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We present a novel theoretical framework for statistical analysis of Large-scale problems that builds on the Robbins compound decision approach. We present a hierarchical Bayesian approach for implementing this framework and illustrate its application to simulated data.

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On a complex manifold, complex flows induce complete holomorphic vector fields. However, only very seldomly a vector field integrates into a flow. In general, it is difficult to say whether a holomorphic vector field on a non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a vector field to be complete. This opened the door for a local study of complete holomorphic vector fields on complex manifolds. In this series of talks we will explore some of these results.
What I will talk about is for the greater part contained or summarized in the articles in the bibliography. Their introductions might be useful as a first reading on the subject.
On a complex manifold, complex flows induce complete holomorphic vector fields. However, only very seldomly a vector field integrates into a flow. In general, it is difficult to say whether a holomorphic vector field on a non-compact manifold is complete or not (vector fields on compact manifolds are always complete). Some twenty-five years ago, Rebelo realized an exploited the fact that there are local (and not just global!) obstructions for a ...

34M05 ; 57S20 ; 32C99

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The Cremona group is the group of birational transformations of the projective plane. Even if this group comes from algebraic geometry, tools from geometric group theory have been powerful to study it. In this talk, based on a joint work with Christian Urech, we will build a natural action of the Cremona group on a CAT(0) cube complex. We will then explain how we can obtain new and old group theoretical and dynamical results on the Cremona group.

14E07 ; 20F65 ; 20F67

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Virtualconference  Effectively generating RAAGs in MCGs
Runnels, Ian (Auteur de la Conférence) | CIRM (Editeur )

Given a (mostly arbitrary) collection of mapping classes on a surface S, we find an explicit constant N such that their Nth powers generate a right-angled Artin subgroup of the mapping class group MCG(S).

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Parabolic subgroups are the fundamental building blocks of Artin groups. These subgroups are isomorphic copies of smaller Artin groups nested inside a given Artin group. In this talk, I will discuss questions surrounding how parabolic subgroups sit inside Artin groups and how they interact with each other. I will show that, in an FC type Artin group, the intersection of two finite type parabolic subgroups is a parabolic subgroup. I will also discuss how parabolic subgroups might be used to construct a simplicial complex for Artin groups similar to the curve complex for mapping class groups. This talk will focus on using geometric techniques to generalize results known for finite type Artin groups to Artin groups of FC type.
Parabolic subgroups are the fundamental building blocks of Artin groups. These subgroups are isomorphic copies of smaller Artin groups nested inside a given Artin group. In this talk, I will discuss questions surrounding how parabolic subgroups sit inside Artin groups and how they interact with each other. I will show that, in an FC type Artin group, the intersection of two finite type parabolic subgroups is a parabolic subgroup. I will also ...

20F65 ; 20F36

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The famous Hanna Neumann Conjecture (now the Friedman--Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to 'quantify' this bound with respect to the rank of the join of H and K, the subgroup generated by H and K. In this talk I describe what is known about the set of realizable values (rank of join, rank of intersection) for arbitrary H, K, and about my recent results in this direction. In particular, we resolve the remaining open case (m=4) of Guzman’s `Group-Theoretic Conjecture’ in the affirmative. This has some interesting corollaries for the geometry of hyperbolic 3-manifolds. Our methods rely on recasting the topological pushout of core graphs in terms of the Dicks graphs introduced in the context of his Amalgamated Graph Conjecture. This allows to translate the question of existence of a pair of subgroups H,K with prescribed ranks of joins and intersections into graph theoretic language, and completely resolve it in some cases. In particular, we completely describe the locus of realizable values of ranks in the case when the rank of one of the subgroups H,K equals two.
The famous Hanna Neumann Conjecture (now the Friedman--Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to 'quantify' this bound with respect to the rank of the join of H and K, the subgroup generated by H and K. In this talk I describe what is known about the set of realizable values (rank of join, rank of intersection) for arbitrary ...

20E05 ; 20E07 ; 20F65 ; 57M07

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A well-known result of Davis-Januszkiewicz is that every right-angled Artin group (RAAG) is commensurable to some rightangled Coxeter group (RACG). In this talk we consider the converse question: which RACGs are commensurable to some RAAG? To do so, we investigate some natural candidate RAAG subgroups of RACGs and characterize when such subgroups are indeed RAAGs. As an application, we show that a 2-dimensional, one-ended RACG with planar defining graph is quasiisometric to a RAAG if and only if it is commensurable to a RAAG. This talk is based on work joint with Pallavi Dani.
A well-known result of Davis-Januszkiewicz is that every right-angled Artin group (RAAG) is commensurable to some rightangled Coxeter group (RACG). In this talk we consider the converse question: which RACGs are commensurable to some RAAG? To do so, we investigate some natural candidate RAAG subgroups of RACGs and characterize when such subgroups are indeed RAAGs. As an application, we show that a 2-dimensional, one-ended RACG with planar ...

20F65 ; 57M07 ; 20F55

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We will discuss an analogy between the structure of fibrings of 3-manifolds and free-by-cyclic groups; we will focus on effective computabilility. This is joint work with Giles Gardam.

20F65 ; 20E36

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A spin structure on a closed surface $S$ of genus $g \geq 2$ is a covering of the unit tangent bundle of $S$ witch restricts to a standard covering of the fiber. Such a spin structure has a parity, even or add. The spin mapping class is the stabilizer of such a spin structure in the mapping class group of $S$. We use a subgraph of the curve graph to construct an explicit generating set of the spin mapping class group consisting of Dehn twists about a system of $2g-1$ simple closed curves.
A spin structure on a closed surface $S$ of genus $g \geq 2$ is a covering of the unit tangent bundle of $S$ witch restricts to a standard covering of the fiber. Such a spin structure has a parity, even or add. The spin mapping class is the stabilizer of such a spin structure in the mapping class group of $S$. We use a subgraph of the curve graph to construct an explicit generating set of the spin mapping class group consisting of Dehn twists ...

20F65 ; 20F34 ; 20F28

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The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.
The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry ...

20F65 ; 20F67 ; 20E06 ; 57M07 ; 57M10

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One of the main examples of Artin groups are braid groups. We can use powerful topological methods on braid groups that come from the action of braid on the curve complex of the n-puctured disk. However, these methods cannot be applied in general to Artin groups. In this talk we explain how we can construct a complex for Artin groups, which is an analogue to the curve complex in the braid case, by using parabolic subgroups.

20F36 ; 20F65 ; 57M07

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The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered.
In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by discussing ongoing work in the area.
This talk contains some joint work with C.Abbott, D.Osin and A.Rasmussen.
The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered.
In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by ...

20F65 ; 20F67 ; 20E08

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Virtualconference  Shortcut graphs and groups
Hoda, Nima (Auteur de la Conférence) | CIRM (Editeur )

Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), cocompactly cubulated groups, hyperbolic groups, Coxeter groups and the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. I will discuss some of these examples as well as some general constructions and properties of shortcut graphs and groups.
Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), cocompactly cubulated ...

20F65 ; 20F67 ; 05C12

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A group is said to be coherent if every finitely generated subgroup is finitely presented. This property is enjoyed by free groups, and the fundamental groups of surfaces and 3-manifolds. A group that is not coherent is incoherent, and it is very interesting to try and understand which groups are coherent. We will discuss some of the geometric and topological aspects of this question, particularly quasi-convexité and algebraic fibers. We show that free-by-free and surface-by-free groups are incoherent, when the rank and genus are at least 2. The proof uses an understanding of fibers and also the RFRS property. this is joint work with Robert Kropholler and Stefano Vidussi.
A group is said to be coherent if every finitely generated subgroup is finitely presented. This property is enjoyed by free groups, and the fundamental groups of surfaces and 3-manifolds. A group that is not coherent is incoherent, and it is very interesting to try and understand which groups are coherent. We will discuss some of the geometric and topological aspects of this question, particularly quasi-convexité and algebraic fibers. We show ...

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Virtualconference  Dynamics of Jouanolou foliation
Deroin, Bertrand (Auteur de la Conférence) | CIRM (Editeur )

I will report on some joint work with Aurélien Alvarez, which shows that the Jouanolou foliation in degree two is structurally stable, and that it has a non-trivial domain of discontinuity. This result is opposed to a series of results beginning in the sixties with the works of Hudai-Verenov and Ilyashenko.

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I will describe a remarkable family of higher dimensional foliations generalizing the equations studied by Darboux, Halphen, Ramanujan, and many others, and discuss some related geometric problems motivated by number theory.

14D23 ; 14K99 ; 37F75 ; 11J81 ; 11G18

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We consider a holomorphic (singular) foliation F on a projective manifold X and a group G of birational transformations of X which preserve F (i.e. it permutes the set of leaves). We say that the transverse action of G is finite if some finite index subgroup of G fixes each leaf of F.In a joint work with J. V. Pereira, E. Rousseau and F. Touzet, we show finiteness for the group of birational transformations of general type foliations with tame singularities and transverse finiteness for (non-virtually euclidean) transversely projective foliations. In this talk I will focus on the latter result, time permitting, I will show how the presence of a transverse structure (projective, hyperbolic, spherical...) and the analysis of the resulting monodromy representation allow to reduce to the case of modular foliations on Shimura varieties and to conclude.
We consider a holomorphic (singular) foliation F on a projective manifold X and a group G of birational transformations of X which preserve F (i.e. it permutes the set of leaves). We say that the transverse action of G is finite if some finite index subgroup of G fixes each leaf of F.In a joint work with J. V. Pereira, E. Rousseau and F. Touzet, we show finiteness for the group of birational transformations of general type foliations with tame ...

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To any algebraic differential equation, one can associate a first-order structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions.To describe the structure associated to a given algebraic (nonlinear) differential equation (E), typical questions are:- Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations?- Is it possible to express the general solutions of (E) from successive resolutions of algebraic differential equations of lower order than (E)?- Given distinct initial conditions for (E), under which conditions are the solutions associated to these initial conditions algebraically independent?In my talk, I will discuss in this setting one of the first examples of non-completely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative model-theoretic description of the associated structure based on the global hyperbolic dynamical properties identified by Anosov in the 70’s for the geodesic motion in negative curvature.
To any algebraic differential equation, one can associate a first-order structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions.To describe the structure associated to a given algebraic (nonlinear) differential equation (E), typical questions are:- Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations?- Is it possible to ...

12H05 ; 37D40 ; 53D25 ; 53C22

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It is thought that the classifications and constructions of holomorphic Poisson structures are worth studying. The classification when the Picard rank is 2 or higher is unknown. In this talk, we will introduce the classification of holomorphic Poisson structures with the reduced degeneracy divisor that have only simple normal crossing singularities, on the product of Fano variety of Picard rank 1. This claims that such a Poisson manifold must be a diagonal Poisson structure on the product of projective spaces, so this is a generalization of Lima and Pereira's study. The talk will also include various examples, classifications, and problems of high-dimensional holomorphic Poisson structures.
It is thought that the classifications and constructions of holomorphic Poisson structures are worth studying. The classification when the Picard rank is 2 or higher is unknown. In this talk, we will introduce the classification of holomorphic Poisson structures with the reduced degeneracy divisor that have only simple normal crossing singularities, on the product of Fano variety of Picard rank 1. This claims that such a Poisson manifold must be ...

53D17 ; 14J45 ; 14C17

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By the work of Brunella and McQuillan, it is known that smooth foliated surfaces of general type with only canonical singularities admit a unique canonical model. It is then natural to wonder if these canonical models have a good moduli theory and, in particular, if they admit a moduli functor.In this talk, I will show that the canonical models and their minimal partial du Val resolutions are bounded.

14C20 ; 14E99 ; 32M25

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