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Audiovisual Mathematics Library

Audiovisual Mathematics Library

Topology 172 results

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Which geodesic flows are left-handed? - Dehornoy, Pierre (Author of the conference) | CIRM H

Post-edited

Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot[-]
Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows ...[+]

37C27 ; 37C15 ; 37C10 ; 57M25

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Variations on an example of Hirzebruch - Stover, Matthew (Author of the conference) | CIRM H

Multi angle

In '84, Hirzebruch constructed a very explicit noncompact ball quotient manifold in the process of constructing smooth projective surfaces with Chern slope arbitrarily close to 3. I will discuss how this and some closely related ball quotients are useful in answering a variety of other questions. Some of this is joint with Luca Di Cerbo.

14M27 ; 32Q45 ; 57M50

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2y

The $\mathbb{L}^ \bullet$-Homology fundamental class for singular spaces and the stratified Novikov conjecture - Banagl, Markus (Author of the conference) | CIRM H

Post-edited

An oriented manifold possesses an L-homology fundamental class which is an integral refinement of its Hirzebruch L-class and assembles to the symmetric signature. In joint work with Gerd Laures and James McClure, we give a construction of such an L-homology fundamental class for those oriented singular spaces, which are integral intersection homology Poincaré spaces. Our approach constructs a morphism of ad theories from intersection Poincaré bordism to L-theory. We shall indicate an application to the stratified Novikov conjecture. The latter has been treated analytically by Albin, Leichtnam, Mazzeo and Piazza.[-]
An oriented manifold possesses an L-homology fundamental class which is an integral refinement of its Hirzebruch L-class and assembles to the symmetric signature. In joint work with Gerd Laures and James McClure, we give a construction of such an L-homology fundamental class for those oriented singular spaces, which are integral intersection homology Poincaré spaces. Our approach constructs a morphism of ad theories from intersection Poincaré ...[+]

55N33 ; 57R67 ; 57R20 ; 57N80 ; 19G24

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On real algebraic knots and links - Orevkov, Stepan (Author of the conference) | CIRM H

Multi angle

I will present the following results on real algebraic spatial curves:
(1) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves of genus 0 or 1 up to degree 6 up to rigid isotopy.
(2) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves with maximal encomplexed writhe up to (not rigid yet) isotopy.
(3) Classification of smooth spatial real algebraic curves of genus 0 with two irreducible components up to degree 6 up to rigid isotopy, in particular, the first (as far as know) example of two spatial real algebraic curves which are isotopic, have equal degree, genus and encomplexed writhe of each irreducible component but not rigidly isotopic.[-]
I will present the following results on real algebraic spatial curves:
(1) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves of genus 0 or 1 up to degree 6 up to rigid isotopy.
(2) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves with maximal encomplexed writhe up to (not rigid yet) isotopy.
(3) Classification of smooth spatial real algebraic curves of genus 0 with ...[+]

14P25

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Real curves and a Klein TQFT - Georgieva, Penka (Author of the conference) | CIRM H

Multi angle

The local Gromov-Witten theory of curves studied by Bryan and Pandharipande revealed strong structural results for the local GW invariants, which were later used by Ionel and Parker in the proof of the Gopakumar-Vafa conjecture. In this talk I will report on a joint work in progress with Eleny Ionel on the extension of these results to the real setting.

14N35 ; 53D45

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2y

Expected topology of a random subcomplex in a simplicial complex - Welschinger, Jean-Yves (Author of the conference) | CIRM H

Post-edited

I will explain how to bound from above and below the expected Betti numbers of a random subcomplex in a simplicial complex and get asymptotic results under infinitely many barycentric subdivisions. This is a joint work with Nermin Salepci. It complements previous joint works with Damien Gayet on random topology.

52Cxx ; 60C05 ; 60B05 ; 55U10

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Shuffles of trees - Hoffbeck, Eric (Author of the conference) | CIRM H

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We study a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. Our notion of shuffle is motivated by the theory of operads and occurs in the theory of dendroidal sets. We give several equivalent descriptions of the shuffles, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. This is a joint work with Ieke Moerdijk.[-]
We study a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. Our notion of shuffle is motivated by the theory of operads and occurs in the theory of dendroidal sets. We give several equivalent descriptions of the shuffles, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. This is a ...[+]

55U10 ; 18D50 ; 05C05

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2y

Homotopy theory of strict $\omega$-categories and its connections with homology of monoids - Lecture 1 - Métayer, François (Author of the conference) | CIRM H

Post-edited

In the first part, we describe the canonical model structure on the category of strict $\omega$-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as $\omega$-categories freely generated by polygraphs and introduce the key notion of polygraphic resolution. Finally, by considering a monoid as a particular $\omega$-category, this polygraphic point of view will lead us to an alternative definition of monoid homology, which happens to coincide with the usual one.[-]
In the first part, we describe the canonical model structure on the category of strict $\omega$-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as $\omega$-categories freely generated by polygraphs and introduce the key notion of polygraphic resolution. Finally, by considering a monoid as a particular $\omega$-category, this polygraphic point of view will lead us to an alternative definition ...[+]

18D05 ; 18G55 ; 18G50 ; 18G10

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2y

An integration approach to the Toeplitz square peg problem - Tao, Terence (Author of the conference) | CIRM H

Post-edited

The Toeplitz square peg problem asks if every simple closed curve in the plane inscribes a square. This is known for sufficiently regular curves (e.g. polygons), but is open in general. We show that the answer is affirmative if the curve consists of two Lipschitz graphs of constant less than 1 using an integration by parts technique, and give some related problems which look more tractable.

55N45

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2y

Structure of hyperbolic manifolds - Lecture 1 - Purcell, Jessica (Author of the conference) | CIRM H

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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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