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- Lectures on the Geometry of Numbers
- Geometry of numbers
- [PDF] Lectures on dynamics, fractal geometry, and metric number theory - Semantic Scholar
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I will explain some of these applications including Siu's rigidity of negatively curved Kahler manifolds and the Corlette-Gromov-Schoen rigidity of representations theorem, as well as the very recent results of Markovic and Benoist-Hulin about the existence and uniqueness of harmonic maps between rank-1 symmetric spaces. I will explain why in general the Caratheodory and Teichmueller metrics do not agree on Teichmueller spaces and why this yields a proof of the convexity conjecture of Siu.
Moreover, I will illustrate how deep theorems in Teichmueller dynamics play an important role in classifying Teichmueller discs where the two metrics agree. Geometric analogues of these profound facts have been of great interest over the years.
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We will discuss effective versions of these theorems for hyperbolic 3-manifolds and for rational maps. The class of quadratic polynomials to which our theorem applies forms an open dense subset of the Mandelbroat set conjectually. We will also explain how our theorems fit in this dictionary.
This talk is based on joint work with Winter, and in part with Margulis and Mohammadi. By Mostow rigidity, there are only countably many hyperbolic 3-manifolds of finite volume.
Lectures on the Geometry of Numbers
In such a manifold, a geodesic plane is either closed or dense; this dichotomy was proved by Ratner and Shah independently around 30 years ago. There was little progress on this question for a hyperbolic 3-manifold of infinite volume until recently. The main obstacle in infinite volume case is scarcity of recurrence of unipotent flows, without which we cannot use unipotent dynamics.
The key ingredient of this construction is to show that circular slices of a Sierpinski curve of positive modulus inherit positivity of the modulus. They give rise to one of first examples of a fractal in the plane. A beautiful theorem of Descartes in implies that if the initial four circles have integral curvatures, then all the circles in the packing have integral curvatures, as observed by Soddy, a Nobel laureate in Chemistry.
This remarkable integrality feature gives rise to several natural Diophantine questions about integral Apollonian packings such as "how many circles have prime curvatures? The main obstacle in infinite volume case is scarcity of recurrence of unipotent flows, without which we cannot use unipotent dynamics.
The key ingredient of this construction is to show that circular slices of a Sierpinski curve of positive modulus inherit positivity of the modulus. They give rise to one of first examples of a fractal in the plane.
Geometry of numbers
A beautiful theorem of Descartes in implies that if the initial four circles have integral curvatures, then all the circles in the packing have integral curvatures, as observed by Soddy, a Nobel laureate in Chemistry. This remarkable integrality feature gives rise to several natural Diophantine questions about integral Apollonian packings such as "how many circles have prime curvatures?
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Various notions of "evenness" play a fundamental role in algebraic topology. Spaces with only even cells tend to have more readily computed and conceptual cohomology, and cohomology theories which take non-zero values only on even spheres are the heart of the "chromatic" approach to stable homotopy which links algebraic topology and algebraic geometry. In the most geometric cases, we have both, meaning that geometric objects related to vector bundles and bordism are shockingly well-behaved, with the cells and homotopy groups themselves encoding rich information.
Equivariantly, we have many different notions of spheres and cells built out of representations, and these give other kinds of notions for "even".
[PDF] Lectures on dynamics, fractal geometry, and metric number theory - Semantic Scholar
In this series of talks, I'll discuss a version of even for equivariant spectra which behaves very much like the classical cases. This allows us to understand the spaces that make up the equivariant bordism theory used in the proof of the Kervaire Invariant One problem, to give a conceptual description of the Steenrod algebra for various equivariant cohomology theories, and to explain certain duality phenomena observed in spectral algebraic geometry. See more information here. Abstract : Complex projective space plays a fundamental role in algebraic topology as a space which simultaneously represents line bundles and the second cohomology group with integral coefficients.
This space has an additional extremely useful feature: it has cells only in even dimensions and homotopy only in even dimensions. This makes many algebraic topology constructions exceptionally easy. More surprisingly, Wilson in his thesis showed that the even spaces in the complex bordism spectrum have only even cells and only even homotopy groups. Help us improve our products. Sign up to take part.
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A Nature Research Journal. The literary Society at Eton has, we believe, adopted this plan on very many occasions; recently it will be remembered that Mr. Gladstone addressed the society on Homer. Russell's lecture is a full one, and on the lines which it follows, a useful one.
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But geometry is equally the development of a method pervading nature; its mastery gives man a power to govern matter. The training which enables him to comprehend the mechanism of the universe, enables him also to make creations of his own in harmony with those greater designs of which his own are but a small portion.
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These two uses of geometric education the one purely gymnastic, the other practical and technic, may be so combined that each shall aid and not impede the other. The order, number, and measure which pervade the universe can be easily brought within the scope of elementary education, and so form the fit preparation for scientific observation and experiment in later life, by means of which the standard of application of abstract truths to matter and events in human life are determined and made familiar.