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Nakamura Lab
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A lab that explores algebraic varieties through minimal model theory and singularity theory.
AboutAbout the lab
Finding well-behaved representatives of algebraic varieties
Associate Professor Yusuke Nakamura's lab studies birational geometry, a branch of algebraic geometry.
Birational geometry divides complex algebraic varieties into birational equivalence classes, and aims to find a well-behaved representative in each class.
One method for finding such representatives is the minimal model program. The lab aims to solve the termination of flips problem, which concerns whether this program terminates, using singularity theory.
The lab is also interested in applications of minimal model theory, and it also conducts research in pure combinatorics that has no direct connection to algebraic geometry.
Key terms
- Birational geometry
- A branch of geometry that studies complex algebraic varieties by dividing them into birational equivalence classes and finding a well-behaved representative in each class
- Minimal model program
- A method that repeatedly contracts subvarieties to reach a well-behaved representative (a minimal model)
- Termination of flips
- An open problem of showing that the operation called a flip cannot be repeated infinitely in the minimal model program
ResearchResearch themes
The lab focuses on birational geometry, a branch of algebraic geometry. It studies the foundational theory of minimal models and is also interested in its applications, and it conducts research in combinatorics as well.
*Illustrative imageTermination of flips
The minimal model program is a method that reaches a minimal model by repeatedly contracting subvarieties. Proving that flips cannot be repeated infinitely is an open problem, and the lab approaches it through singularity theory.
*Illustrative imageRational points of Fano varieties over finite fields
As an application of minimal model theory, the lab has worked on extending Esnault's result (2003), which shows that a smooth Fano variety defined over a finite field always has a rational point, to the three-dimensional case with singularities.
*Illustrative imageEhrhart theory for periodic graphs
Growing out of a personal interest, the lab also does research in pure combinatorics on extending Ehrhart theory to periodic graphs. Ehrhart theory has an interpretation in terms of the cohomology of toric varieties, but such an interpretation has not yet been found for periodic graphs.
MembersFaculty
- Associate ProfessorYusuke Nakamura
VoicesStudent voices
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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: “Willmore Surface sculpture Durham” Chemical Engineer / CC BY-SA 4.0 (cropped)
- Termination of flips: “Algebraic surfaces” Ashton Bradley Aspir8 / CC BY-SA 4.0 (cropped)
- Rational points of Fano varieties over finite fields: “Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface” Ulrich Derenthal, Oliver Labs / CC BY-SA 3.0 (cropped)
- Ehrhart theory for periodic graphs: “Goldberg polyhedron 5 3” Tomruen / CC BY-SA 3.0 (cropped)
- Related lab (Arithmetic Geometry and Algebraic Geometry Lab): “Algebraic–Geometry1” Jean Constant / CC BY-SA 4.0 (cropped)
- Related lab (Complex Geometry Lab): “Triangle general surface” Gregors / CC BY-SA 3.0 (cropped)
- Related lab (Hishida Lab): “F1 CFD Simulation” Cofaser / CC BY-SA 4.0 (cropped)
- Related lab (Izumi Lab): “Lensed Galaxy Arc in Galaxy Cluster RCS” NASA Hubble / CC BY 2.0 (cropped)
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