Metric Algebraic Geometry

Breiding, Paul, Kohn, Kathlén, Sturmfels, Bernd

Metric Algebraic Geometry
  • 出版商: Birkhauser Boston
  • 出版日期: 2024-02-28
  • 售價: $2,190
  • 貴賓價: 9.5 折 $2,081
  • 語言: 英文
  • 頁數: 215
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3031514610
  • ISBN-13: 9783031514616

    • 下單後立即進貨 (約1週~2週)

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商品描述

Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.

After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.

This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.

This is an openaccess book.

商品描述(中文翻譯)

度量代數幾何結合了代數幾何和微分幾何的概念。在傳統基礎上，它提供了21世紀的實用工具。許多應用問題圍繞著度量問題，例如相對於距離的優化。

在簡短介紹19世紀平面曲線幾何之後，我們轉向用實數多項式方程表示的問題。解集是實代數多樣性。我們的許多度量問題出現在數據科學、優化和統計學中。這些問題包括在機器學習中最小化Wasserstein距離、最大似然估計、計算曲率或最小化到多樣性的歐幾里德距離。

本書針對廣泛的研究人員和學生群體，可用於研究生水平的一學期課程。主要先決條件是扎實的本科數學基礎，特別是代數和幾何學。

本書是開放存取的。

作者簡介

Paul Breiding is professor for mathematical methods in data science at the University of Osnabrück, an Emmy-Noether Research Group Leader and member of the Academy of Sciences and Literature Mainz. In 2021 he received the Early Career Prize of the SIAM Activity Group on Algebraic Geometry. His interests lie in numerical and random algebraic geometry. He is one of the developers of the software HomotopyContinuation.jl.

Kathlén Kohn is a tenure-track assistant professor at KTH in Stockholm. Her research investigates the underlying geometry in computer-vision, machine-learning and statistical problems, using algebraic methods. For her research in computer vision, she received the Best Student Paper Award at the International Conference on Computer Vision (ICCV) in 2019 and the Swedish L'Oréal-Unesco for Women in Science prize in 2023.
After many years at UC Berkeley, Bernd Sturmfels now serves as a director at the Max-Planck Institute for Mathematics in the Sciences in Leipzig, Germany, where he leads the Nonlinear Algebra group. He has published 10 books and 300 articles, and he mentored 60 doctoral students, plus countless postdocs. His interests range from algebraic geometry and combinatorics to statistics, optimization and physics.

作者簡介(中文翻譯)

Paul Breiding是奧斯納布呂克大學數據科學數學方法的教授，也是Emmy-Noether研究小組的領導者，並且是馬因茲科學與文學學院的成員。他在2021年獲得了SIAM代數幾何活動小組的早期職業獎。他的興趣在於數值和隨機代數幾何。他是軟體HomotopyContinuation.jl的開發者之一。

Kathlén Kohn是斯德哥爾摩皇家理工學院的助理教授，並且是一位終身職位的助理教授。她的研究探討計算機視覺、機器學習和統計問題中的基礎幾何，並使用代數方法。她在計算機視覺研究中，於2019年獲得了國際計算機視覺會議（ICCV）的最佳學生論文獎，並在2023年獲得了瑞典L'Oréal-Unesco婦女科學獎。

在經過多年在加州大學伯克利分校的經歷後，Bernd Sturmfels現在擔任德國萊比錫的馬克斯·普朗克數學科學研究所的主任，並領導非線性代數小組。他已經出版了10本書和300篇文章，並指導了60名博士生和無數的博士後研究員。他的興趣涵蓋了代數幾何、組合數學、統計學、優化和物理學。

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