Survey Of Diophantine Geometry
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Abstract : Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Following a clue from a paper of Manin, we then speculate on the possibility of understanding the algebraic relations among periods via Galois groups of arithmetic differential equations. Abstract : One of the main themes of algebraic geometry is to classify algebraic varieties and to study various geometric properties of each of the interesting classes. Classical theories of curves and surfaces give a beautiful framework of classification theory.
Recent developments provide more details in the case of dimension three. We are going to introduce the three-dimensional story and share some expectations for even higher dimensions. Abstract : One of the main goals of Diophantine geometry is linking the geometric properties of an algebraic varieties with the distribution of the integral or rational points on it. Broad conjectures by Lang, Vojta and Campana predicts the degeneracy resp. We shall discuss these conjectures by working out in details some concrete cases.
Especially, degeneracy results based on Diophantine approximation techniques will be proved. Abstract : I will survey concepts that are near to K-stability and which have origins in toric geometry. A main goal will be to explain their role in measuring arithmetic complexity of rational points, for example questions in Diophantine approximation for projective varieties. There are also important connections to measures of growth and positivity of line bundles. Our proof builds on previous work by Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image sheaves.
As an application, we recover a higher-dimensional Diophantine approximation theorem of K. Roth-type due to D. McKinnon and M. Roth with a significantly shortened proof, while simultaneously extending the scope of the use of Seshadri constants in this context in a natural way. We will also discuss recent progress regarding applications to the degeneracy of integral points on varieties in the complement of divisors.
Fundamentals of Diophantine Geometry
This is joint work with Aaron Levin. Abstract : Brody hyperbolic projective varieties over the complex numbers are extremely rich in properties. For instance, such varieties have only finitely many automorphisms, and every surjective endomorphism is actually an automorphism of finite order. In these lecture series, we will see how to establish some of these properties, and thereby verify many of the predictions made by the conjectures of Green-Griffiths and Lang.
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Abstract : In , Bugeaud, Corvaja, and Zannier gave an upper bound for the greatest common divisor gcd a n -1,b n -1 , where a and b are fixed integers and n varies over the positive integers. In contrast to the elementary statement of their result, the proof required deep results from Diophantine approximation. I will discuss a higher-dimensional generalization of their result and some recent related results with Julie Wang in Nevanlinna theory.
In the same spirit we will discuss some arithmetic properties of possibly transcendental holomorphic sections in semi-abelian schemes joint with Corvaja and Zannier.
We will then find some new problems. Abstract : In this short talk, I will discuss some results developing the ideas of Parshin in using hyperbolic and homotopy methods to study integral points on abelian varieties over one dimensional complex function fields.
Abstract : If X is a smooth variety with ample cotangent bundle, then X will be hyperbolic. Thus, ampleness of the cotangent bundle is a strong hyperbolicity property. Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Continued fractions. Category List of topics List of recreational topics Wikibook Wikversity. Areas of mathematics.
Category theory Information theory Mathematical logic Philosophy of mathematics Set theory. Abstract Elementary Linear Multilinear. Calculus Real analysis Complex analysis Differential equations Functional analysis. Combinatorics Graph theory Order theory Game theory. Arithmetic Algebraic number theory Analytic number theory Diophantine geometry.
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Fundamentals of Diophantine Geometry
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Atiyah Gromov. In recent times, powerful new geometric ideas and methods have been developed by means of which important new arithmetical theorems and related results have been found and proved and some of these are not easily proved otherwise. Further, there has been a tendency to clothe the old results, their extensions, and proofs in the new geometrical language. Sometimes, however, the full implications of results are best described in a geometrical setting.
Lang has these aspects very much in mind in this book, and seems to miss no opportunity for geometric presentation. This accounts for his title "Diophantine Geometry.