梅兰妮-马切特-伍德 数学家

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梅兰妮-马切特-伍德
数学家 | 2022级
从算术统计学的角度解决数论中的基础性问题。



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标题
数学家
工作单位
哈佛大学数学系
工作地点
马萨诸塞州剑桥市
年龄
获奖时41岁
重点领域
数学、统计和概率
网址
哈佛大学。梅兰妮-马切特-伍德
拉德克利夫高级研究学院。梅兰妮-马切特-伍德
2022年9月26日出版
关于梅兰妮的工作
梅兰妮-马切特-伍德是一位研究纯数学基础问题的数学家。她擅长数论（整数的属性和关系）和代数几何（研究几何结构，如曲线和曲面，用多项式方程描述）。通过结合广泛的数学方法，她揭示了看待数字基本属性的新方法。

伍德的研究通常是由算术统计学中的问题激发的。许多数学研究集中在为猜想开发证明，但在某些情况下，检查一个假设在所有可能的情况下都是真的并不可行。例如，确定素数的一些属性依赖于无限多的数字的行为（例如，双子素数猜想就是这种情况，该猜想认为有无限多的素数相距2，如11和13）。伍德对这类问题采取了算术统计的方法。她为数论对象（如素数）开发了概率模型，以揭示它们的平均行为方式。伍德和合作者为椭圆曲线的等级分布开发了这样一个模型。等级是一种衡量标准，它传达了关于该曲线方程的解中有多少是有理数的信息。等级较高的曲线具有更大、更复杂的有理数解集。以前人们一直认为等级是无界的，这意味着应该可以找到具有任意高等级的曲线。然而，伍德和她的合作者预测，存在有限数量的等级大于21的椭圆曲线，使得有理数上的椭圆曲线的等级集合是有界的。

在其他工作中，伍德引入了新的技术来确定随机图的沙堆群（围绕临界点自我组织的动力系统模型）的行为。而她对科恩-伦斯特拉启发法（一套关于二次数域类群分布的猜想）的研究，推进了对这些类群结构和更普遍的情况家族的理解。伍德正在揭示自然数的新特性，这些特性与其他数学猜想和定理有关，从而为今后数论的新发现创造了条件。

个人简历
梅兰妮-马切特-伍德在杜克大学获得学士学位（2003年），在剑桥大学获得数学高级研究证书（2004年），并在普林斯顿大学获得博士学位（2009年）。她是美国数学研究所的研究员（2009-2017），并在斯坦福大学（2009-2011）、威斯康星大学麦迪逊分校（2011-2019）和加利福尼亚大学伯克利分校（2019-2020）担任教职。2020年，她成为哈佛大学的教授和拉德克利夫高级研究所的拉德克利夫校友教授。伍德在各种权威期刊上发表过文章，包括《数学发明》（Inventiones Mathematicae）、《美国数学会杂志》、《杜克大学数学杂志》和《欧洲数学会杂志》。

在梅兰妮的话语中


一位面带微笑的白人妇女坐在一张桌子前，在书架前摆放着文件。


"数字和它们的特性是人类最古老和最普遍的兴趣之一。然而，数字蕴含着更多的秘密，我们仍在努力揭示。揭开这些神秘的面纱需要新的视角，而且往往发生在我们发现数学不同部分之间令人惊讶的联系时"。



Melanie Matchett Wood
Mathematician | Class of 2022
Addressing foundational questions in number theory from the perspective of arithmetic statistics.


Portrait of Melanie Matchett Wood
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Title
Mathematician
Affiliation
Department of Mathematics, Harvard University
Location
Cambridge, Massachusetts
Age
41 at time of award
Area of Focus
Mathematics, Statistics, and Probability
Website
Harvard University: Melanie Matchett Wood
Radcliffe Institute for Advanced Study: Melanie Matchett Wood
Published September 26, 2022
ABOUT MELANIE’S WORK
Melanie Matchett Wood is a mathematician investigating foundational questions in pure mathematics. She specializes in number theory (the properties and relationships of whole numbers) and algebraic geometry (the study of geometric structures, such as curves and surfaces, that are described using polynomial equations). By combining a breadth of mathematical approaches, she reveals new ways to see fundamental properties of numbers.

Wood’s research is often motivated by questions in arithmetic statistics. Much of mathematics research focuses on developing proofs for conjectures, but in some cases, it is not feasible to check that a hypothesis is true for all possible cases. For example, determining some properties of prime numbers relies on the behavior of infinitely many numbers (such is the case, for example, for the twin prime conjecture, which posits that there are an infinite number of prime numbers that are 2 apart, such as 11 and 13). Wood takes an arithmetic statistics approach to such problems. She develops probabilistic models for number theoretic objects (such as prime numbers) to reveal how they will behave on average. Wood and collaborators developed such a model for the distribution of ranks of elliptic curves. The rank is a measurement that conveys information about how many of the solutions to that curve’s equation are rational numbers. Curves with higher ranks have larger and more complicated sets of rational solutions. It has been previously assumed that rank is unbounded, meaning it should be possible to find curves with arbitrarily high ranks. However, Wood and her collaborators predicted that there are a finite number of elliptic curves with rank greater than 21, making the set of ranks of elliptic curves over the rational numbers bounded.

In other work, Wood introduced new techniques to determine the behavior of sandpile groups (models of dynamical systems that self organize around a critical point) of random graphs. And her research on the Cohen-Lenstra heuristics (a set of conjectures about the distribution of class groups of quadratic number fields) has advanced understanding of the structure of these groups and for more general families of situations. Wood is revealing new properties of natural numbers that are relevant to other mathematical conjectures and theorems, thereby setting the stage for new discoveries in number theory in the future.

BIOGRAPHY
Melanie Matchett Wood received a BS (2003) from Duke University, a Certificate of Advanced Study in Mathematics (2004) from the University of Cambridge, and a PhD (2009) from Princeton University. She was a researcher with the American Institute of Mathematics (2009–2017) and held faculty positions at Stanford University (2009–2011), the University of Wisconsin at Madison (2011–2019), and the University of California at Berkeley (2019–2020). In 2020, she became a professor at Harvard University and Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. Wood has published in a variety of leading journals, including Inventiones Mathematicae, Journal of the American Mathematical Society, Duke Mathematical Journal, and Journal of the European Mathematical Society.

IN MELANIE'S WORDS


A smiling White woman sits at a table with paperwork in front of a shelf of books


”Numbers and their properties are one of the most ancient and universal interests of humanity. Yet numbers hold more secrets that we are still working to reveal. Unlocking these mysteries requires new perspectives and often happens when we discover surprising connections between different parts of mathematics.”