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2 y
In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-dimensional knot theory. However, the theory extends to other ambient four-manifolds, and we will pay particular attention to the setting of complex curves in simple complex surfaces, where the theory produces surprisingly satisfying pictures and leads to interesting results about trisections of complex surfaces.
This talk is based on various joint works with Dave Gay, Peter Lambert-Cole, and Alex Zupan.
[-] In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-...
[+] 57Q45 ; 57M25 ; 57M50
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y
In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.
[-] In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...
[+] 57M27 ; 57M50 ; 57M25
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y
In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.
[-] In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...
[+] 57M27 ; 57M50 ; 57M25
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y
Fock and Goncharov give parameterisations of two different types of moduli spaces of properly convex real projective structures. I'll discuss a number of observations made about these parameterisations, the geometric structures that are parameterised by them, their relationship with representations into SL(3,R), canonical cell decompositions, and compactifications. This includes joint work with Alex Casella, Robert Haraway, Robert Löwe and Dominic Tate.
[-] Fock and Goncharov give parameterisations of two different types of moduli spaces of properly convex real projective structures. I'll discuss a number of observations made about these parameterisations, the geometric structures that are parameterised by them, their relationship with representations into SL(3,R), canonical cell decompositions, and compactifications. This includes joint work with Alex Casella, Robert Haraway, Robert Löwe and ...
[+] 57M50
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y
Various surgery operations on dimension four begin with a 4–manifold $X$ and an embedded surface $S$, then remove a neighborhood of $S$ and replace it with something else to produce an interesting new 4–manifold. In a few standard surgery constructions, especially the Gluck twist operation, I will show how, given a trisection diagram of $X$ with decorations that describe the embedded surface $S$, to produce a trisection diagram for the new 4–manifold.
This is joint work with Jeff Meier.
[-] Various surgery operations on dimension four begin with a 4–manifold $X$ and an embedded surface $S$, then remove a neighborhood of $S$ and replace it with something else to produce an interesting new 4–manifold. In a few standard surgery constructions, especially the Gluck twist operation, I will show how, given a trisection diagram of $X$ with decorations that describe the embedded surface $S$, to produce a trisection diagram for the new ...
[+] 57M50 ; 57R45 ; 57R65 ; 57R17
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y
Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a PD(3) pair.
This is joint work with Genevieve Walsh.
[-] Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a ...
[+] 57M07 ; 20F67 ; 20F65 ; 57M50
English
