Mar 27, 2017
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
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 Abstract
 
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09:30 AM  10:30 AM


Galois theory of period and the AndréOort conjecture
Yves Andre (Centre National de la Recherche Scientifique (CNRS))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The idea of a Galois theory of periods comes from an insight of Grothendieck. I shall briefly outline of this (conjectural) theory, then sketch the path which led me from it to the AO conjecture, as well as some paths backward. Principally polarized abelian varieties of dimension g are parametrized by the algebraic variety A_g, those with prescribed extra "symmetries" by special subvarieties of A_g, and those with maximal symmetry (complex multiplication) by special points. The AO conjecture characterizes special subvarieties of A_g by the density of their special points. It has been proven last year, after two decades of collaborative efforts putting together many different areas
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


Motivic Gamma Functions and recursion
Spencer Bloch (Retired)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss a program of V. Golyshev and collaborators to develop a theory of motivic gamma functions. These are Mellin transforms of period integrals. They satisfy the same recursive relations as solutions of the Picard Fuchs equation.
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12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
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02:00 PM  03:00 PM


Elliptic multiple zeta values and periods
Nils Matthes (MaxPlanckInstitut für Mathematik)

 Location
 MSRI: Simons Auditorium
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03:00 PM  03:30 PM


Tea Break

 Location
 MSRI: Atrium
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03:30 PM  04:30 PM


Period Polynomial Relations among Double Zeta Values and Various Generalizations
Ding Ma (Duke University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk, I will introduce the famous result by GanglKanekoZagier about a family of period polynomial relations among double zeta value of even weight. Then I will generalize their result in various ways, from which we can see the appearance of periods of newforms in low levels. At the end, I will give a generalization of the EichlerShimuraManin correspondence to the case of the space of newforms of level 2 and 3 and a certain period polynomial space
 Supplements



Mar 28, 2017
Tuesday

09:30 AM  10:30 AM


Why you should care about motives
Annette HuberKlawitter (AlbertLudwigsUniversität Freiburg)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
Period numbers are complex numbers defined as values of integrals over rational algebraic differential forms over domains of intergration satisfying certain algebraicity conditions, again over the rational or algebraic numbers. Theere are several definitions in the literature, but luckily they all give the same set, even a countable algebra containing all algebraic numbers. We take it as given that periods are very interesting. In this expository talk we want to explain how the language of motives is intrinsically related to periods. They are best understood as entries of the comparison matrices between singular cohomology and algebraic de Rham cohomology not only of algebraic varities, but more generally of motives. This point of view gives a lot of structure to the period algebra. The category of motives itself is characterised as representations of a proalgebraic group, the motivic Galois group. This generalises the ordinary Galois group. At least conjecturally it is related to the period algebra in the same way as the Galois group is related to the field of algebraic numbers. Along the way we will give an introduction to Nori's definition of the abelian category of motives
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


Outer space, symplectic derivations of free Lie algebras and modular forms
Karen Vogtmann (University of Warwick)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk I will describe the connection, discovered by Kontsevich, between symplectic derivations of a free Lie algebra and the “symmetric space” for the group Out(F_n) of outer automorphisms of a free group. The latter is known as Outer space, and can be described as a space of free actions of F_n on metric simplicial trees. The fact that the quotients of such actions are finite graphs leads to a combinatorial understanding of this space which can be used to gain cohomological information about both the group Out(F_n) and the Lie algebra of symplectic derivations. One surprising outcome is a way of constructing cohomology classes from classical modular forms, as described in joint work with Conant and Kassabov. No prior knowledge of Outer space or Kontsevich’s theorem will be assumed
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


A Galois theory of exponential periods
Javier Fresán (ETH Zürich)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Exponential periods form a class of complex numbers containing the special values of the gamma and the Bessel functions, the Euler constant and other interesting numbers which are not expected to be periods in the usual sense. However, they appear as coefficients of the comparison isomorphism between two cohomology theories associated to varieties with a regular function: the de Rham cohomology of a connection with irregular singularities and the socalled “rapid decay” cohomology. I will explain how this point of view allows one to construct a Tannakian category of exponential motives and to produce Galois groups which conjecturally govern all algebraic relations among these numbers. The focus will be on examples and open questions rather than on the more abstracts aspects of the theory. This is a joint work with Peter Jossen.
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03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
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 Abstract
 
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03:30 PM  04:30 PM


Galois actions on operads
Geoffroy Horel (Université de Paris XIII (ParisNord))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The GrothendieckTeichmüller group is a profinite group that contains the absolute Galois group of the rational numbers and is conjecturally isomorphic to it. In this talk I will explain how one can understand this group using the homotopy theory of operads. This is joint work with Pedro Boavida de Brito and Marcy Robertson
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
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 Abstract
 
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Mar 29, 2017
Wednesday

09:00 AM  10:00 AM


The GoldmanTuraev Lie bialgebra and the KashiwaraVergne problem
Anton Alekseev (Université de Genève)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
It is conjectured that several graded Lie algebras coming up in different fields of mathematics coincide: the GrothendieckTeichmueller Lie algebra grt related to the braid group in 3d topology, the double shuffle Lie algebra ds in the theory of multiple zeta values and the KashiwaraVergne Lie algebra kv in Lie theory. We are adding one more piece to this puzzle: it turns out that the KashiwaraVergne Lie algebra plays an important role in the GoldmanTuraev theory defined in terms of intersections and selfintersections of curves on 2manifolds. This allows to define the KashiwaraVergne problem for surfaces of arbitrary genus. In particular, we focus on the genus one case and discuss the relation between elliptic kv and elliptic grt. The talk is based on a joint work with N. Kawazumi, Y. Kuno and F. Naef
 Supplements


10:00 AM  10:30 AM


Break

 Location
 MSRI: Atrium
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 Abstract
 
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10:30 AM  11:30 AM


A stabilizer interpretation de double shuffle Lie algebras
Benjamin Enriquez (Université de Strasbourg)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We recall the main results of double shuffle theory: the cyclotomicanalogues of MZVs (of order $N\geq 1$) satisfy a collection of relations arising from the study of their combinatorics, and also from their identifications with periods. The scheme arising from these relations is a torsor Under a prounipotent algebraic group $\mathrm{DMR}_0$. This is a subgroup of the group $\mathrm{Out}^*$ of invariant tangential outer automorphisms of a free Lie algebra, equipped with an action of $\mu_N$. The Lie algebra $\mathfrak{dmr}_0$ of $\mathrm{DMR}_0$ is a subspace of the Lie algebra $\mathrm{out}^*$, defined by a pair of shuffle relations (Racinet) and containing the GrothendieckTeichmüller Lie algebra or its analogues(Furusho). We show that the harmonic coproduct may be viewed as an element of a module over $\mathrm{out}^*$, and that $\mathfrak{dmr}_0$ then identifies with the stabilizer Lie algebra of this element. A similar identification concerning $\mathrm{DMR}_0$ enables one to construct a "Betti" version of the harmonic coproduct, and to identify the scheme arising from double shuffle relations as the set of elements of $\mathrm{Out}^*$ taking the harmonic coproduct to its "Betti" version
 Supplements


11:30 AM  12:30 PM


the operad structure of $\overline{M_{0,n+1}}({\mathbb{R}})$
Anton Khoroshkin (Higher School of Economics)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
The category of representations over a quantum group $U_q(g)$ form a braided tensor category that produces an action of the (pure) braid groups on tensor products. Respectively, the category of crystals (which is a limit for q tends to zero) form a coboundary category together with an action of (pure) cacti group on tensor products. The little discs operad is an operad whose space of $n$ary operations is the EilenbergMaclein space of the pure braid groups with $n$ braids. Correspondingly, the real locus of the DeligneMumford compactification of the moduli space of stable rational curves with marked points assemble an operad of the EilenbergMaclein spaces of pure cacti groups. I will present the detailed description of the latter operad as well as its deformation theory and relationships with the little discs operad, graph complexes and GrothendieckTeichmuller Lie algebra
 Supplements



Mar 30, 2017
Thursday

09:30 AM  10:30 AM


Quotients of Kontsevich's "Lie" Lie algebra
Jim Conant (University of Tennessee)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We study two quotients of the Lie Lie algebra (the Lie algebra of symplectic derivations of the free Lie algebra), namely the abelianization and the the quotient by the Lie algebra generated by degree 1 elements. The abelianization has a very close connection to the homology of groups of automorphism groups of free groups, whereas the second is the socalled "Johnson cokernel," the cokernel of the Johnson homomorphism defined for mapping class groups of punctured surfaces
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


Motivic Euler numbers and an arithmetic count of the lines on a cubic surface.
Kirsten Wickelgren (Georgia Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines depends on the surface, but Segre showed that a certain signed count is always 3. We extend this count to an arbitrary field using A1homotopy theory: we define an Euler number in the GrothendieckWitt group for a relatively oriented algebraic vector bundle as a sum of local degrees, and then generalize the count of lines to a cubic surface over an arbitrary field. This is joint work with Jesse Leo Kass
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Galois theory for motivic cyclotomic multiple zeta values
Claire Glanois (MaxPlanckInstitut für Mathematik)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Cyclotomic multiple zeta values (CMZV), are an interesting first bunch of examples of periods and a fruitful recent approach is to look at their motivic version (MCMZV), which are motivic periods of the fundamental groupoid of ℙ1 ∖ {0, μN, ∞}. Notably, MCMZV have a Hopf comodule structure, dual of the action of the motivic Galois group on these specific motivic periods; the explicit combinatorial formula of the coaction (Goncharov, Brown) enables, via the period map (isomorphism under Grothendieck’s period conjecture), to deduce results on CMZV. We will here highlight how to apply some Galois descents ideas to the study of these motivic periods and look at how periods of the fundamental groupoid of ℙ1 ∖ {0, μN', ∞} are embedded into periods of π1(ℙ1 ∖ {0, μN, ∞}), when N′  N, via a few examples
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
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 Abstract
 
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03:30 PM  04:30 PM


Motives arising from higher homotopy theory of hyperplane arrangements.
Deepam Patel (Purdue University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will discuss a construction of motives arising from higher homotopy groups of hyperplane arrangements in the setting of Nori's category of motives
 Supplements


05:00 PM  06:00 PM


Motives and derivations of free Lie algebras
Richard Hain (Duke University)

 Location
 MSRI: Simons Auditorium
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 Abstract
 
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Mar 31, 2017
Friday

09:30 AM  10:30 AM


Motivic Galois groups, following Ayoub and Nori
Martin Gallauer (University of California, Los Angeles)

 Location
 MSRI: Simons Auditorium
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 Abstract
 
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10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


padic periods via perfectoid spaces
Kiran Kedlaya (University of California, San Diego)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Given the interpretation of classical periods as matrix coefficients arising in the comparison between singular and de Rham cohomology of complex algebraic varieties, it is natural to view as a padic analogue the comparison between etale, crystalline, and de Rham cohomology of algebraic varieties. We describe some new results and perspectives on padic comparison isomorphisms emerging from recent developments in the theory of perfectoid spaces. These include a new direct cohomological realization of the crystalline comparison isomorphism (by BhattMorrowScholze), and the discovery of "abstract instances" of comparison isomorphisms corresponding to asyetunknown families of motives over Shimura varieties (by LiuZhu).
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Iterated padic integrals and rational points on curves
Jennifer Balakrishnan (Boston University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss some new relationships between iterated padic line integrals (Coleman integrals), motivated by the problem of explicitly finding rational points on curves. This is joint work with Netan Dogra
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03:00 PM  03:30 PM


Tea

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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03:30 PM  04:30 PM


A Galois theory of supercongruences
Julian Rosen (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A supercongruence is a congruence between rational numbers modulo a power of a prime. Many supercongruences are known for rational approximations of periods, and in particular for finite truncations of the multiple zeta value series. In this talk, I will explain how the Galois theory of multiple zeta values leads to a Galois theory of supercongruences.
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