K-groups in all degrees. Quillen's algebraic K-theory made use of a quite different principle. Nevertheless, Bott periodicity, seemingly so topological, has a variety of purely algebraic manifestations, which shed light on the sense in which an algebraically-defined space has a homotopy type.
GDOW02 3rd April 2013
15:00 to 16:00
The Boardman-Vogt tensor product of operadic bimodules
(Joint work with Bill Dwyer.) The Boardman-Vogt tensor product of operads endows the category of operads with a symmetric monoidal structure that codifies interchanging algebraic structures. In this talk I will explain how to lift the Boardman-Vogt tensor product to the category of composition bimodules over operads. I will also sketch two geometric applications of the lifted B-V tensor product, to building models for spaces of long links and for configuration spaces in product manifolds.
GDOW02 3rd April 2013
16:30 to 17:30
Two models for infinity-operads
Following the foundational work of Joyal and Lurie, the theory of infinity-categories is now widely being used. There are two ways of extending the theory to "infinity-operads", one of them [CM] based on the theory of dendroidal sets, the other [HA] on the theory of cocartesian fibrations between infinity-categories, and ever since the appearance of the first versions of [HA] it was conjectured that the two theories are equivalent. In this lecture I will present an outline of a proof of this conjecture [HHM].

[CM] D.-C. Cisinski, I. Moerdijk, Dendroidal sets as models for homotopy operads, Journal of Topology 2011

[HA] J. Lurie, Higher Algebra, book available at http://www.math.harvard.edu/~lurie/

[HMM] G. Heuts, V. Hinich, I. Moerdijk, The equivalence of the dendroidal model and Lurie's model for infinity operads, in preparation.

GDOW02 4th April 2013
09:30 to 10:30
Symplectic and Poisson structures in the derived setting
This is a report on an ongoing project joint with T. Pantev, M. Vaquié and G. Vezzosi.

The purpose of this talk is to present the notions of n-shifted symplectic and n-shifted Poisson structures on derived algebraic stacks and to explain their relevance for the study of moduli spaces. I will start by the notion of n-shifted symplectic structures and present some existence results insuring that they exist in many examples. In a second part I will present the notion of n-shifted Poisson structures and explain its relation with the geometry of branes (maps from spheres to a fixed target). To finish I will explain what deformation quantization is in the n-shifted context.

GDOW02 4th April 2013
11:00 to 12:00
V Dotsenko Givental action is gauge symmetry in a homotopy Lie algebra
In this talk, I shall explain the mystery of hypercommutative algebras (=genus 0 cohomological field theories): it has been known for a while that the space of hypercommutative algebra structures on a given vector space possesses an unusually large group of symmetries. I shall explain how that appears naturally as a group of gauge symmetries of Maurer--Cartan elements in a certain homotopy Lie algebra. The existence of that homotopy algebra is still a result of several wonderful coincidences, but somehow I shall explain precisely what coincidences one should be hunting for in order for that sort of phenomena to occur.
GDOW02 4th April 2013
13:30 to 14:30
Computability and Zipf's Law: operadic perspective
The classical model of computability is the theory of partial recursive functions. Church's thesis postulates the "universality" of this model, and a vast corpus of other approaches confirms this thesis. Partial recursive functions is the minimal subset of partial functions containing a list of elementary functions and stable wrt another list of basic operations. One part of my talk is dedicated to the operad generated by basic operations, and possibly larger algebras over this operad formalizing also oracle assisted computations. Another part will deal with applications of computability and complexity to the creation of a mathematical model of Zipf's law: empirical probability measure observable on a vast amount of data, starting with distribution of words in texts.
GDOW02 4th April 2013
15:00 to 16:00
Strong homotopy (bi)algebras, homotopy coherent diagrams and derived deformations
Spaces of homotopy coherent diagrams or of strong homotopy (s.h.) algebras (for arbitrary monads) can be realised by right-deriving sets of diagrams or of algebras. This description involves a model category generalising Leinster's homotopy monoids.

For any monad on a simplicial category, s.h. algebras thus form a Segal space. A monad on a category of deformations then yields a derived deformation functor. There are similar statements for bialgebras, giving derived deformations of schemes or of Hopf algebras.

GDOW02 4th April 2013
16:30 to 17:30
Higher string topology
We given an algebraic and geometric approach to string topology operation parametrized by Sullivan diagrams, producing in particular non-trivial higher degree operations associated to families of surfaces of any genus or any number of boundary components.
GDOW02 5th April 2013
09:30 to 10:30
The nerve of a differential graded algebra
The nerve of a differential graded algebra is a quasicategory: in this talk, we explain what the quasi-iso-morphisms look like in this quasicategory. If the dg algebra A is a dg Banach algebra concentrated in dimensions i>-n , the nerve of A is a Lie n-stack, that is, a quasicategory enriched in Banach analytic spaces. We show that Kuranishi's approach yields a finite dimensional Lie n-stack parametrizing deformations of a complex of holomorphic vector bundles on a compact complex manifold of length n. This is joint work with Kai Behrend.
GDOW02 5th April 2013
11:00 to 12:00
Deformations of the En-operad
We show that the cohomology of the deformation complex of the En operad may be expressed through M. Kontsevich's graph cohomology.
GDOW02 5th April 2013
13:30 to 14:30
Moduli spaces of geometric structures on surfaces and modular operads
It is well-known that the moduli space of Riemann surfaces with nonempty boundary is homotopy equivalent to the moduli space of metric ribbon graphs. We generalize this to a large class of geometric structures on surfaces, including unoriented, Spin, r-spin, and principal G-bundles, etc. For any class of geometric structures that can be described in terms of sections a suitable sheaf of spaces on a surface, we define a moduli space and show that it is homotopy equivalent to a moduli space of graphs with appropriate decorations at the vertices. The construction rests on the contractibility of the arc complex and can be interpreted in terms of derived modular envelopes of cyclic operads.
GDOW02 5th April 2013
15:00 to 16:00
Gauge theory, loop groups, and string topology
In this talk I will describe a joint project with John Jones, in which we relate the gauge theory of a principal bundle G --> P --> M to the string topology spectrum of the principal bundle. This spectrum has, as its homology the homology of the corresponding adjoint bundle. In this study G can be any topological group. In the universal case when P is contractible, the adjoint bundle is equivalent to the free loop space, LM, and this spectrum realizes the original Chas-Sullivan string topology homological structure. One of our main results is to identify the group of units the string topology spectrum, and to relate it to the gauge group of the original bundle. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections, to the setting of principal fiberwise spectra over a manifold, and show how it allows us to do explicit calculations. We also show how, in the universal case, an action of a Lie group on a manifold yields a representation of the loop group on the string topology spectrum. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum of a principal bundle is the "linearization" of the gauge group.
GDOW03 8th April 2013
09:30 to 10:30
Divergence and super-divergence cocycles on the Grothendieck-Teichmueller Lie algebra
The Grothendieck-Teichmueller Lie algebra grt can be viewed as a Lie subalgebra of derivations of the free Lie algebra in two generators. We use this observation to define two cocycles: the divergence cocycle on grt and the super-divergence cocycle on its even part. The divergence cocycle serves to define the Kashiwara-Vergne Lie algebra which is conjecturally isomorphic to grt. The super-divergence cocycle plays a role in the Rouviere's theory of symmetric spaces, and it is conjectured to be an injective map on the even part of grt.
GDOW03 8th April 2013
11:00 to 12:00
Honorary MZVs and modular forms
$\sum_{a>b>c>d>e>0}(-1)^{b+d}/(a^3b^6c^3d^6e^3)$ is an alternating sum with weight 21 and depth 5, yet has a conjectural expression as a Q-linear combination of multiple zeta values (MZVs) that includes a sum of depth 7. The MZV Data Mine [arXiv:0907.2557] contains many other examples of "honorary MZVs", i.e. alternating sums that are reducible to MZVs, sometimes at the expense of an increase of depth by an even integer. This talk concerns the conjecture that the enumeration of MZVs that are not reducible to MZVs of lesser depth, yet are reducible to alternating sums of lesser depth, is generated by an enumeration of modular forms.
GDOW03 8th April 2013
13:30 to 14:30
Towards higher dimensional analogues of the torsor of Drinfeld's associators
The purpose of this talk is to give a description of the set of homotopy classes of formality quasi-isomorphisms for a Sullivan model of the Little n-discs operads, where we consider any n>1.

The Sullivan model of a topological operad combines a commutative dg-algebra structure, reflecting the rational homotopy of the spaces underlying the operad, and a cooperad structure, reflecting the composition structures of the operad. I will explain the definition of an obstruction spectral sequence for the formality of these Sullivan models of operads.

I will give a description of the obstruction spectral sequence associated to the little discs operads, and I will explain the connection with the definition of the Drinfeld associators.

GDOW03 8th April 2013
15:00 to 16:00
Multiple Dedekind Zeta Functions
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider them as generalizations of Euler's multiple zeta values to arbitrary number fields. Over imaginary quadratic fields MDZV capture in particular multiple Eisenstein series [ZGK]. We give an analogue of multiple Eisenstein series over real quadratic field. And an alternative definition of values of multiple Eisenstein-Kronecker series [G]. Each of them as a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation and can be written as infinite sums, and have an analytic continuation. Finally, we prove that the multiple residue of a multiple Dedekind zeta function at (1,...,1) is a period.
GDOW03 8th April 2013
16:30 to 17:30
Associators and representations of braids
>From their very construction 20 years ago as elements of a torsor for the Grothendieck-Teichmueller group, Drinfeld associators provide representations of the (usual) braid groups. The study of representation-theoretic consequences of this construction reverberate into questions on associators, and provide conjectural views on what to expect from generalizations of associators to other types of braid groups, where the symmetric group is replaced by more general reflection groups. This talk will review some of these questions as well as the state of the art in this area.
GDOW03 9th April 2013
09:30 to 10:30
Multiple zeta values and quasi-symmetric functions
In the past 15 years many insights about multiple zeta values (MZVs) have come from the observation that they are homomorphic images of elements of the Hopf algebra QSym of quasi-symmetric functions. I will discuss some aspects of homomorphisms from QSym to the reals related to MZVs, including connections with topology and recent results on MZVs of even arguments.
GDOW03 9th April 2013
11:00 to 12:00
Y Ohno On multiple zeta-star values
Multiple zeta-star values are defined with non-strict inequalities in the summation range while the multiple zeta values are defined with strict inequalities. To understand the structure of multiple zeta algebra, we study relations among multiple zeta-star values. In this talk, I will present some families of relations among multiple zeta-star values.
GDOW03 9th April 2013
13:30 to 14:30
Cyclotomic p-adic multi-zeta values
We will define the generalization of p-adic multi-zeta values to roots of unity and compute these values for depth less than or equal to 2. The method is to solve the fundamental differential equation satisfied by the crystalline frobenius using rigid analytic methods and to study the coefficients of its power series expansion.
GDOW03 9th April 2013
15:00 to 16:00
A shuffle product formula for generalized iterated integrals
In generalized iterated integrals, one can integrate complex powers of certain holomorphic 1-forms on Riemann surfaces. In this talk, I will present a shuffle product formula on such integrals. Applications will include expressions of Dedekind zeta functions of abelian number fields as series of certain polyzeta functions, as well as identities involving the Riemann zeta function.
GDOW03 9th April 2013
17:00 to 18:00
D Zagier RVP Lecture: From modular forms to finite groups
Modular forms occur in many different parts of number theory and other fields of mathematics. In this talk we will discuss two places where generalised modular forms appear in connection with the theory of finite groups. One is the occurrence of "quasimodular forms" in the representation theory of symmetric groups (and in related questions concerning coverings of Riemann surfaces), the other, the fairly recent discovery of an unsuspected connection between the theory of "mock modular forms" and the representation theory of one of the sporadic simple groups.
GDOW03 10th April 2013
09:00 to 10:00
R Hain Higher Genus Polylogarithms
Are there polylogarithms in higher genus? Classical polylogarithms are defined on P1-{0, 1,1}, which is the moduli spaceM0,4 of 4-pointed genus 0 curves. The elliptic polylogarithms of Beilinson and Levin are defined on M1,1, the moduli space of elliptic curves and on M1,2, the punctured universal elliptic curve over it. In this talk I will give a uniform definition of polylogarithms of all genera which specializes to these in genera 0 and 1. I will then explain that there are are countable many polylogarithms in genus 2 — though they appear to be less interesting than elliptic polylogarithms — and that, when g > 2, there are very few. The upside of this “rigidity” of higher genus moduli spaces is that one can construct a theory of characteristic classes of rational points of curves of genus g > 2.
GDOW03 10th April 2013
10:30 to 11:30
C Dupont The Hopf algebra of dissection polylogarithms
Grothendieck's theory of motives has given birth to a conjectural Galois theory for periods. Replacing the periods with their motivic avatars, one gets an algebra of motivic periods that are acted upon by a motivic Galois group. Recently, the computation of this action for multiple zeta values has been studied and used by Deligne, Goncharov and Brown among others. In this talk we will introduce a family of periods indexed by some combinatorial objects called dissection diagrams, and compute the action of the motivic Galois group on their motivic avatars. This generalizes the case of (generic) iterated integrals on the punctured complex plane. We will show that the motivic action is given by a very simple combinatorial Hopf algebra.
GDOW03 10th April 2013
11:45 to 12:45
Periods of modular forms and relations in the fundamental Lie algebra of Universal Mixed Elliptic Motives
Hain and Matsumoto have defined a category of so-called universal mixed elliptic motives, universal in the sense that such objects should be thought of as living over the moduli of all elliptic curves. They have shown that this category is neutral Tannakian. An interesting question then is understand explicitly the fundamental Lie algebra of this category. We make some progress in this direction, by proving a result about relations between a minimal set of generators for this Lie algebra. In particular, we find that periods of modular forms are closely connected to these relations. This work is closely related to older work of Schneps, and it also appears that there may be some connection to work of Gangl-Kaneko-Zagier.
GDOW03 11th April 2013
09:30 to 10:30
Anatomy of the motivic Lie algebra
The motivic Lie algebra is contained in the Grothendieck-Teichmuller Lie algebra, and is isomorphic to the free graded Lie algebra with one generator in every odd degree >1. Using motivic MZV's one can define canonical generators for this algebra, but their arithmetic properties are very mysterious.

In this talk, I will explain how elements of the motivic Lie algebra admit a kind of Taylor expansion with a rich internal structure. This is closely connected with the theory of modular forms, universal elliptic motives, and some other unexpected algebraic objects.

GDOW03 11th April 2013
11:00 to 12:00
Motivic Superstring Amplitudes
We review the recent advances in open superstring N-point tree-level computations. One basic ingredient is a basis of (N-3)! generalized Gaussian hypergeometric functions (generalized Euler integrals) encoding all string effects through their dependence on the string tension $\alpha'$. The structure of the open superstring amplitudes is analyzed. We find a striking and elegant form, which allows to disentangle its $\alpha'$ power series expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the $\alpha'$-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the $\alpha'$-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a transcendental function. In terms of elements of this Hopf algebra the alpha'-expansion assumes a very simple and symmetric form, which carries all the relevant information.
GDOW03 11th April 2013
13:30 to 14:30
O Schnetz Proof of the zig-zag conjecture
In quantum field theory primitive Feynman graphs give--via the period map--rise to renormalization scheme independent contribution to the beta function. While the periods of many Feynman graphs are multiple zeta values there exists the distinguished family of zig-zag graphs whose periods were conjectured in 1995 by Broadhurst and Kreimer to be certain rational multiples of odd single zetas.

In joint work with F. Brown it was possible in 2012 to prove the zig-zag conjecture using the theory of graphical functions, single valued multiple polylogarithms and a theorem by D. Zagier on multiple zeta values of the form zeta(2,...,2,3,2,...,2).

GDOW03 11th April 2013
15:00 to 16:00
Polylogarithms, Multiple Zeta Values and Superstring Amplitudes
Superstring Theory is one of the most promising candidates for a quantum theory of gravity. Its interactions involve all kinds of multiple zeta values, and I will present how they emerge at the leading order (so-called tree level) of open string scattering amplitudes. The expansion of the underlying generalized Gaussian hypergeometric functions in the string tension gives rise to iterated integrals within the [0,1] interval. I will discuss polylogarithm manipulations required for their systematic reduction to multiple zeta values.
GDOW03 12th April 2013
09:30 to 10:30
LC Schneps Elliptic Grothendieck-Teichmueller theory
I'll talk about the definition by Benjamin Enriquez of an elliptic (genus one) version of the Grothendieck-Teichmueller Lie algebra grt. I will compare it to the work of Hain, Matsumoto and Pollack on the fundamental Lie algebra of the category of elliptic mixed Tate motives. I will define conjectural analogous elliptic double shuffle and Kashiwara-Vergne Lie algebras. Finally, I will discuss a structure conjecture that emerges from work of Pollack, and prove the first case of it.
GDOW03 12th April 2013
11:00 to 12:00
Grothendieck-Teichmuller Groups in the Combinatorial Anabelian Geometry
By a result of Harbater and Schneps, the Grothendieck-Teichmuller groups may be regarded as natural objects in the study of the combinatorial anabelian geometry. In this talk, we discuss some results on the Grothendieck-Teichmuller groups that relate to the phenomenon of the tripod synchronization. In particular, I explain the surjectivity of the tripod homomorphism and a non-surjectivity result on the combinatorial cuspidalization.
GDOW03 12th April 2013
13:30 to 14:30
On multiple L-functions and their p-adic analogues
This is based on my current joint work with Y.Komori, K.Matsumoto and H.Tsumura. The first half of my talk is to review and discuss basic properties of complex multiple L-functions and those of their special values at integer points. Based on them, in the second half, I will explain our construction of p-adic multiple L-functions which generalizes that of Kubota-Leopoldt's p-adic L-function and then will show their several basic properties particularly on their special values at both positive and negative integer points.
GDO 18th April 2013
14:00 to 16:00
Finite multiple zeta values
GDO 18th April 2013
16:00 to 18:00
B Enriquez A lower bound of a sub-quotient of the Lie algebra associated to Grothendieck-Teichmüller group
We show that the filtration given by the central descending series of the commutator of the free Lie algebra on two generators x,y induces by a filtration of the graded Lie algebra grt_1 associated to the Grothendieck-Teichmüller group. The degree 0 part of the associated graded space has already been computed (by the collaborator of the author). We get here a lower bound for the degree 1 part; more precisely, this graded space splits into a sum of homogeneous components, on which we get a filtration and we give a lower bound for the dimensions of each sub-quotient.

The proof uses the construction of a vector space included in certain Lie sub-algebras of extensions between abelian Lie algebras, and reduces the problem to a question of commutative algebras, which is treated with invariant theory and results of Ihara, Takao, and Schneps on the quadratic relations between elements of the degree 1 part associated to grt_1 for the depth filtration (corresponding to the y-degree). As a corollary, we give another proof of a statement of Ecalle describing the sub-space of the degree 2 part of the same graded space.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons