LAB LOGA page on LabLog, a review site (not the lab's official page)
Algebraic Geometry, Differential Geometry, and Complex Geometry Lab
Machine-translated · View original
The lab uses partial differential equations to study the geometric properties of algebraic varieties.
AboutAbout the lab
Using equations to understand canonical metrics on manifolds.
Prof. Hisamoto uses partial differential equations to study the geometric properties of algebraic varieties.
Prof. Hisamoto is particularly interested in the relationship between canonical metrics on manifolds and K-stability.
Recently, Prof. Hisamoto has also become interested in Fano varieties that are not K-stable, and works on research using parabolic partial differential equations.
In the seminar for undergraduate thesis research, students learn Riemann surfaces and the theory of complex manifolds; in the small-group class in graduate school, students work on problems matching their own interests and write a paper.
Key terms
- Algebraic variety
- A figure expressed as the set of solutions of several polynomial equations; a generalization of curves and surfaces to higher dimensions
- Kähler metric
- A special ruler on a complex manifold for measuring lengths and angles, which fits well with the complex structure
- K-stability
- A notion of stability that holds when the Donaldson-Futaki invariant is positive for every degeneration; it is related to the existence of canonical metrics
- Fano variety
- An algebraic variety whose curvature is uniformly oriented in the positive direction
ResearchResearch themes
The lab uses nonlinear partial differential equations, such as the Monge-Ampère equation, to study the geometric properties of algebraic varieties. The main focus is the relationship between canonical metrics on manifolds and K-stability.
*Illustrative imageCanonical metrics and stability
The lab studies when a manifold algebraically embedded in projective space admits a canonical Kähler metric. In higher dimensions, smoothness alone is not enough; one needs to degenerate the manifold to non-smooth ones and examine how it behaves.
*Illustrative imageUniform stability
In joint research, the concept of uniform stability was introduced, and it was shown that the existence of a Kähler-Einstein metric implies this stronger form of stability. The delta invariant, which measures its degree, is actively studied from the viewpoint of algebraic geometry.
*Illustrative imageOptimal degenerations and parabolic equations
Recently, the lab has focused on Fano varieties that are not K-stable. An optimal degeneration that minimizes the invariant is believed to exist, and it corresponds to a global-in-time solution of a parabolic partial differential equation. The aim is to extract properties of the original variety from this degeneration.
MembersFaculty
- Associate ProfessorTomoyuki Hisamoto
VoicesStudent voices
Ratings and reviews reflect the experiences of individual current or former students, not the lab's official views. Whether a lab has reviews, or how many, says nothing about the lab itself.
Students who take part in an interview may receive a thank-you gift worth ¥1,000, regardless of what they say.
Student ratings (0 students)
--
--
--
--
--
--
Student voices answer questions like these
- What's great about this lab?
- What did you learn after joining?
- What's the vibe, and what is the professor like?
- Who is this lab for?
- Where do graduates go?
Your experience can help younger students choose their lab. Your name won't be shown.
Write a lab reviewAnonymous · Your draft is saved if you stopRelatedLabs similar to Algebraic Geometry, Differential Geometry, and Complex Geometry Lab
Differential Geometry, Complex Geometry, Dynamical Systems, and Global Analysis LabSchool of Science, Department of MathematicsModuli spaces of solutions to nonlinear partial differential…
Fourier Analysis and Partial Differential Equations LabSchool of Science, Department of MathematicsTheory of nonlinear partial differential equations. Nonlinea…
Naito LabSchool of Science, Department of MathematicsStudies variational problems arising from differential geome…
Analytic and Algebraic Number Theory LabSchool of Science, Department of MathematicsMultiple zeta values and the theory of modular forms; algebr…The information on this page is compiled by the LabLog team from the lab's public information and interviews. For corrections or removal requests, please contact LabLog, not the lab.
Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Peano intersection” Salix alba / CC BY-SA 4.0 (cropped)
- Canonical metrics and stability: “Differential geometry of manifolds of figures 1977 cover” 作者不詳 / Public domain (cropped)
- Uniform stability: “Algebraic geometry” P.wormer / CC BY-SA 3.0 (cropped)
- Optimal degenerations and parabolic equations: “De Long Room - DPLA - 4a293bd28e3b53c081af01f5355f855c” Brackett, J. Raymond / Public domain (cropped)
- Related lab (微分幾何・複素幾何・力学系・
大域解析学 （松尾 信一郎） ): “Minimal surface curvature planes-nl” ChristiaanPR / CC BY-SA 4.0 (cropped) - Related lab (フーリエ解析・偏微分方程式
（加藤 淳） ): “Ocean waves water” Jon Sullivan / Public domain (cropped) - Related lab (微分幾何・離散幾何解析・偏微分方程式・
数値解析 （内藤 久資） ): “Carbon lattice diamond” GYassineMrabetTalk✉ This structure was created with PyMOL. / Public domain (cropped) - Related lab (解析的整数論・代数的整数論
（ヘンリク・バッハマン） ): “ZetaFunction” Gaussianer / Public domain (cropped)
About LabLog
LabLog is a lab review app for students only. Student members at each university interview students in labs and publish what they say. Student voices can only be read by students who sign up with a university email address. Operated by bestiee Inc.
Learn more about LabLog ↗Search engines and AI may reference the basic lab info and rating summaries on this page and the description of LabLog. Student voices are for student members only. Automated collection by programs (scraping, or input to or training of AI) is prohibited under Article 8 of the Terms of Service. If you need data for research or analysis, please contact us.