Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何和代数方法
基本信息
- 批准号:RGPIN-2018-03880
- 负责人:
- 金额:$ 2.99万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2020
- 资助国家:加拿大
- 起止时间:2020-01-01 至 2021-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The main objective of the proposed research program is to develop new and improved techniques to attack arithmetic problems in discrete geometry and additive combinatorics.
Several central problems in different parts of mathematics can be translated into questions in discrete geometry. In many cases, such connections were discovered relatively recently due to pioneering works of great mathematicians like Jean Bourgain, Tim Gowers, and Terry Tao. As it turned out there are classical problems in discrete geometry which have direct impacts in harmonic analysis, combinatorics, number theory, and theoretical computer science. This field even has its unique Mathematics Subject Classification code: 52C10, Erdos problems and related topics of discrete geometry. Many of these problems can be formulated as counting (bounding) the maximum number of incidences between lines/curves/planes/surfaces and points. (We say that a point is incident to a line (or curve or surface) if the point lies on the line.) Incidence bounds provide inside information about the arithmetic-geometric structure of the underlying field. A well known example - and an important part of my research - is the sum-product problem: given a finite set of integers, A, is it possible that both the sumset, A+A, and the product set, A*A, are small? (The sumset and product set are the set of pairwise sums and products of elements from A.) For example if A is the set of the first n natural numbers then the sumset is small, it has cardinality 2n-1, while the product set is almost quadratic in n=|A|. If A is a geometric progression then the product set is small, but then the sumset is quadratic in |A|. Erdos and Szemeredi is conjectured that |A+A|+|A*A|>|A|2-epsilon, where epsilon goes to zero as |A| goes to infinity. All significant improvements in this problem have come from (discrete) geometry, by understanding the connections between the geometry of the plane and the arithmetic of the underlying field.
The particular problem I will focus on is Erdos' Unit Distances Problem: What is the maximum number of unit distances among pairs of n points on the plane? Erdos conjectured that the upper bound on unit distances is n1+epsilon, where epsilon goes to zero as n goes to infinity. This is a 70 year old problem in which the best upper bound, n4/3, was given more than 30 years ago. I plan to improve this upper bound. There are examples of metrics similar to the Euclidean, where the number of unit distances is n4/3 , so any possible improvement should use more than the combinatorics of unit circle arrangements. Improving this bound might seem to be overly ambitious, but recent developments in using algebraic methods to tackle similar problems make the plan look more feasible.
提出的研究计划的主要目标是开发新的和改进的技术来解决离散几何和加性组合中的算术问题。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
数据更新时间:{{ journalArticles.updateTime }}
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Solymosi, Jozsef其他文献
Near optimal bounds for the Erdos distinct distances problem in high dimensions
- DOI:
10.1007/s00493-008-2099-1 - 发表时间:
2008-01-01 - 期刊:
- 影响因子:1.1
- 作者:
Solymosi, Jozsef;Vu, Van H. - 通讯作者:
Vu, Van H.
Solymosi, Jozsef的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
{{ truncateString('Solymosi, Jozsef', 18)}}的其他基金
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何与代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2022
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何与代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2021
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何和代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2019
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何和代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2018
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Additive Combinatorics
加法组合学
- 批准号:
261014-2013 - 财政年份:2017
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Additive Combinatorics
加法组合学
- 批准号:
261014-2013 - 财政年份:2016
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Additive Combinatorics
加法组合学
- 批准号:
261014-2013 - 财政年份:2015
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Additive Combinatorics
加法组合学
- 批准号:
261014-2013 - 财政年份:2014
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Additive Combinatorics
加法组合学
- 批准号:
261014-2013 - 财政年份:2013
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Problems in additive combinatorics
加法组合数学中的问题
- 批准号:
261014-2008 - 财政年份:2012
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
相似国自然基金
Lienard系统的不变代数曲线、可积性与极限环问题研究
- 批准号:12301200
- 批准年份:2023
- 资助金额:30.00 万元
- 项目类别:青年科学基金项目
对RS和AG码新型软判决代数译码的研究
- 批准号:61671486
- 批准年份:2016
- 资助金额:60.0 万元
- 项目类别:面上项目
同伦和Hodge理论的方法在Algebraic Cycle中的应用
- 批准号:11171234
- 批准年份:2011
- 资助金额:40.0 万元
- 项目类别:面上项目
相似海外基金
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何与代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2022
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何与代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2021
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何和代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2019
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Geometric Numerical Integration Methods for Differential-Algebraic Equations and Their Application to Evolutionary Equations
微分代数方程的几何数值积分方法及其在演化方程中的应用
- 批准号:
19K23399 - 财政年份:2019
- 资助金额:
$ 2.99万 - 项目类别:
Grant-in-Aid for Research Activity Start-up
Geometric and algebraic methods in Erdos type problems
鄂尔多斯型问题的几何和代数方法
- 批准号:
RGPIN-2018-03880 - 财政年份:2018
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Grants Program - Individual
Study on Statistical Methods based on Geometric and Algebraic Structures
基于几何和代数结构的统计方法研究
- 批准号:
17K12651 - 财政年份:2017
- 资助金额:
$ 2.99万 - 项目类别:
Grant-in-Aid for Young Scientists (B)
Algebraic and Geometric Methods in Data Analysis
数据分析中的代数和几何方法
- 批准号:
1702395 - 财政年份:2017
- 资助金额:
$ 2.99万 - 项目类别:
Continuing Grant
Integrated Geometric and Algebraic Multigrid Methods
综合几何和代数多重网格方法
- 批准号:
1522615 - 财政年份:2015
- 资助金额:
$ 2.99万 - 项目类别:
Continuing Grant
Study of nonlinear geometric problems by methods of algebraic analysis
用代数分析方法研究非线性几何问题
- 批准号:
23654047 - 财政年份:2011
- 资助金额:
$ 2.99万 - 项目类别:
Grant-in-Aid for Challenging Exploratory Research
Algebraic and geometric methods in switched control system analysis and design
开关控制系统分析与设计中的代数和几何方法
- 批准号:
DP110102704 - 财政年份:2011
- 资助金额:
$ 2.99万 - 项目类别:
Discovery Projects