Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Geometry of Hyperbolas01:30

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
Extended Versions of Green’s Theorem01:27

Extended Versions of Green’s Theorem

Green’s Theorem connects the circulation of a vector field around a closed curve with the behavior of the field across the region enclosed by that curve. It provides a way to replace a line integral around a boundary with a double integral over the interior region, making it especially useful in plane geometry, fluid flow, and vector calculus.Although Green’s Theorem is often introduced using simple regions without gaps, it can also be applied to regions made from several simple parts. This...
Collisions in Multiple Dimensions: Problem Solving01:06

Collisions in Multiple Dimensions: Problem Solving

In multiple dimensions, the conservation of momentum applies in each direction independently. Hence, to solve collisions in multiple dimensions, we should write down the momentum conservation in each direction separately. To help understand collisions in multiple dimensions, consider an example.
A small car of mass 1,200 kg traveling east at 60 km/h collides at an intersection with a truck of mass 3,000 kg traveling due north at 40 km/h. The two vehicles are locked together. What is the...
Graphical Representation of Inequalities01:28

Graphical Representation of Inequalities

The graph of the equation where y equals x squared forms a curve known as a parabola. This curve acts as a boundary in the coordinate plane, dividing it into distinct regions based on the relative position of points.When the equality sign in the equation is replaced with an inequality—such as greater than, less than, greater than or equal to, or less than or equal to—the graphical representation changes from a single curve into a broader shaded area that signifies the set of all points...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

IdopNetworks: How to infer the individualized genetic architecture of genomics for precision medicine.

Drug discovery today·2026
Same author

Topological Data Analysis in Materials Science: Principles, Machine Learning Integration, and Application Landscapes.

Chemical reviews·2026
Same author

Statistical learning of stochastic complex systems via the Yau-Yau nonlinear filter.

Innovation (Cambridge (Mass.))·2026
Same author

A deep-learning framework for brain tumor segmentation via three-dimensional mass-preserving geometric transformation.

Brain informatics·2026
Same author

An end-to-end generalizable deep learning framework to comprehensively analyze transcriptional regulation.

Nature communications·2026
Same author

Graph statistics theory of individualized quantitative genetics under haplotype-resolved genome assembly.

Proceedings of the National Academy of Sciences of the United States of America·2026

Related Experiment Video

Updated: Jun 27, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

A geometric approach to problems in birational geometry.

Chen-Yu Chi1, Shing-Tung Yau

  • 1Department of Mathematics, Harvard University, Cambridge, MA 02138, USA. cychi@math.harvard.edu

Proceedings of the National Academy of Sciences of the United States of America
|November 27, 2008
PubMed
Summary

Birational invariants, specifically pseudonormed spaces of pluricanonical forms, can determine birational equivalence for projective varieties of general type. This work provides a Torelli-type theorem for birational geometry.

More Related Videos

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads
07:58

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads

Published on: July 25, 2025

Related Experiment Videos

Last Updated: Jun 27, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads
07:58

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads

Published on: July 25, 2025

Area of Science:

  • Algebraic Geometry
  • Complex Geometry
  • Differential Geometry

Background:

  • Birational invariants are crucial for classifying algebraic varieties.
  • Pluricanonical forms and their associated metric structures (pseudonormed spaces) are key invariants.
  • A fundamental question concerns whether isometric pseudonormed spaces imply the existence of an isometric birational map.

Purpose of the Study:

  • To investigate if isometric pseudonormed spaces between two projective varieties of general type guarantee a birational map inducing these isometries.
  • To establish a Torelli-type theorem for birational equivalence in this context.

Main Methods:

  • Utilizing the theory of pluricanonical systems on algebraic varieties.
  • Employing metric structures on spaces of differential forms.
  • Developing techniques to relate birational maps to isometries of pseudonormed spaces.

Main Results:

  • A positive answer is provided for projective varieties of general type.
  • It is shown that isometric pseudonormed spaces imply the existence of a birational map inducing these isometries.
  • This result establishes a strong connection between metric properties and birational classification.

Conclusions:

  • The study confirms a Torelli-type theorem for birational equivalence.
  • Pseudonormed spaces of pluricanonical forms are powerful invariants for distinguishing and relating algebraic varieties.
  • This work deepens the understanding of birational geometry through metric invariants.