In model theory—a branch of mathematical logic—a minimal structure is an infinite one-sorted structure such that every subset of its domain that is definable with parameters is either finite or cofinite. A strongly minimal theory is a complete theory all models of which are minimal. A strongly minimal structure is a structure whose theory is strongly minimal. Thus a structure is minimal only if the parametrically definable subsets of its domain cannot be avoided, because they are already parametrically definable in the pure language of equality.Strong minimality was one of the early notions in the new field of classification theory and stability theory that was opened up by Morley's theorem on totally categorical structures. The nontrivial standard examples for strongly minimal theories are the one-sorted theories of infinite-dimensional vector spaces, and the theories ACFp of algebraically closed fields. As the example ACFp shows, the parametrically definable subsets of the square of the domain of a minimal structure can be relatively complicated ("curves"). More generally, a subset of a structure that is defined as the set of realizations of a formula φ(x) is called a minimal set if every parametrically definable subset of it is either finite or cofinite. It is called a strongly minimal set if this is true even in all elementary extensions. A strongly minimal set, equipped with the closure operator given by algebraic closure in the model-theoretic sense, is an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. Totally categorical theories are controlled by a strongly minimal set; this fact explains (and is used in the proof of) Morley's theorem. Boris Zilber conjectured that the only pregeometries that can arise from strongly minimal sets are those that arise in vector spaces, projective spaces, or algebraically closed fields. This conjecture was refuted by Ehud Hrushovski, who developed a method known as "Hrushovski construction" to build new strongly minimal structures from finite structures. (Wikipedia).
Strong minimality for Painleve equations and Fuchsian equations Strong minimality is a central notion in model theory which has an interpretation in differential algebra as a functional transcendence statement. We will talk about some new proofs of strong minimality for differential equat
From playlist DART X
Strongly minimal groups in o-minimal structures - K. Peterzil - Workshop 3 - CEB T1 2018
Kobi Peterzil (Haifa) / 27.03.2018 Strongly minimal groups in o-minimal structures Let G be a definable two-dimensional group in an o-minimal structure M and let D be a strongly minimal expansion of G, whose atomic relations are definable in M. We prove that if D is not locally modular t
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
This is Why Minimalism is a Thing
In this video I talk about why minimalism is a thing people follow in today's society. There is a reason, and it's a good one.
From playlist Inspiration and Life Advice
Ampleness in strongly minimal structures - K. Tent - Workshop 3 - CEB T1 2018
Katrin Tent (Münster) / 30.03.2018 Ampleness in strongly minimal structures The notion of ampleness captures essential properties of projective spaces over fields. It is natural to ask whether any sufficiently ample strongly minimal set arises from an algebraically closed field. In this
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Hans Schoutens: O-minimalism: the first-order properties of o-minimality
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Algebra
Minimal logic, or minimal calculus, is an intuitionistic and paraconsistent logic, that rejects both the Law of Excluded Middle (LEM) as well as the Principle Of Explosion (Ex Falso Quodlibet, EFQ). https://en.wikipedia.org/wiki/Minimal_logic https://en.wikipedia.org/wiki/Principle_of_exp
From playlist Logic
Towards Strong Minimality and the Fuchsian Triangle Groups - J. Nagloo - Workshop 3 - CEB T1 2018
Joel Nagloo (City University of New York) / 29.03.2018 Towards Strong Minimality and the Fuchsian Triangle Groups From the work of Freitag and Scanlon, we have that the ODEs satisfied by the Hauptmoduls of arithmetic subgroups of SL2(Z) are strongly minimal and geometrically trivial. A c
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
F. Coda Marques - Morse theory and the volume spectrum
In this talk I will survey recent developments on the existence theory of closed minimal hypersurfaces in Riemannian manifolds, including a Morse-theoretic existence result for the generic case.
From playlist 70 ans des Annales de l'institut Fourier
Gabriele NEBE - Lattices, Perfects lattices, Voronoi reduction theory, modular forms, ...
Lattices, Perfects lattices, Voronoi reduction theory, modular forms, computations of isometries and automorphisms The talks of Coulangeon will introduce the notion of perfect, eutactic and extreme lattices and the Voronoi's algorithm to enumerate perfect lattices (both Eulcidean and He
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
Functional Transcendence via Model Theory - Joel Ronnie Nagloo
CAARMS Topic: Functional Transcendence via Model Theory Speaker: Joel Ronnie Nagloo Affiliation: Bronx Community College - CUNY Date: July 12, 2018 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Matthew DeVilbiss, University of Illinois at Chicago
October 28, Matthew DeVilbiss, University of Illinois at Chicago Generic Differential Equations are Strongly Minimal
From playlist Fall 2021 Online Kolchin Seminar in Differential Algebra
Title: Differential Fields—A Model Theorist's View May 2016 Kolchin Seminar Workshop
From playlist May 2016 Kolchin Seminar Workshop
Ax-Lindemann-Weierstrass Theorem (ALW) for Fuchsian automorphic functions - Joel Nagloo
Joint IAS/Princeton University Number Theory Seminar Topic: Ax-Lindemann-Weierstrass Theorem (ALW) for Fuchsian automorphic functions Speaker: Joel Nagloo Affiliation: City University of New York Date: January 21, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Virginie Ehrlacher - Sparse approximation of the Lieb functional in DFT with moment constraints
Recorded 28 March 2023. Virginie Ehrlacher of the École Nationale des Ponts-et-Chaussées presents "Sparse approximation of the Lieb functional in DFT with moment constraints (joint work with Luca Nenna)" at IPAM's Increasing the Length, Time, and Accuracy of Materials Modeling Using Exasca
From playlist 2023 Increasing the Length, Time, and Accuracy of Materials Modeling Using Exascale Computing
What is quantum mechanics? A minimal formulation (Seminar) by Pierre Hohenberg
29 December 2017 VENUE : Ramanujan Lecture Hall, ICTS , Bangalore This talk asks why the interpretation of quantum mechanics, in contrast to classical mechanics is still a subject of controversy, and presents a 'minimal formulation' modeled on a formulation of classical mechanics. In bot
From playlist US-India Advanced Studies Institute: Classical and Quantum Information