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Effort to birationally classify algebraic varieties
geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex
Minimal_model_program
Surgery operation in minimal model program
in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any
Flip_(algebraic_geometry)
Type of metric in Riemannian geometry
problem, which is called the algebrization conjecture via analytical minimal model program. Suppose ( X , g ) {\displaystyle (X,g)} is a Riemannian manifold
Kähler–Einstein_metric
Mathematics award
curvature bounds." 2019 Chenyang Xu – "For major advances in the minimal model program and applications to the moduli of algebraic varieties." Karim Adiprasito
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Kurdish mathematician
proof of boundedness of Fano varieties and contributions to the minimal model program". In his office at the university, Birkar has two photographs of
Caucher_Birkar
Chinese mathematician
University. Xu is known for his work in birational geometry, the minimal model program, and the K-stability of Fano varieties. After completing his PhD
Chenyang_Xu
Mathematical model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety AK defined over the field of fractions K of a
Néron_model
Japanese mathematician (born 1951)
of Mori's results to dimensions higher than three is called the minimal model program and is an active area of research in algebraic geometry. He has
Shigefumi_Mori
Study of complex manifolds and several complex variables
of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli spaces sets the field apart from
Complex_geometry
conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X {\displaystyle X} with
Abundance_conjecture
Formal semantics of logic programming languages
coincides with the minimal Herbrand model of the program. The fixpoint semantics suggest an algorithm for computing the minimal Herbrand model: Start with the
Syntax and semantics of logic programming
Syntax_and_semantics_of_logic_programming
Singularities of algebraic varieties
(1980). Terminal singularities are important in the minimal model program because smooth minimal models do not exist in the desired generality, and hence
Canonical_singularity
Simplest supersymmetric extension to the Standard Model
The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical
Minimal Supersymmetric Standard Model
Minimal_Supersymmetric_Standard_Model
Generalizations of codimension-1 subvarieties of algebraic varieties
Project, Tag 0AFW. "Chapter 2. Preliminaries". Foundations of the minimal model program. Mathematical Society of Japan Memoirs. 2017. pp. 16–47. doi:10
Divisor_(algebraic_geometry)
Type of commutative ring in mathematics
Cohen–Macaulay. Toric varieties over any field are Cohen–Macaulay. The minimal model program makes prominent use of varieties with klt (Kawamata log terminal)
Cohen–Macaulay_ring
Topics referred to by the same term
an ancient Egyptian mathematical papyrus Matrilysin, an enzyme Minimal model program, a branch of birational geometry Million progressive motile (million
MMP
Hungarian mathematician
Princeton University. Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic
János_Kollár
Algebraic geometry scheme
ISBN 978-3-540-03603-6, MR 0222093 Kollár, János (2013), Singularities of the Minimal Model Program, Cambridge University Press, ISBN 978-1-107-03534-8, MR 3057950
Gorenstein_scheme
Field of algebraic geometry
especially in traditional problems in algebraic geometry. Famously the minimal model program was used to construct moduli spaces of varieties of general type
Birational_geometry
Algebraic structure with addition, multiplication, and division
in higher dimensions is referred to as birational geometry. The minimal model program attempts to identify the simplest (in a certain precise sense) algebraic
Field_(mathematics)
Concept in algebraic geometry
definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of
Fano_variety
1, "Book on Moduli of Surfaces". Mori's minimal model program The minimal model program is a research program aiming to do birational classification of
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Geometric projective variety
named so because it behaves nicely from the point of view of Mori's minimal model program. Any quasi-smooth projective spherical variety (in particular, any
Mori_dream_space
Complex manifolds in mathematics
affirmatively the problem of whether they are Cohen–Macaulay. The minimal model program started by Shigefumi Mori proved that the sheaf in the definition
Relative_canonical_model
Concept in algebraic geometry
map, where k is any sufficiently divisible positive integer. The minimal model program proposed that the canonical ring of every smooth or mildly singular
Canonical_bundle
perspective is crucial in birational geometry (in particular in Mori's minimal model program). Let X {\displaystyle X} be a projective variety and N S ¯ ( X
Contraction_morphism
Algebraic geometry
positive dimension. The ones arising from extremal contractions in the minimal model program are called Mori fibrations or Mori fiber spaces (for Shigefumi Mori)
Fano_fibration
Geometric space whose points represent algebro-geometric objects of some fixed kind
See also Shimura variety. Using techniques arising out of the minimal model program, moduli spaces of varieties of general type were constructed by
Moduli_space
Concept in algebraic geometry
fundamental to the classification of algebraic varieties and the minimal model program. Kodaira vanishing fails over fields of positive characteristic
Coherent_sheaf_cohomology
Algebraic variety of dimension two
in a sense, and give rise to much of their interesting geometry. Minimal model program Dolgachev, I.V. (2001) [1994], "Algebraic surface", Encyclopedia
Algebraic_surface
model of a logic program P then A is the head of one of the rules of P. Minimality Any stable model of a logic program P is minimal among the models of
Stable_model_semantics
Japanese mathematician
geometry. Kawamata was involved in the development of the minimal model program in the 1980s. The program aims to show that every algebraic variety is birational
Yujiro_Kawamata
Public research university in England
the boundedness of fano varieties and for contributions to the minimal model program". Frederick Kipping, professor of chemistry (1897–1936), made the
University_of_Nottingham
Gives general conditions under which sheaf cohomology groups with indices > 0 are zero
273–278, MR 0541027 Fujino, Osamu (2009). "Introduction to the log minimal model program for log canonical pairs". arXiv:0907.1506 [math.AG]. Sommese, Andrew
Kodaira_vanishing_theorem
Shading language
for one that supports only minimal pixel shaders. By creating a pixel shader for each of these profiles a supporting program enlarges the number of supported
High-Level_Shader_Language
Algebraic variety in a projective space
equivalence relation Hilbert scheme Lefschetz hyperplane theorem Minimal model program Kollár & Moduli, Ch I. Shafarevich, Igor R. (1994), Basic Algebraic
Projective_variety
also Fano leads to strong links with birational geometry and the minimal model program, providing a number of alternative characterisations of K-stability
K-stability_of_Fano_varieties
Kind of partial function between algebraic varieties
Function field of an algebraic variety Resolution of singularities Minimal model program Log structure Hartshorne, Robin (1977), Algebraic Geometry, Berlin
Rational_mapping
Vanishing theorem for multiplier ideals
ISBN 978-3-540-22531-7. Fujino, Osamu (2011). "Fundamental Theorems for the Log Minimal Model Program". Publications of the Research Institute for Mathematical Sciences
Nadel_vanishing_theorem
Unirational variety Ruled variety Kodaira dimension Canonical ring Minimal model program Intersection theory Intersection number Chow ring Chern class Serre's
List of algebraic geometry topics
List_of_algebraic_geometry_topics
fellow János Kollár – received Cole Prize for contributions to the minimal model program for threefolds in algebraic geometry Graeme Milton – received SIAM
List of University of Utah people
List_of_University_of_Utah_people
Mathematical classification of surfaces
(In the more modern terminology of the minimal model program, a smooth projective surface X would be called minimal if its canonical line bundle KX is nef
Enriques–Kodaira classification
Enriques–Kodaira_classification
Declarative logic programming language
the program, if the atoms in the body of the rule are in the set, then so is the head. The model-theoretic semantics define the minimal Herbrand model to
Datalog
Structure over a vocabulary defined solely by syntactical properties
\sigma } . A Herbrand structure that is a model of a theory T {\displaystyle T} is called a Herbrand model of T {\displaystyle T} . For a constant symbol
Herbrand_structure
Programming language
Minimal BASIC is a dialect of the BASIC programming language developed as an international standard. The effort started at ANSI in January 1974, and was
Minimal_BASIC
Linguistic research program proposed by Noam Chomsky
Chomsky. Following Imre Lakatos's distinction, Chomsky presents minimalism as a program, understood as a mode of inquiry that provides a conceptual framework
Minimalist_program
Process by which software is developed
Slow programming As part of the larger slow movement, emphasizes careful and gradual work without (or minimal) time pressures. Slow programming aims to
Software_development_process
and only if κ(V)=0 and q(V)=3 and its generalization so on. The minimal model program might be led from this conjecture. Iitaka, Shigeru (1970), "On D-dimensions
Iitaka_dimension
Singularities of algebraic varieties
ISBN 978-0-691-12923-5, MR 2289519 Kollár, János (2013), Singularities of the Minimal Model Program, Cambridge University Press, doi:10.1017/CBO9781139547895, ISBN 978-1-107-03534-8
Normal_crossing_singularity
Russian mathematician (born 1950)
base is a smooth minimal rational surface. Since the late 80's Shokurov began to contribute to the development of the Minimal model program (MMP). In 1984
Vyacheslav_Shokurov
Project to classify Fano varieties
four dimensions, they may be decomposable (in the sense of the minimal model program) into as few as a few thousand "building blocks". List of complex
Periodic_table_of_shapes
General-purpose programming language
instructions, all with minimal runtime support. Despite its low-level capabilities, the language was designed to encourage cross-platform programming. A standards-compliant
C_(programming_language)
Surface that locally minimizes its area
Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface
Minimal_surface
British mathematician
various aspects of algebraic geometry, such as: singularities in the minimal model program; compactification of moduli spaces; the rationality of orbit spaces
Nicholas_Shepherd-Barron
Theorem in algebraic geometry
Kebekus, Stefan (2013). "Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks". Handbook of Moduli II. Advanced
Bogomolov–Sommese vanishing theorem
Bogomolov–Sommese_vanishing_theorem
British automotive technology company
deployment programs in secondary markets, including Southeast Asia. In early 2026, the company introduced a safety framework detailing automated minimal risk
Carziqo
Pump-action shotgun
Olin Corporation, starting in 1964. It was redesignated the Model 1300 in 1978 with minimal changes. Production ceased in 2006 when the U.S. Repeating
Winchester_Model_1200
Programming language
composed with other Datalog program values. The minimal model of a Datalog program value can be computed and is itself a Datalog program value. In this way, Flix
Flix_(programming_language)
Minimum weapons needed to deter an adversary
In nuclear strategy, minimal deterrence, also known as minimum deterrence and finite deterrence, is an application of deterrence theory in which a state
Minimal_deterrence
Programming paradigm based on formal logic
standard) model of the program. For Horn clause programs, there always exists such a standard model: It is the unique minimal model of the program. Informally
Logic_programming
Compartmental and kinetic modeling software
SAAM II (Simulation, Analysis, and Modeling, version 2.0) is a computer program used for compartmental modeling and systems analysis in the life sciences
SAAM_II
Family of programming languages
allowed multiple users to edit and run BASIC programs simultaneously on remote terminals. This general model became popular on minicomputer systems like
BASIC
Person who writes computer software
is an author of computer source code – someone with skill in computer programming. The professional titles software developer and software engineer may
Programmer
Measure of algorithmic complexity
complexity is the minimal description length of any program, and denoted C ( x ) {\displaystyle C(x)} while the prefix-free complexity is the minimal description
Kolmogorov_complexity
Assessment framework and assessor certification program
The Cybersecurity Maturity Model Certification (CMMC) is an assessment framework and assessor certification program designed for a variety of standards
Cybersecurity Maturity Model Certification
Cybersecurity_Maturity_Model_Certification
threefold is an algebraic variety of dimension 3. The Mori program showed that 3-folds have minimal models. Algebraic curve Algebraic surface Cubic threefold
3-fold
Security by granting only essential access
the Principle of Least Privilege (PoLP), also known as the Principle of Minimal Privilege (PoMP) or the Principle of Least Authority (PoLA), requires that
Principle_of_least_privilege
Area of mathematical logic
called strong minimality: A theory T is called strongly minimal if every model of T is minimal. A structure is called strongly minimal if the theory of
Model_theory
Prototype-based programming language
object-oriented programming language inspired by Smalltalk, Self, Lua, Lisp, Act1, and NewtonScript. Io has a prototype-based object model similar to those
Io_(programming_language)
Aspect of consciousness research
correlates of consciousness programmed as a neural network. Stanislas Dehaene and Jean-Pierre Changeux introduced this model in 1986. It is associated with
Models_of_consciousness
American software company
startup accelerator program. In its early years, the company grew with minimal outside investment and operated under a lean business model. Since 2021, it
Vanta_(company)
Book by Brian Kernighan and Dennis Ritchie
introduced the "Hello, World!" program, which prints only the text "hello, world" as an illustration of a minimal working C program. Since then, many texts have
The_C_Programming_Language
Processes to automate content creation on YouTube
revenue-generating system that operates efficiently with minimal direct intervention from the channel owner. This model emphasizes the technical aspects of channel
YouTube_automation
AI that generates content
techniques beyond Markov models. Harold Cohen developed and exhibited works produced by AARON, a pioneering computer program designed to autonomously
Generative_AI
Computation model defining an abstract machine
completeness is the ability for a model of computation or a system of instructions to simulate a Turing machine. A programming language that is Turing complete
Turing_machine
Model or diagram describing interrelated things
entity–relationship model (or ER model) describes interrelated things of interest in a specific domain of knowledge. A basic ER model is composed of entity
Entity–relationship_model
Concerned with the notion of stability in model theory
reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories
Stable_theory
German military officer (1891–1945)
Otto Moritz Walter Model (IPA: [ˈmoːdəl]; 24 January 1891 – 21 April 1945) was a German Generalfeldmarschall during World War II. Although he was a hard-driving
Walter_Model
Programming language
concurrency. While the communicating-processes model is favored in Go, it is not the only one: all goroutines in a program share a single address space. This means
Go_(programming_language)
Mathematical optimization concept
Equilibrium Model and Dual Linear Program". CRAN - R Project. Retrieved 2023-06-26. Detailed documentation on the General Equilibrium Model and Dual Linear
Dual_linear_program
Framework for communication protocols used in IP networking
OSI model (presentation and session layers). According to the TCP/IP model, such functions are the realm of libraries and application programming interfaces
Internet_protocol_suite
Programming paradigm
the probabilistic facts, the resulting logic program is stratified, it has a unique minimal Herbrand model which can be seen as the unique interpretation
Probabilistic logic programming
Probabilistic_logic_programming
Computer architecture where code and data share a common bus
The von Neumann architecture—also known as the von Neumann model or Princeton architecture—is a computer architecture based on the First Draft of a Report
Von_Neumann_architecture
Population models of evolutionary algorithms
The population model of an evolutionary algorithm (EA) describes the structural properties of its population to which its members are subject. A population
Population model (evolutionary algorithm)
Population_model_(evolutionary_algorithm)
First Lady of the United States (2017–2021; since 2025)
(born Melanija Knavs; April 26, 1970) is a Slovenian and American former model serving as the first lady of the United States since 2025, a role she previously
Melania_Trump
Mathematical model in economics
_{k=h+1}^{t}i_{h}d_{k}+F(j-1)\right] \atop s_{t}+F(t-1)}\right]} denote the minimal cost program for periods 1 to t. If at period t* the minimum in F(t) occurs for
Dynamic_lot-size_model
Early object-oriented programming language
the minimal program in Simula, measured by the size of the source code. It consists of one thing only; a dummy statement. However, the minimal program is
Simula
Checking software against expectations
separate program module or library. Sanity testing determines whether it is reasonable to proceed with further testing. Smoke testing consists of minimal attempts
Software_testing
Piece of software accessing a server service
A client is part of a client–server model, which is still used today. Clients and servers may be computer programs run on the same machine and connect
Client_(computing)
Proactive cyber defense activity
on automated reporting and does little or no routine data collection. Minimal – At Level 1 maturity, an organization incorporates threat intelligence
Threat_hunting
Problem in computer science
Turing-complete model of computation. This model of computation includes all programs in Turing-equivalent programming languages. Given a program and an input
Halting_problem
Mental illness with multiple personality states
trauma model, viewing the disorder as an organic response to severe childhood trauma. Critics of the trauma model support the sociogenic (fantasy) model of
Dissociative identity disorder
Dissociative_identity_disorder
Line of smartphones by Apple
year. Since then, Apple has annually released new iPhone models and iOS versions; the latest models are the iPhone 17, the higher-end iPhone 17 Pro and 17
IPhone
Machine to increase pressure of gas by reducing its volume
maintain since less components are exposed to wear, and only generate minimal vibration. But, their initial cost is higher, require highly precise CNC
Compressor
Software company
Y Combinator startup accelerator program. As a low-code/no-code platform, Zapier is intended for users with minimal to moderate technical knowledge. As
Zapier
Models constructed from paper
Paper models, also called card models or papercraft, are models constructed mainly from sheets of heavy paper, paperboard, card stock, or foam. This may
Paper_model
Type of logical formula
if P is true in M. The minimal model semantics of Horn clauses is the basis for the stable model semantics of logic programs. Constrained Horn clauses
Horn_clause
In theoretical computer science the random-access stored-program (RASP) machine model is an abstract machine used for the purposes of algorithm development
Random-access stored-program machine
Random-access_stored-program_machine
Human-readable data serialization language
communications applications as Extensible Markup Language (XML) but has a minimal syntax that intentionally differs from Standard Generalized Markup Language
YAML
Language for controlling a computer
input could be added while the program was running. The languages developed at this time therefore are designed for minimal interaction. After the invention
Programming_language
MINIMAL MODEL-PROGRAM
MINIMAL MODEL-PROGRAM
Surname or Lastname
English
English : from an Old German personal name, Godilo, Godila.German (Gödel) : from a pet form of a compound personal name beginning with the element gÅd ‘good’ or god, got ‘god’.Variant of Godl or Gödl, South German variants of Gote, from Middle High German got(t)e, gö(t)te ‘godfather’.Jewish (Ashkenazic) : from the Yiddish male personal name Godl, a pet form of God, a variant of biblical Gad.
Boy/Male
Egyptian
To model.
Boy/Male
Muslim
Sample, Model, Paragon
Girl/Female
English, Hindu, Indian, Marathi
Small Daughter
Boy/Male
Muslim
Model, Example
Male
Yiddish
Pet form of Yiddish Mordche, MOTEL means "devotee of Marduk."Â
Girl/Female
Christian & English(British/American/Australian)
Model or Pattern
Girl/Female
Hebrew
From the tower.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Marathi
Enjoyment
Boy/Male
Latin
Swarthy.
Girl/Female
Hindu, Indian, Traditional
Model; Idea
Female
Yiddish
(×”Ö¸×דֶעל) Pet form of Yiddish Hode, HODEL means "myrtle tree."
Boy/Male
Australian, French
Famous Ruler
Boy/Male
Hindu
Model state of india
Girl/Female
Arabic, Muslim
Example; Model; Demo
Boy/Male
Anglo Saxon
Wealthy.
Boy/Male
Arabic, Muslim
Sample; Model; Paragon
Boy/Male
Arabic, Muslim
Model; Example
Girl/Female
British, English, German, Russian
Supper
Surname or Lastname
English (Surrey)
English (Surrey) : unexplained. Compare Moad.
MINIMAL MODEL-PROGRAM
MINIMAL MODEL-PROGRAM
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a place in the parish of Kirkham, Lancashire, named from Old Norse byname Svartr ‘black’ + Old Norse brekka ‘slope’.
Girl/Female
Arabic, Australian, Malaysian, Muslim
Abstinent
Boy/Male
Indian, Punjabi, Sikh
Lord's Light
Boy/Male
Hindu
Boy/Male
Muslim/Islamic
Dignity Splendour
Boy/Male
Indian, Tamil
Worship; Warrior in Prayer
Boy/Male
Hindi
White horse.
Male
Russian
(Олег) Russian form of Scandinavian Helge, OLEG means "dedicated to the gods; holy."
Girl/Female
British, English, Greek
Follower of Christ
Boy/Male
American, Australian, Celtic, Chinese, Christian, Greek, Irish
Calm; Healer; Tranquil
MINIMAL MODEL-PROGRAM
MINIMAL MODEL-PROGRAM
MINIMAL MODEL-PROGRAM
MINIMAL MODEL-PROGRAM
MINIMAL MODEL-PROGRAM
n.
The scale as affected by the various positions in it of the minor intervals; as, the Dorian mode, the Ionic mode, etc., of ancient Greek music.
n.
Something intended to serve, or that may serve, as a pattern of something to be made; a material representation or embodiment of an ideal; sometimes, a drawing; a plan; as, the clay model of a sculpture; the inventor's model of a machine.
n.
Manner of doing or being; method; form; fashion; custom; way; style; as, the mode of speaking; the mode of dressing.
a.
Impregnated with minerals; as, mineral waters.
a.
Indicating, or pertaining to, some mode of conceiving existence, or of expressing thought.
a.
Of or pertaining to minerals; consisting of a mineral or of minerals; as, a mineral substance.
a.
Of or pertaining to a mode or mood; consisting in mode or form only; relating to form; having the form without the essence or reality.
a.
Of or relating to animals; as, animal functions.
n.
Prevailing popular custom; fashion, especially in the phrase the mode.
v. t.
To plan or form after a pattern; to form in model; to form a model or pattern for; to shape; to mold; to fashion; as, to model a house or a government; to model an edifice according to the plan delineated.
a.
Of or pertaining to a sine; employing, or founded upon, sines; as, a sinical quadrant.
v. i.
Anything which is neither animal nor vegetable, as in the most general classification of things into three kingdoms (animal, vegetable, and mineral).
pl.
of Minimus
pl.
of Minimum
n.
Anything which serves, or may serve, as an example for imitation; as, a government formed on the model of the American constitution; a model of eloquence, virtue, or behavior.
a.
Suitable to be taken as a model or pattern; as, a model house; a model husband.
v. i.
To make a copy or a pattern; to design or imitate forms; as, to model in wax.
a.
Consisting of the flesh of animals; as, animal food.