Matroids in Applied and Computational Algebra

应用和计算代数中的拟阵

基本信息

  • 批准号:
    EP/R023379/1
  • 负责人:
  • 金额:
    $ 46.16万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Fellowship
  • 财政年份:
    2018
  • 资助国家:
    英国
  • 起止时间:
    2018 至 无数据
  • 项目状态:
    已结题

项目摘要

Matroids are novel combinatorial objects that generalise and unify several concepts of independence, such as the ones in vector spaces and graphs. They appear in coding theory and optimisation, and have well-studied connections to statistics and computer science. In 1948, Tutte, a British mathematician and codebreaker, associated a polynomial to each matroid which contains many interesting properties of the matroid. Many problems about Tutte polynomials are still unsolved. In the past few years, several breakthroughs happened in the field using algebraic geometry tools. On the other hand, the rich connections between these fields also led to new insights on important problems in algebra and geometry.Graphic matroids are the first natural class and feature prominently in my own previous research, where I uncovered surprising connections to new areas such as divisor theory, system reliability theory and neuroscience.In this project, I propose to investigate new methods and solve several important open problems. This leads to many applications. For example, here are some important problems of various fields that will be studied using matroids in this project.1. A graph can be viewed as an analogue of an algebraic curve. A divisor on a graph is anassignments of integers to its vertices. One can think of it as each vertex having a number of ``tokens". We define a game in which at each step a vertexis chosen and lends a token to each of its neighbours or borrows one from each. Two divisors are equivalent if there is a sequence of moves taking one to the other. The mathematical structures arising from this process are very rich. In this settingthere is an analogue of the classical Riemann-Roch theorem, and a canonical algebraicobject (a variety) encoding equivalences of divisors on it which is closely related to the graphicmatroid. I plan to solve several important problems, such as establishing algebraic propertiesof this variety, by studying the Tutte polynomial.2. Consider a network in which every edge has an associated probability of being operational. Onecan think of the vertices as a set of people who pass messages among themselves and edges as communicationlinks among them. Computing various reliability notions of messaging in such a network has many applications.A well-studied case arises when a person is fixed as a source and multiple people as targets, and the objective is to find the probability of the source being able to communicate with the targets. The network reliability is obtained by plugging special values in the Tutte polynomial of the associated matroid. Computing reliability is also related to algebraic properties mentioned in the previous part. Therefore, positive results in each of them will directly impact the other. Computing reliability is a hard problem in computer science. Given that networks arising from applications usually have special properties,as part of this project, I attempt to unify, and characterise families of networks in which reliability can be computedefficiently and find algorithms.3. A neural network is a graph modeling different regions of a brain as vertices. In one common setting, a potential is associated to each vertex and when a vertex's potential is increased above a certain threshold, it distributes the excess potential with its neighbours, who might in turn continue the same process. A network is in a critical state if it is stable but a small external stimulus is able to make it unstable. The network then continues with a series of potential transfers (avalanche) until it reaches a stable state. Criticality has been shown to provide useful biological information about the brain. Combinatorial properties of critical states are reminiscent of matroids. As part of this project, I will formally define the matroid containing all these information and use it as a tool to study avalanche size distributions in neural networks.
拟阵是一种新颖的组合对象,它推广和统一了几个独立的概念,如向量空间和图中的概念。它们出现在编码理论和优化中,并与统计学和计算机科学有着广泛研究的联系。1948年,英国数学家和密码破译家Tutte将一个多项式与每个拟阵联系起来,其中包含拟阵的许多有趣性质。关于Tutte多项式的许多问题仍然没有解决。在过去的几年里,利用代数几何工具在这一领域取得了一些突破。另一方面,这些领域之间丰富的联系也带来了对代数和几何中重要问题的新见解。图形拟阵是第一个自然类,在我之前的研究中,我发现了与除数理论、系统可靠性理论和神经科学等新领域的惊人联系。在这个项目中,我提议探索新的方法,并解决几个重要的开放问题。这导致了许多应用。例如,以下是本项目中将使用拟阵研究的各个领域的一些重要问题。图可以看作是代数曲线的类比。图上的除数是整数对其顶点的赋值。我们可以把它看作是每个顶点都有许多“令牌”。我们定义了一个博弈,在这个博弈中,在每一步都选择一个顶点,并将一个令牌借给它的每个邻居或从每个邻居那里借一个。如果有一系列的移动将一个移到另一个,则两个除数是等价的。这一过程产生的数学结构非常丰富。在这一背景下,有一个类似经典的Riemann-Roch定理,以及一个与图拟阵密切相关的标准代数对象(变种),它编码了与图拟阵密切相关的因子的等价。我计划通过研究Tutte多项式来解决几个重要的问题,例如建立这种簇的代数性质。考虑这样一个网络,在该网络中,每条边都有相关的运行概率。人们可以把顶点看作是一组相互传递信息的人,而边则是它们之间的通信纽带。在这样的网络中计算消息传递的各种可靠性概念有很多应用,当一个人被固定为一个源,多个人作为目标,目标是求出源能够与目标通信的概率时,就出现了一个很好的研究案例。网络可靠性是通过在相应拟阵的Tutte多项式中插入特定值来获得的。计算可靠性还与前面部分提到的代数性质有关。因此,它们中的每一个的积极结果都将直接影响到另一个。计算可靠性是计算机科学中的一个难题。考虑到由应用引起的网络通常具有特殊的性质,作为本项目的一部分,我试图统一和刻画可以有效地计算可靠性并找到算法的网络族。神经网络是将大脑的不同区域建模为顶点的图形。在一种常见的设置中,每个顶点都有一个电势,当一个顶点的电势增加到某个阈值以上时,它会将多余的电势分配给它的邻居,这些邻居可能会继续同样的过程。如果一个网络是稳定的,那么它就处于临界状态,但一个很小的外部刺激就能使它变得不稳定。然后,网络继续进行一系列潜在传输(雪崩),直到达到稳定状态。临界性已被证明可以提供有关大脑的有用生物信息。临界态的组合性质使人联想到拟阵。作为这个项目的一部分,我将正式定义包含所有这些信息的拟阵,并将其用作研究神经网络中雪崩大小分布的工具。

项目成果

期刊论文数量(10)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Conditional probabilities via line arrangements and point configurations
通过线排列和点配置的条件概率
  • DOI:
    10.1080/03081087.2021.1912693
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    1.1
  • 作者:
    Clarke O
  • 通讯作者:
    Clarke O
Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties
标准单项式理论和 Grassmannians 和 flag 变种内 Richardson 变种的环面退化
  • DOI:
    10.48550/arxiv.2009.03210
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Bonala N
  • 通讯作者:
    Bonala N
Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
格罗布纳简并、格拉斯曼代数和泛簇代数族
  • DOI:
    10.3842/sigma.2021.059
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Bossinger L
  • 通讯作者:
    Bossinger L
Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian
标准单项式理论和格拉斯曼阶理查森簇的环面退化
Standard monomial theory and toric degenerations of Richardson varieties in flag varieties
旗品种中理查森品种的标准单项式理论和复曲面退化
  • DOI:
    10.48550/arxiv.2103.16197
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Bonala N
  • 通讯作者:
    Bonala N
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Fatemeh Mohammadi其他文献

Development and Psychometric Evaluation of the Men’s Worry about Their Wives’ High Risk Pregnancy Questionnaire
男性对妻子高危妊娠的担忧问卷的制定及心理测量评估
The trend of atherogenic indices in patients with type 2 diabetes after bariatric surgery: a national cohort study
2型糖尿病患者减重手术后致动脉粥样硬化指数的变化趋势:一项全国性队列研究
  • DOI:
    10.1016/j.soard.2024.10.022
  • 发表时间:
    2025-04-01
  • 期刊:
  • 影响因子:
    3.800
  • 作者:
    Arsalan Seyedi;Soghra Rabizadeh;Faeze Abbaspour;Sahar Karimpour Reyhan;Nasrin Asgari Soran;Ali Nabipoor;Amirhossein Yadegar;Fatemeh Mohammadi;Rana Hashemi;Reihane Qahremani;Elahe Saffari;Sajedeh Riazi;Fatemeh Sarv;Manouchehr Nakhjavani;Abdolreza Pazouki;Alireza Esteghamati
  • 通讯作者:
    Alireza Esteghamati
Simulation study of solvent effect on competitive degradation of aggregation of homo- and hetero- amyloid-β dimers
  • DOI:
    10.1016/j.molliq.2024.126658
  • 发表时间:
    2025-01-01
  • 期刊:
  • 影响因子:
  • 作者:
    Hamed Zahraee;Seyed Shahriar Arab;Elahe Parvaee;Fatemeh Mohammadi;Khalil Abnous;Seyed Mohammad Taghdisi;Zahra Khoshbin
  • 通讯作者:
    Zahra Khoshbin
A novel approach to clean energy: combining solar chimney and Forgo heat exchangers for enhanced combined cycle power generation
一种清洁能源的新方法:将太阳能烟囱与福戈热交换器相结合,以提高联合循环发电
  • DOI:
    10.1016/j.tsep.2025.103787
  • 发表时间:
    2025-08-01
  • 期刊:
  • 影响因子:
    5.400
  • 作者:
    Fatemeh Mohammadi;Ali Jahangiri;Mohammad Ameri
  • 通讯作者:
    Mohammad Ameri
Combinatorics of Essential Sets for Positroids
Positroids 基本集的组合
  • DOI:
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    0.9
  • 作者:
    Fatemeh Mohammadi;Francesca Zaffalon
  • 通讯作者:
    Francesca Zaffalon

Fatemeh Mohammadi的其他文献

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