Mar 27, 2017
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Supplements
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09:30 AM - 10:30 AM
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Galois theory of period and the André-Oort conjecture
Yves Andre (Centre National de la Recherche Scientifique (CNRS))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Motivic Gamma Functions and recursion
Spencer Bloch (Retired)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Elliptic multiple zeta values and periods
Nils Matthes (Max-Planck-Institut für Mathematik)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Tea Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Period Polynomial Relations among Double Zeta Values and Various Generalizations
Ding Ma (Duke University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In this talk, I will introduce the famous result by Gangl-Kaneko-Zagier 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 Eichler-Shimura-Manin correspondence to the case of the space of newforms of level 2 and 3 and a certain period polynomial space
- Supplements
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Mar 28, 2017
Tuesday
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09:30 AM - 10:30 AM
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Why you should care about motives
Annette Huber-Klawitter (Albert-Ludwigs-Universität Freiburg)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 pro-algebraic 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Outer space, symplectic derivations of free Lie algebras and modular forms
Karen Vogtmann (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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A Galois theory of exponential periods
Javier Fresán (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 so-called “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.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Galois actions on operads
Geoffroy Horel (Université de Paris XIII (Paris-Nord))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The Grothendieck-Teichmüller 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
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Supplements
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Mar 29, 2017
Wednesday
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09:00 AM - 10:00 AM
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The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem
Anton Alekseev (Université de Genève)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
It is conjectured that several graded Lie algebras coming up in different fields of mathematics coincide: the Grothendieck-Teichmueller 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 Kashiwara-Vergne Lie algebra kv in Lie theory. We are adding one more piece to this puzzle: it turns out that the Kashiwara-Vergne Lie algebra plays an important role in the Goldman-Turaev theory defined in terms of intersections and self-intersections of curves on 2-manifolds. This allows to define the Kashiwara-Vergne 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
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10:00 AM - 10:30 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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A stabilizer interpretation de double shuffle Lie algebras
Benjamin Enriquez (Université de Strasbourg)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 Grothendieck-Teichmüller 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
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11:30 AM - 12:30 PM
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the operad structure of $\overline{M_{0,n+1}}({\mathbb{R}})$
Anton Khoroshkin (Higher School of Economics)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 Eilenberg-Maclein space of the pure braid groups with $n$ braids. Correspondingly, the real locus of the Deligne-Mumford compactification of the moduli space of stable rational curves with marked points assemble an operad of the Eilenberg-Maclein 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 Grothendieck-Teichmuller Lie algebra
- Supplements
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Mar 30, 2017
Thursday
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09:30 AM - 10:30 AM
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Quotients of Kontsevich's "Lie" Lie algebra
Jim Conant (University of Tennessee)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 so-called "Johnson cokernel," the cokernel of the Johnson homomorphism defined for mapping class groups of punctured surfaces
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Motivic Euler numbers and an arithmetic count of the lines on a cubic surface.
Kirsten Wickelgren (Duke University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 A1-homotopy theory: we define an Euler number in the Grothendieck-Witt 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
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Galois theory for motivic cyclotomic multiple zeta values
Claire Glanois (Max-Planck-Institut für Mathematik)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Motives arising from higher homotopy theory of hyperplane arrangements.
Deepam Patel (Purdue University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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05:00 PM - 06:00 PM
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Motives and derivations of free Lie algebras
Richard Hain (Duke University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Supplements
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Mar 31, 2017
Friday
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09:30 AM - 10:30 AM
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Motivic Galois groups, following Ayoub and Nori
Martin Gallauer (University of California, Los Angeles)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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p-adic periods via perfectoid spaces
Kiran Kedlaya (University of California, San Diego)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 p-adic analogue the comparison between etale, crystalline, and de Rham cohomology of algebraic varieties. We describe some new results and perspectives on p-adic 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 Bhatt-Morrow-Scholze), and the discovery of "abstract instances" of comparison isomorphisms corresponding to as-yet-unknown families of motives over Shimura varieties (by Liu-Zhu).
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Iterated p-adic integrals and rational points on curves
Jennifer Balakrishnan (Boston University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will discuss some new relationships between iterated p-adic 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
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Tea
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- Location
- SLMath: Eisenbud Auditorium
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- Supplements
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03:30 PM - 04:30 PM
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A Galois theory of supercongruences
Julian Rosen (University of Michigan)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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.
- Supplements
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