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Topology 175 results

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Khovanov-Seidel braids representation - Queffelec, Hoel (Author of the conference) | CIRM H

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Khovanov and Seidel introduced in the early 2000's an action of the braid group by autoequi-valences on the homotopy category of projective modules over the zig-zag algebra. This categorical action descends to the Burau representation, one of the most famous braid representations, but unlike the classical story, the lifting is faithful. It is interesting to notice that simultaneously, the Burau representation was also extended into a faithful finite-dimensional linear representation by Lawrence, Krammer and Bigelow, proving the linearity of the braid group.
I will review the basic constructions, both at the level of vector representations and at the ca-tegorical level. We will discuss possible extensions of these from classical braids (type A) to larger Artin-Tits groups, spherical or not, and try to relate Khovanov-Seidel's construction to Soergel bimodules and categorified quantum groups. I will also try to emphasize several metric aspects that appear in an elegant way from the categorical setting, with an emphasis on Bridgeland's stability conditions. Along the way, I would like to list several open questions and problems that I care about, hoping that someone in the audience will come up with a good idea.[-]
Khovanov and Seidel introduced in the early 2000's an action of the braid group by autoequi-valences on the homotopy category of projective modules over the zig-zag algebra. This categorical action descends to the Burau representation, one of the most famous braid representations, but unlike the classical story, the lifting is faithful. It is interesting to notice that simultaneously, the Burau representation was also extended into a faithful ...[+]

20F36 ; 18G35 ; 20F65

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Diagram groups and their geometry - lecture 1 - Skipper, Rachel (Author of the conference) | CIRM H

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In these talks, we will discuss a family of groups called diagram groups, studied extensively by Guba and Sapir and others. These depend on semigroup presentations and turn out to have many good algorithmic properties. The first lecture will be a survey of diagram groups, including several examples and gen-eralizations. The second lecture will take a geometric approach, understanding these groups through median-like geometry.

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Stone duality and its formalization - van Gool, Sam (Author of the conference) | CIRM H

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This talk has a dual aim: to provide a mathematical overview of Stone duality theory, and to invite collaboration on its Lean formalization.
Stone duality is an algebraic way of looking at profinite topologies. A profinite set is a compact, T2, totally disconnected space, or, equivalently, a topological space which can be obtained as the projective limit of finite discrete spaces. Stone proved in the 1930s that the category of profinite sets is dually equivalent to that of Boolean algebras, and, more generally, that the category of spectral spaces is dually equivalent to that of bounded distributive lattices. I will explain how spectral spaces can be advantageously understood as profinite posets, also known as Priestley spaces. I will also point to more modern research that takes Stone duality further, and may touch upon some mathematical contexts where it pops up, notably topos theory and condensed mathematics.
Elements of Stone duality theory have been formalized in Lean over the past few years, and I will report on some of the most recent progress. I will also propose a number of concrete formalization goals at various levels of estimated difficulty, to provide the audience with some potential project ideas for this week.[-]
This talk has a dual aim: to provide a mathematical overview of Stone duality theory, and to invite collaboration on its Lean formalization.
Stone duality is an algebraic way of looking at profinite topologies. A profinite set is a compact, T2, totally disconnected space, or, equivalently, a topological space which can be obtained as the projective limit of finite discrete spaces. Stone proved in the 1930s that the category of profinite sets is ...[+]

06D50 ; 06F30 ; 68V15

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Margulis-Zimmer's super-rigidity - Lee, Homin (Author of the conference) | CIRM H

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We introduce Margulis' and Zimmer's superrigidity. Statements give heuristics in Zimmer program, that is higher rank lattice actions on smooth manifolds. After we state the statement, we mainly focus how it interacts with group actions. Finally, we will also discuss about open questions.

22E40 ; 57M60

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This talk begins with examples of rigid and non-rigid geometric structures, followed by an in-depth discussion of the Fundamental Theorem of Riemannian Geometry, on existence and uniqueness of a torsion-free connection compatible with a Riemannian metric. This result is interpreted as giving a framing on the orthonormal frame bundle uniquely determined by the metric. It is seen to be a consequence of the vanishing of the first prolongation of the orthogonal Lie algebra.[-]
This talk begins with examples of rigid and non-rigid geometric structures, followed by an in-depth discussion of the Fundamental Theorem of Riemannian Geometry, on existence and uniqueness of a torsion-free connection compatible with a Riemannian metric. This result is interpreted as giving a framing on the orthonormal frame bundle uniquely determined by the metric. It is seen to be a consequence of the vanishing of the first prolongation of ...[+]

53B20 ; 53B05 ; 22F05

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For a class of fundamental groups of closed oriented hyperbolic 3-manifolds acting on their Gromov boundary, we compute the K-theory of the associated crossed products in terms of the first homology group of the manifold. Using classification results of purely infinite C*-algebras, we conclude that there exist infinitely many pairwise nonisomorphic torsion-free hyperbolic groups acting on their boundary, for which all crossed products are isomorphic. As in all these cases the boundary is homeomorphic to the 2-sphere, we find infinitely many pairwise non-conjugate Cartan subalgebras with spectrum $S^2$ in such crossed products. This is joint work with Johannes Ebert and Julian Kranz.[-]
For a class of fundamental groups of closed oriented hyperbolic 3-manifolds acting on their Gromov boundary, we compute the K-theory of the associated crossed products in terms of the first homology group of the manifold. Using classification results of purely infinite C*-algebras, we conclude that there exist infinitely many pairwise nonisomorphic torsion-free hyperbolic groups acting on their boundary, for which all crossed products are ...[+]

46L35 ; 37B05

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I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. To obtain it, we prove a parametrized version of the classical result, due to Kadeishvili and Stasheff, that the cohomology of a Poincaré duality space carries a cyclic C-infinity algebra structure. I will also discuss how this higher structure can be used to relate seemingly disparate problems in commutative algebra and differential topology: on one hand, the problem of putting multiplicative structures on minimal free resolutions and, on the other hand, the question of whether a given Poincaré duality fibration can be promoted to a smooth manifold bundle.[-]
I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. To obtain it, we prove a parametrized version of the classical result, due to Kadeishvili and Stasheff, that the cohomology of a Poincaré duality space carries a cyclic C-infinity algebra structure. I will also discuss how this higher structure can be used to relate ...[+]

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On analytic exponential functors on free groups - Vespa, Christine (Author of the conference) | CIRM H

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Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. For example, Hochschild-Pirashvili homology for a wedge of circles or Jacobi diagrams in handlebodies give rise to interesting functors on gr. Some of these natural examples satisfy further properties: they are analytic and/or exponential. Pirashvili proves that the category of exponential contravariant functors from gr to the category k-Mod of k-modules is equivalent to the category of cocommutative Hopf algebras over k. Powell proves an equivalence between the category of analytic contravariant functors from gr to k-Mod, and the category of linear functors on the linear PROP associated to the Lie operad when k is a field of characteristic 0. In this talk, after explaining these two equivalences of categories, I will explain how they interact with each other. (This is a joint work with Minkyu Kim).[-]
Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. For example, Hochschild-Pirashvili homology for a wedge of circles or Jacobi diagrams in handlebodies give rise to interesting functors on gr. Some of these natural examples satisfy further properties: they are analytic and/or exponential. Pirashvili proves that the category of exponential contravariant ...[+]

18A25 ; 16T05 ; 18M70

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There is a family of symplectic representations of the braid groups given by the "integral reduced Burau representation". I will explain a calculation of the stable homology of the braid groups with coefficients in this Burau representation, composed with any algebraic rational representation of the symplectic group. The answer has important consequences in analytic number theory. (Joint with Bergström-Diaconu-Westerland.)

14H10 ; 55P48 ; 20F36 ; 18M70

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Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.[-]
Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a ...[+]

18M70 ; 18N40 ; 22E60 ; 55P62 ; 55U10 ; 14D15 ; 14D23

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