🌐 Mathematical Logic (منطق ریاضی)
Mathematical logic is the study of the strengths and limitations of formal languages, proofs, and algorithms and their relationships to mathematical structures. It also aims to address foundational issues in mathematics and investigates the power of mathematical reasoning itself. The various subfields of this area are connected through their study of foundational notions: sets, proof, computation, and models. The period from the 1930s through the 1970s saw great progress in logic. Mathematical logic is a broad area encompassing proof theory, computability theory, set theory, and model theory. These areas are joined by their focus on the interplay between expressibility, definability and provability. Logic relates to theoretical computer science through computability theory and proof theory, to algebra, number theory, and algebraic geometry through model theory, and to analysis and ergodic theory through set theory and infinite combinatorics.
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The exciting and active areas of logic today are set theory, model theory, and connections with computer science. Set theory addresses various ways to axiomatize mathematics, with implications for understanding the properties of sets having large infinite cardinalities and connections with the axiomatization of mathematics. Model theory investigates particular mathematical theories, such as complex algebraic geometry, and has been used to settle open questions in these areas. Theoretical computer science developed partially out of logic and Computational complexity, as part of theoretical computer science, is deeply connected to questions in proof theory and computability theory as well as to related areas including combinatorics and probability. The core goal of computational complexity is to determine the limits of computation; this includes some of the most fundamental open questions in mathematics and theoretical computer science, including the P versus NP question.
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#منطق_ریاضی
Mathematical logic is the study of the strengths and limitations of formal languages, proofs, and algorithms and their relationships to mathematical structures. It also aims to address foundational issues in mathematics and investigates the power of mathematical reasoning itself. The various subfields of this area are connected through their study of foundational notions: sets, proof, computation, and models. The period from the 1930s through the 1970s saw great progress in logic. Mathematical logic is a broad area encompassing proof theory, computability theory, set theory, and model theory. These areas are joined by their focus on the interplay between expressibility, definability and provability. Logic relates to theoretical computer science through computability theory and proof theory, to algebra, number theory, and algebraic geometry through model theory, and to analysis and ergodic theory through set theory and infinite combinatorics.
📚 @topmathlearn 📚
The exciting and active areas of logic today are set theory, model theory, and connections with computer science. Set theory addresses various ways to axiomatize mathematics, with implications for understanding the properties of sets having large infinite cardinalities and connections with the axiomatization of mathematics. Model theory investigates particular mathematical theories, such as complex algebraic geometry, and has been used to settle open questions in these areas. Theoretical computer science developed partially out of logic and Computational complexity, as part of theoretical computer science, is deeply connected to questions in proof theory and computability theory as well as to related areas including combinatorics and probability. The core goal of computational complexity is to determine the limits of computation; this includes some of the most fundamental open questions in mathematics and theoretical computer science, including the P versus NP question.
📚 @topmathlearn 📚
#منطق_ریاضی
❤5
🌐 Topology (توپولوژی)
Topology is concerned with the intrinsic properties of the shapes of spaces and it studies properties of spaces that are invariant under deformations. Topology is the qualitative study of shapes and spaces by identifying and analyzing features that are unchanged when the object is continuously deformed. The modern field of topology draws from a diverse collection of core areas of mathematics. Much of basic topology is most profitably described in the language of algebra i.e. groups, rings, modules, and exact sequences. General topology now forms one of the foundations of modern analysis and geometry. The notion of moduli space was invented by Riemann in the 19th century to encode how Riemann surfaces vary in families; today, the study of geometric and homotopy-theoretic aspects of moduli spaces is an important subject with strong ties to algebraic and symplectic geometry. It also leads to interesting dynamical systems and group theory. More algebraic aspects of topology study homotopy theory and algebraic K-theory, and their applications to geometry and number theory. Topology itself has many active branches, many of which are intimately connected to other areas of mathematics and science: knot theory (biology, physics, algebra), algebraic topology (differential geometry, algebra, data analysis, physics, chemistry, engineering, robotics), geometric group theory (algebra).
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A “topology” is precisely the minimum structure on a set that allows one to even define what “continuous” means. Continuity, which refers to changes that may “stretch” or “fold” but never “tear”, is a fundamental concept in mathematics and science. By establishing the “minimal” requirements to understand continuity, topology has applications in almost every branch of mathematics and science, such as analysis (analytical constructions such as differential forms play a crucial role in topology), differential geometry and partial differential equations (through the modern subject of gauge theory), algebraic geometry (for instance, through the topology of algebraic varieties), combinatorics (knot theory), and theoretical physics (general relativity and the shape of the universe, string theory). In addition, topology can strikingly be used to study a wide variety of more "applied" areas, ranging from the structure of large data sets to the geometry of DNA.
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One class of spaces which plays a central role in mathematics, and whose topology is extensively studied, are the n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space. The case of manifolds of dimension n=1 is straightforward, and the case where n=2 was understood thoroughly in the 19th century. Moreover, intense activity in the 1960’s (including the pioneering work of Browder, Milnor, Novikov, and Smale) expresses the topology of manifolds of dimension n>4 in terms of an elaborate but purely algebraic denoscription. The study of manifolds of dimension n=3 and 4 is quite different from the higher-dimensional cases; and, though both cases n=3 and 4 are quite different in their overall character, both are generally referred to as low-dimensional topology. Low-dimensional topology is currently a very active part of mathematics, benefiting greatly from its interactions with the fields of partial differential equations, differential geometry, algebraic geometry, modern physics, representation theory, number theory, and algebra.
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#توپولوژی
Topology is concerned with the intrinsic properties of the shapes of spaces and it studies properties of spaces that are invariant under deformations. Topology is the qualitative study of shapes and spaces by identifying and analyzing features that are unchanged when the object is continuously deformed. The modern field of topology draws from a diverse collection of core areas of mathematics. Much of basic topology is most profitably described in the language of algebra i.e. groups, rings, modules, and exact sequences. General topology now forms one of the foundations of modern analysis and geometry. The notion of moduli space was invented by Riemann in the 19th century to encode how Riemann surfaces vary in families; today, the study of geometric and homotopy-theoretic aspects of moduli spaces is an important subject with strong ties to algebraic and symplectic geometry. It also leads to interesting dynamical systems and group theory. More algebraic aspects of topology study homotopy theory and algebraic K-theory, and their applications to geometry and number theory. Topology itself has many active branches, many of which are intimately connected to other areas of mathematics and science: knot theory (biology, physics, algebra), algebraic topology (differential geometry, algebra, data analysis, physics, chemistry, engineering, robotics), geometric group theory (algebra).
📚 @topmathearn 📚
A “topology” is precisely the minimum structure on a set that allows one to even define what “continuous” means. Continuity, which refers to changes that may “stretch” or “fold” but never “tear”, is a fundamental concept in mathematics and science. By establishing the “minimal” requirements to understand continuity, topology has applications in almost every branch of mathematics and science, such as analysis (analytical constructions such as differential forms play a crucial role in topology), differential geometry and partial differential equations (through the modern subject of gauge theory), algebraic geometry (for instance, through the topology of algebraic varieties), combinatorics (knot theory), and theoretical physics (general relativity and the shape of the universe, string theory). In addition, topology can strikingly be used to study a wide variety of more "applied" areas, ranging from the structure of large data sets to the geometry of DNA.
📚 @topmathlearn 📚
One class of spaces which plays a central role in mathematics, and whose topology is extensively studied, are the n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space. The case of manifolds of dimension n=1 is straightforward, and the case where n=2 was understood thoroughly in the 19th century. Moreover, intense activity in the 1960’s (including the pioneering work of Browder, Milnor, Novikov, and Smale) expresses the topology of manifolds of dimension n>4 in terms of an elaborate but purely algebraic denoscription. The study of manifolds of dimension n=3 and 4 is quite different from the higher-dimensional cases; and, though both cases n=3 and 4 are quite different in their overall character, both are generally referred to as low-dimensional topology. Low-dimensional topology is currently a very active part of mathematics, benefiting greatly from its interactions with the fields of partial differential equations, differential geometry, algebraic geometry, modern physics, representation theory, number theory, and algebra.
📚 @topmathlearn 📚
#توپولوژی
❤5
For the case of manifolds of dimension n=3, a conjectural classification picture emerged in the 1970s, thanks to the work of William Thurston, in terms of symmetric geometries. Specifically, Thurston conjectured that every three-manifold can be decomposed canonically into pieces, each of which can be endowed with one of eight possible geometries. Elements of this vast picture are presently unfolding thanks to the Ricci Flow equations introduced by Richard Hamilton, which have been used by Grigory Perelman to solve the century-old Poincaré conjecture, and have also shed light on Thurston’s more general geometrization conjecture. The central role of Thurston’s conjecture in three-manifold topology has helped place hyperbolic geometry, the richest of the eight geometries, into the research forefront.
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The case of manifolds of dimension n=4 remains the most elusive. In view of the foundational results of Freedman, understanding manifolds up to their topological equivalence is a theory which is similar in character to the higher-dimensional manifold theory. However, the theory of differentiable four-manifolds is quite different. The subject was fundamentally transformed by the pioneering work of Simon Donaldson, who was studying moduli spaces of solutions to certain partial differential equations which came from mathematical physics. Studying algebro-topological properties of these moduli spaces, Donaldson came up with very interesting smooth invariants for four-manifolds which demonstrated the unique and elusive character of smooth four-manifold topology. In the case where the underlying manifold is Kähler, these moduli spaces also admit an interpretation in terms of stable bundles, and hence shed light on the differential topology of smooth algebraic surfaces. Since Donaldson’s work, the physicists Seiberg and Witten introduced another smooth invariant of four-manifolds. Since then, the study of four-manifolds and their invariants has undergone several further exciting developments, tying them deeply with ideas from symplectic geometry and pseudo-holomorphic curves, and hence forming further bridges with algebraic and symplectic geometry, but also connecting them more closely with knot theory and three-manifold topology.
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#توپولوژی
📚 @topmathlearn 📚
The case of manifolds of dimension n=4 remains the most elusive. In view of the foundational results of Freedman, understanding manifolds up to their topological equivalence is a theory which is similar in character to the higher-dimensional manifold theory. However, the theory of differentiable four-manifolds is quite different. The subject was fundamentally transformed by the pioneering work of Simon Donaldson, who was studying moduli spaces of solutions to certain partial differential equations which came from mathematical physics. Studying algebro-topological properties of these moduli spaces, Donaldson came up with very interesting smooth invariants for four-manifolds which demonstrated the unique and elusive character of smooth four-manifold topology. In the case where the underlying manifold is Kähler, these moduli spaces also admit an interpretation in terms of stable bundles, and hence shed light on the differential topology of smooth algebraic surfaces. Since Donaldson’s work, the physicists Seiberg and Witten introduced another smooth invariant of four-manifolds. Since then, the study of four-manifolds and their invariants has undergone several further exciting developments, tying them deeply with ideas from symplectic geometry and pseudo-holomorphic curves, and hence forming further bridges with algebraic and symplectic geometry, but also connecting them more closely with knot theory and three-manifold topology.
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#توپولوژی
❤5
🌐 Algebraic Topology (توپولوژی جبری)
Algebraic topology is the interplay between “continuous” and “discrete” mathematics. Continuous mathematics is formulated in its general form in the language of topological spaces and continuous maps. Discrete mathematics is used to express the concepts of algebra and combinatorics. The notion of shape is fundamental in mathematics. Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus. Algebraic methods become important in topology when working in many dimensions, and increasingly sophisticated parts of algebra are now being employed. In algebraic topology, we investigate spaces by mapping them to algebraic objects such as groups, and thereby bring into play new methods and intuitions from algebra to answer topological questions. For example, the arithmetic of elliptic curves, which was at the heart of Andrew Wiles' solution of the Fermat conjecture, has been lifted into topology, giving new and very powerful tools for the study of geometric objects.
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Classical algebraic topology consists of the construction and use of functors from some category of topological spaces into an algebraic category, say of groups. But one can also postulate that global qualitative geometry is itself of an algebraic nature. Consequently there are two important view points from which one can study algebraic topology: homology and homotopy. Homology, invented by Henri Poincaré, is without doubt one of the most ingenious and influential inventions in mathematics. The basic idea of homology is that we start with a geometric object (a space) which is given by combinatorial data (a simplicial complex). Then the linear algebra and boundary relations determined by these data are used to produce homology groups. Homotopy theory begins again with the construction of functors from topology to algebra. But this approach is important from another view point. Homotopy theory shows that the category of topological spaces has itself a kind of (hidden) algebraic structure. The development of algebraic topology led to the development of many of the key concepts of algebra, including homological algebra, category theory, Lie groups/algebras, and K-theory.
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#توپولوژی_جبری
Algebraic topology is the interplay between “continuous” and “discrete” mathematics. Continuous mathematics is formulated in its general form in the language of topological spaces and continuous maps. Discrete mathematics is used to express the concepts of algebra and combinatorics. The notion of shape is fundamental in mathematics. Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus. Algebraic methods become important in topology when working in many dimensions, and increasingly sophisticated parts of algebra are now being employed. In algebraic topology, we investigate spaces by mapping them to algebraic objects such as groups, and thereby bring into play new methods and intuitions from algebra to answer topological questions. For example, the arithmetic of elliptic curves, which was at the heart of Andrew Wiles' solution of the Fermat conjecture, has been lifted into topology, giving new and very powerful tools for the study of geometric objects.
📚 @topmathlearn 📚
Classical algebraic topology consists of the construction and use of functors from some category of topological spaces into an algebraic category, say of groups. But one can also postulate that global qualitative geometry is itself of an algebraic nature. Consequently there are two important view points from which one can study algebraic topology: homology and homotopy. Homology, invented by Henri Poincaré, is without doubt one of the most ingenious and influential inventions in mathematics. The basic idea of homology is that we start with a geometric object (a space) which is given by combinatorial data (a simplicial complex). Then the linear algebra and boundary relations determined by these data are used to produce homology groups. Homotopy theory begins again with the construction of functors from topology to algebra. But this approach is important from another view point. Homotopy theory shows that the category of topological spaces has itself a kind of (hidden) algebraic structure. The development of algebraic topology led to the development of many of the key concepts of algebra, including homological algebra, category theory, Lie groups/algebras, and K-theory.
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#توپولوژی_جبری
❤5
🌐 Analysis (آنالیز)
Analysis is at the very core of mathematics and covers a wide range of different subjects, studying limit processes, inequalities, integration, differentiation, measure theory, and analytic functions. It is replete with a roster of giants - Newton, Leibnitz, Cauchy, Riemann, Fourier, Hardy, Weierstrass, Gauss, Stokes, Poincaré - and is the original home for topics which have become major branches of mathematics in their own right - topology, differential equations, differential geometry, dynamical systems. Analysis also provides both the theoretical and the practical basis for the mathematics of the sciences and engineering, not to mention other areas in mathematics itself, from analytic number theory to computational mathematics. Applications and connections to other fields, both within and outside mathematics, are numerous. Calculus and the theory of real and complex continuous functions are among the crowning achievements of science. The field of mathematical analysis continues the development of that theory today to give even greater power and generality. Functions are representations of relations between sets, and in particular are useful for representing the changing states of a system: the velocity of a projectile, the frequencies present in a sound signal, the color of a pixel in a digital image, or the prices of a portfolio of stocks. The mathematical field of analysis seeks to formulate methods to analyze quantitatively the change exhibited by the outputs of functions with respect to their inputs, as a way of distilling important information about the underlying systems, such as the way stock prices change over time. Today, analysis forms the foundation of several highly active areas of mathematics, with powerful applications throughout mathematics as well as in the natural sciences and industry.
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Harmonic analysis seeks to decompose functions into their component "harmonics" or waves; that such a decomposition is possible for broad classes of functions was initially discovered by Fourier in the early 1800s, and continues to have applications in a vast swath of mathematics, ranging from image compression methods via "wavelets" to counting integral solutions to Diophantine equations via the Hardy-Littlewood "circle method". The Fourier transform and Fourier series are examples of topics included in harmonic analysis. Applications are extensive in such fields as signal processing, medical imaging and quantum mechanics. Complex analysis studies the behavior of functions on the complex plane and has deep connections to number theory, and in particular to the distribution of prime numbers, via the Riemann zeta function. Real analysis develops a rigorous theory of integration which extends the familiar notions of calculus to a broader class of functions, and in particular provides a foundation for many concepts in probability. Stochastic analysis seeks to understand the behavior of differential equations and dynamical systems whose parameters or inputs are random and unpredictable; this has many applications in the sciences and engineering, and economics.
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#آنالیز
Analysis is at the very core of mathematics and covers a wide range of different subjects, studying limit processes, inequalities, integration, differentiation, measure theory, and analytic functions. It is replete with a roster of giants - Newton, Leibnitz, Cauchy, Riemann, Fourier, Hardy, Weierstrass, Gauss, Stokes, Poincaré - and is the original home for topics which have become major branches of mathematics in their own right - topology, differential equations, differential geometry, dynamical systems. Analysis also provides both the theoretical and the practical basis for the mathematics of the sciences and engineering, not to mention other areas in mathematics itself, from analytic number theory to computational mathematics. Applications and connections to other fields, both within and outside mathematics, are numerous. Calculus and the theory of real and complex continuous functions are among the crowning achievements of science. The field of mathematical analysis continues the development of that theory today to give even greater power and generality. Functions are representations of relations between sets, and in particular are useful for representing the changing states of a system: the velocity of a projectile, the frequencies present in a sound signal, the color of a pixel in a digital image, or the prices of a portfolio of stocks. The mathematical field of analysis seeks to formulate methods to analyze quantitatively the change exhibited by the outputs of functions with respect to their inputs, as a way of distilling important information about the underlying systems, such as the way stock prices change over time. Today, analysis forms the foundation of several highly active areas of mathematics, with powerful applications throughout mathematics as well as in the natural sciences and industry.
📚 @topmathlearn 📚
Harmonic analysis seeks to decompose functions into their component "harmonics" or waves; that such a decomposition is possible for broad classes of functions was initially discovered by Fourier in the early 1800s, and continues to have applications in a vast swath of mathematics, ranging from image compression methods via "wavelets" to counting integral solutions to Diophantine equations via the Hardy-Littlewood "circle method". The Fourier transform and Fourier series are examples of topics included in harmonic analysis. Applications are extensive in such fields as signal processing, medical imaging and quantum mechanics. Complex analysis studies the behavior of functions on the complex plane and has deep connections to number theory, and in particular to the distribution of prime numbers, via the Riemann zeta function. Real analysis develops a rigorous theory of integration which extends the familiar notions of calculus to a broader class of functions, and in particular provides a foundation for many concepts in probability. Stochastic analysis seeks to understand the behavior of differential equations and dynamical systems whose parameters or inputs are random and unpredictable; this has many applications in the sciences and engineering, and economics.
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#آنالیز
❤5
🌐 PDEs & Dynamical Systems
( معادلات دیفرانسیل جزئی و سیستمهای دینامیکی)
The study of Partial Differential Equations (PDEs) is an interdisciplinary field within mathematics and various applied sciences, such as physics, engineering, biology, finance, and beyond. PDEs are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory.
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The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse. Applying analytic methods to partial differential equations (PDEs) allows researchers to study the evolution of a system that is changing in a manner governed by precise constraints (the differential equations in the name); PDEs may represent many important problems in the physical world, such as the diffusion of heat, fluid flow, or quantum mechanics. The formulation of models is grounded in first principles, often drawing from conservation laws and conducted by skilled practitioners. However, a significant challenge arises due to the inherent complexity of the resulting PDEs, rendering them challenging to solve analytically. Consequently, sophisticated numerical algorithms become essential for solving these equations effectively as well as examining the resulting solutions. Developing suitable numerical schemes, in turn, necessitates a clear understanding of the underlying analytical properties of the equations and potential solutions.
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#معادلات_دیفرانسیل_جزئی
#سیستمهای_دینامیکی
( معادلات دیفرانسیل جزئی و سیستمهای دینامیکی)
The study of Partial Differential Equations (PDEs) is an interdisciplinary field within mathematics and various applied sciences, such as physics, engineering, biology, finance, and beyond. PDEs are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory.
📚 @topmathlearn 📚
The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse. Applying analytic methods to partial differential equations (PDEs) allows researchers to study the evolution of a system that is changing in a manner governed by precise constraints (the differential equations in the name); PDEs may represent many important problems in the physical world, such as the diffusion of heat, fluid flow, or quantum mechanics. The formulation of models is grounded in first principles, often drawing from conservation laws and conducted by skilled practitioners. However, a significant challenge arises due to the inherent complexity of the resulting PDEs, rendering them challenging to solve analytically. Consequently, sophisticated numerical algorithms become essential for solving these equations effectively as well as examining the resulting solutions. Developing suitable numerical schemes, in turn, necessitates a clear understanding of the underlying analytical properties of the equations and potential solutions.
📚 @topmathlearn 📚
#معادلات_دیفرانسیل_جزئی
#سیستمهای_دینامیکی
❤5
🌐 Geometry (هندسه)
The modern discipline of geometry is affecting virtually every branch of mathematics and is in a period of great progress. Geometry combines techniques from apparently unconnected areas, such as the use of analytical tools for answering topological questions, and it has strong relations with quantum field theory and string theory. In modern language, the central object of study in geometry is a manifold, which is an object that may have a complicated overall shape, but such that on small scales it looks like ordinary space of a certain dimension. For example, a 1-dimensional manifold is an object such that small pieces of it look like a line, although in general it looks like a curve rather than a straight line. A 2-dimensional manifold, on small scales, looks like a (curved) piece of paper. An n-dimensional manifold likewise looks locally like an ordinary n-dimensional space. Many old problems are being solved using techniques from the area (notably Perelman’s resolution of the Poincaré conjecture) and new directions are being pioneered. An important theme in this area has been the development and use of sophisticated techniques from the theory of PDEs to study natural equations that arise in geometry. Usually, the set of solutions of a system of partial differential equations has the structure of some high dimensional manifold. Understanding the geometry of this manifold often gives new insight into the nature of these solutions, and to the actual phenomenon that is modeled by the differential equations, whether it comes from physics, economics, engineering, or any other quantitative science.
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The Atiyah-Singer index theorem from the 1960s connects the theory of linear PDEs to topology and geometry. The development of tools for nonlinear PDEs in geometry has been slower but has led to many of the most dramatic developments in mathematics including Donaldson's breakthroughs in the theory of four-manifolds using the Yang-Mills equations of high-energy physics. The study of lower dimensional manifolds (dimensions four or less) has particular significance to theoretical physics and has many applications. Floer homology is a mathematically rigorous way of constructing parts of a quantum field theory. Another important and growing area is the mathematics of general relativity. The Lorentz version of the Einstein equations is now at the cutting edge of our hyperbolic PDE technology. One branch of geometric analysis involves the recovery of a function from its integrals over various domains. A well-known application of this idea is Computed Tomography Scanning (CT scans).
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#هندسه
The modern discipline of geometry is affecting virtually every branch of mathematics and is in a period of great progress. Geometry combines techniques from apparently unconnected areas, such as the use of analytical tools for answering topological questions, and it has strong relations with quantum field theory and string theory. In modern language, the central object of study in geometry is a manifold, which is an object that may have a complicated overall shape, but such that on small scales it looks like ordinary space of a certain dimension. For example, a 1-dimensional manifold is an object such that small pieces of it look like a line, although in general it looks like a curve rather than a straight line. A 2-dimensional manifold, on small scales, looks like a (curved) piece of paper. An n-dimensional manifold likewise looks locally like an ordinary n-dimensional space. Many old problems are being solved using techniques from the area (notably Perelman’s resolution of the Poincaré conjecture) and new directions are being pioneered. An important theme in this area has been the development and use of sophisticated techniques from the theory of PDEs to study natural equations that arise in geometry. Usually, the set of solutions of a system of partial differential equations has the structure of some high dimensional manifold. Understanding the geometry of this manifold often gives new insight into the nature of these solutions, and to the actual phenomenon that is modeled by the differential equations, whether it comes from physics, economics, engineering, or any other quantitative science.
📚 @topmathlearn 📚
The Atiyah-Singer index theorem from the 1960s connects the theory of linear PDEs to topology and geometry. The development of tools for nonlinear PDEs in geometry has been slower but has led to many of the most dramatic developments in mathematics including Donaldson's breakthroughs in the theory of four-manifolds using the Yang-Mills equations of high-energy physics. The study of lower dimensional manifolds (dimensions four or less) has particular significance to theoretical physics and has many applications. Floer homology is a mathematically rigorous way of constructing parts of a quantum field theory. Another important and growing area is the mathematics of general relativity. The Lorentz version of the Einstein equations is now at the cutting edge of our hyperbolic PDE technology. One branch of geometric analysis involves the recovery of a function from its integrals over various domains. A well-known application of this idea is Computed Tomography Scanning (CT scans).
📚 @topmathlearn 📚
#هندسه
❤5
Riemannian geometry is the study of manifolds equipped with the additional structure of a Riemannian metric, which is a rule for measuring lengths of curves and angles between tangent vectors. A Riemannian manifold has curvature, and it is precisely this curvature that makes the laws of classical Euclidean geometry, that we learn in elementary school, to be different. For example, the sum of the interior angles of a triangle on a curved Riemannian manifold can be more or less than π if the curvature is positive or negative, respectively. Symplectic geometry is the study of manifolds equipped with an additional structure called a symplectic form. A symplectic form is in some sense (that can be made precise), the opposite of a Riemannian metric, and symplectic manifolds exhibit very different behaviour from Riemannian manifolds. For example, a famous theorem of Darboux says that all symplectic manifolds are “locally” the same, although globally they can be extremely different. Such a theorem is far from true in Riemannian geometry. Symplectic manifolds arise naturally in physical systems from classical mechanics, and are called “phases spaces” in physics. This branch of geometry is very topological in nature. Complex geometry is the study of manifolds which locally “look like” ordinary n-dimensional spaces that are modeled on the complex numbers rather than the real numbers. Because the analysis of holomorphic (or complex-analytic) functions is much more rigid than the real case (for example, not all real smooth functions are real-analytic) there are many fewer “types” of complex manifolds, and there has been more success in (at least partial) classifications. This field is also very closely related to algebraic geometry.
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#هندسه
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#هندسه
❤5
🌐 Differential Geometry (هندسه دیفرانسیل)
Differential geometry is the study of geometry using differential calculus, and uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra. The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. It is a vast subject that has its roots in both the classical theory of curves and surfaces and in the work of Gauss and Riemann motivated by the calculus of variations. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein’s general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is a broad field of mathematics related and with applications to several areas of mathematics, such as algebra, analysis, mathematical physics, partial differential equations, topology and science, such as biology, chemistry, data analysis, engineering, physics. It is not too far-fetched to argue that differential geometry should be in every mathematician’s arsenal.
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While topologists have famously been said to be unable to tell the difference between a donut and a coffee cup (since one can be continuously deformed into the other), geometers definitely care about shape. Differential geometry explores geometric quantities such as curvature and volume, including how such quantities evolve or flow when one continuously deforms a space using specific geometric constraints. Differential geometry has played an essential role in some of the most difficult mathematical problems in history that, at first glance, seem not to even be problems about geometry. The two most well-known examples are the Poincare Conjecture in topology and Fermat’s Last Theorem in number theory.
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#هندسه_دیفرانسیل
Differential geometry is the study of geometry using differential calculus, and uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra. The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. It is a vast subject that has its roots in both the classical theory of curves and surfaces and in the work of Gauss and Riemann motivated by the calculus of variations. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein’s general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is a broad field of mathematics related and with applications to several areas of mathematics, such as algebra, analysis, mathematical physics, partial differential equations, topology and science, such as biology, chemistry, data analysis, engineering, physics. It is not too far-fetched to argue that differential geometry should be in every mathematician’s arsenal.
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While topologists have famously been said to be unable to tell the difference between a donut and a coffee cup (since one can be continuously deformed into the other), geometers definitely care about shape. Differential geometry explores geometric quantities such as curvature and volume, including how such quantities evolve or flow when one continuously deforms a space using specific geometric constraints. Differential geometry has played an essential role in some of the most difficult mathematical problems in history that, at first glance, seem not to even be problems about geometry. The two most well-known examples are the Poincare Conjecture in topology and Fermat’s Last Theorem in number theory.
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#هندسه_دیفرانسیل
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🌐 Algebraic Geometry (هندسه جبری)
Algebraic geometry is the study of algebraic varieties, which are solution sets of systems of polynomial equations. They are sometimes manifolds but also often have singular points at which they are not smooth. Because they are defined algebraically, there are many more tools available from abstract algebra to study them, and conversely, many questions in pure algebra can be understood better by reformulating the problem in terms of algebraic geometry. Moreover, one can study varieties over any field, not just the real or complex numbers. The objects of study of algebraic geometry are, roughly, the common zeroes of polynomials in one or several variables (algebraic varieties). But because polynomials are so ubiquitous in mathematics, algebraic geometry has always stood at the crossroads of many different fields. Classical questions in algebraic geometry involve the study of particular sets of equations or the geometry of lines and linear spaces. Among the kinds of questions one can ask are enumerative: How many conics in the plane are tangent to a given set of five lines? How many lines are contained in a general surface of degree three in space? Another motivating question has been to try to fit together all algebraic varieties of a given type into a space which is itself an algebraic variety; such spaces are called moduli spaces. Simple examples of this type are projective spaces, which parameterize lines through the origin in a vector space, and their generalizations, Grassmannians, which parameterize linear subspaces of a vector space. In turn, the geometry of a Grassmannian can often be applied to solve an enumerative problem. In other kinds of moduli problems, one attempts to classify all curves, surfaces, or higher dimensional varieties of a certain type; another example is the space of all vector bundles of a given type over a fixed algebraic variety. Then one tries to construct and describe the moduli space of all such objects. Often invariant theory, i.e. the study of all invariant polynomials under the action of a group on a vector space, or a more general algebraic variety, plays a crucial role in the construction.
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Recent developments in high energy physics have led to a host of spectacular results and open problems in complex algebraic geometry. For example, the case where the dimension is one, i.e., the case of algebraic curves, is essentially the study of compact Riemann surfaces. This study has a long history involving calculus, complex analysis, and low dimensional topology. The moduli space of all compact Riemann surfaces has a very rich geometry and enumerative structure, which is an object of much current research, and has surprising connections with fields as diverse as geometric topology in dimensions two and three, nonlinear partial differential equations, and conformal field theory and string theory. Many questions posed by physicists have been solved by using the wealth of techniques developed by algebraic geometers. In turn, physics questions have led to new conjectures and new methods in this very central area of mathematics. Finally, the proposed ten dimensional space-time of string theory involves six very small extra dimensions, which correspond to certain three dimensional algebraic varieties, Calabi-Yau manifolds. These manifolds were already of great interest to mathematicians. Amazing ideas from physics have suggested that Calabi-Yau manifolds come in pairs. The geometry of the so-called mirror manifold of a Calabi-Yau manifold turns out to be connected to classical enumerative questions on the original manifold.
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#هندسه_جبری
Algebraic geometry is the study of algebraic varieties, which are solution sets of systems of polynomial equations. They are sometimes manifolds but also often have singular points at which they are not smooth. Because they are defined algebraically, there are many more tools available from abstract algebra to study them, and conversely, many questions in pure algebra can be understood better by reformulating the problem in terms of algebraic geometry. Moreover, one can study varieties over any field, not just the real or complex numbers. The objects of study of algebraic geometry are, roughly, the common zeroes of polynomials in one or several variables (algebraic varieties). But because polynomials are so ubiquitous in mathematics, algebraic geometry has always stood at the crossroads of many different fields. Classical questions in algebraic geometry involve the study of particular sets of equations or the geometry of lines and linear spaces. Among the kinds of questions one can ask are enumerative: How many conics in the plane are tangent to a given set of five lines? How many lines are contained in a general surface of degree three in space? Another motivating question has been to try to fit together all algebraic varieties of a given type into a space which is itself an algebraic variety; such spaces are called moduli spaces. Simple examples of this type are projective spaces, which parameterize lines through the origin in a vector space, and their generalizations, Grassmannians, which parameterize linear subspaces of a vector space. In turn, the geometry of a Grassmannian can often be applied to solve an enumerative problem. In other kinds of moduli problems, one attempts to classify all curves, surfaces, or higher dimensional varieties of a certain type; another example is the space of all vector bundles of a given type over a fixed algebraic variety. Then one tries to construct and describe the moduli space of all such objects. Often invariant theory, i.e. the study of all invariant polynomials under the action of a group on a vector space, or a more general algebraic variety, plays a crucial role in the construction.
📚 @topmathlearn 📚
Recent developments in high energy physics have led to a host of spectacular results and open problems in complex algebraic geometry. For example, the case where the dimension is one, i.e., the case of algebraic curves, is essentially the study of compact Riemann surfaces. This study has a long history involving calculus, complex analysis, and low dimensional topology. The moduli space of all compact Riemann surfaces has a very rich geometry and enumerative structure, which is an object of much current research, and has surprising connections with fields as diverse as geometric topology in dimensions two and three, nonlinear partial differential equations, and conformal field theory and string theory. Many questions posed by physicists have been solved by using the wealth of techniques developed by algebraic geometers. In turn, physics questions have led to new conjectures and new methods in this very central area of mathematics. Finally, the proposed ten dimensional space-time of string theory involves six very small extra dimensions, which correspond to certain three dimensional algebraic varieties, Calabi-Yau manifolds. These manifolds were already of great interest to mathematicians. Amazing ideas from physics have suggested that Calabi-Yau manifolds come in pairs. The geometry of the so-called mirror manifold of a Calabi-Yau manifold turns out to be connected to classical enumerative questions on the original manifold.
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#هندسه_جبری
❤5
🌐 Applied Mathematics (ریاضی کاربردی)
Applied mathematics is concerned with formulating, analyzing and solving an application problem, whether it arises from physical science or biological science, using mathematical language, theories and techniques. The hallmark of applied mathematics is mathematical modeling, which may be regarded as the art of abstraction that helps us realize the truth. The power of modeling and abstraction has been clearly evidenced by the enormous advancements and progress of physical sciences in the past two centuries. As the solutions to all these application problems hinge on the solutions to their mathematical models, analyzing and solving these problems has been critically important for the resolution of many scientific, engineering, and industrial application problems. However, except in some very simple situations, closed form solutions do not exist even for simple, linear models. As a result, seeking numerical (approximate) solutions becomes the only viable way to find (and to see) the solutions of these mathematical models. The computations involved in solving a mathematical model, in particular, a nonlinear model, are often enormous, which are impossible to carry out by hand. On the other hand, the job is perfect for computers because they are good at number-crunching.
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To realize the potential, a big (and difficult) question which must be answered is how to utilize computers’ superpower of doing arithmetic calculations to compute solutions of mathematical models. The key to this question is to develop reliable and efficient numerical/computational methods and algorithms which can be efficiently implemented on computers. To develop, analyze and implement these enabling methods and algorithms for all kinds of mathematical problems is a scientific field called Computational Mathematics, a sub-field of it, that focuses on partial differential equations (PDEs) related mathematical problems, is known as numerical PDEs, which is one of main focuses in Computational Mathematics. Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.
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Applied mathematics is concerned with formulating, analyzing and solving an application problem, whether it arises from physical science or biological science, using mathematical language, theories and techniques. The hallmark of applied mathematics is mathematical modeling, which may be regarded as the art of abstraction that helps us realize the truth. The power of modeling and abstraction has been clearly evidenced by the enormous advancements and progress of physical sciences in the past two centuries. As the solutions to all these application problems hinge on the solutions to their mathematical models, analyzing and solving these problems has been critically important for the resolution of many scientific, engineering, and industrial application problems. However, except in some very simple situations, closed form solutions do not exist even for simple, linear models. As a result, seeking numerical (approximate) solutions becomes the only viable way to find (and to see) the solutions of these mathematical models. The computations involved in solving a mathematical model, in particular, a nonlinear model, are often enormous, which are impossible to carry out by hand. On the other hand, the job is perfect for computers because they are good at number-crunching.
📚 @topmathlearn 📚
To realize the potential, a big (and difficult) question which must be answered is how to utilize computers’ superpower of doing arithmetic calculations to compute solutions of mathematical models. The key to this question is to develop reliable and efficient numerical/computational methods and algorithms which can be efficiently implemented on computers. To develop, analyze and implement these enabling methods and algorithms for all kinds of mathematical problems is a scientific field called Computational Mathematics, a sub-field of it, that focuses on partial differential equations (PDEs) related mathematical problems, is known as numerical PDEs, which is one of main focuses in Computational Mathematics. Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.
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❤4
The centerpieces of Computational Mathematics are error and speed. Errors are everywhere; they occur in every step of an algorithm (i.e., approximation error) and in every calculation done by a computer (i.e., round-off error). If a beautiful method or algorithm on paper does not have a mechanism to control its error accumulation when implemented on computers, then it is useless. Such an error-control capability is the watershed to distinguish “good” methods and algorithms from “bad” ones. Among good methods and algorithms of the same nature, a faster (in terms of computer execution time) method or algorithm is considered to be a better method or algorithm. Developing good and fast computational methods and algorithms is the heart and soul of Computational Mathematics. Problems in all areas of mathematics, applied science, engineering, economics, medicine and statistics can be posed as mathematical optimization problems. An optimization problem begins with a set of independent variables and often includes conditions or restrictions that define acceptable values of the variables. Such restrictions are known as the constraints of the problem. The other essential component of an optimization problem is a single measure of goodness, termed the objective function, which depends in some way on the variables. The solution of an optimization problem is a set of allowed values of the variables for which the objective function assumes its optimal value. In mathematical terms, this usually involves maximizing or minimizing.
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#ریاضی_کاربردی
📚 @topmathlearn 📚
#ریاضی_کاربردی
❤4