Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize and predict natural phenomena. This is in contrast to experimental physics; the advancement of science depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations. For example, while developing special relativity, Albert Einstein was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was uninterested in the Michelson–Morley experiment on Earth's drift through a luminiferous aether. Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect an experimental result lacking a theoretical formulation. A physical theory is a model of physical events, it is judged by the extent. The quality of a physical theory is judged on its ability to make new predictions which can be verified by new observations.
A physical theory differs from a mathematical theorem in that while both are based on some form of axioms, judgment of mathematical applicability is not based on agreement with any experimental results. A physical theory differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms. A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces. Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that energy are not continuously variable. Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ empirical formulas to agree with experimental results without deep physical understanding.
"Modelers" appear much like phenomenologists, but try to model speculative theories that have certain desirable features, or apply the techniques of mathematical modeling to physics problems. Some attempt to create approximate theories, called effective theories, because developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, reinterpret or generalise extant theories, or create new ones altogether. Sometimes the vision provided by pure mathematical systems can provide clues to how a physical system might be modeled. Theoretical problems that need computational investigation are the concern of computational physics. Theoretical advances may consist in setting aside old, incorrect paradigms or may be an alternative model that provides answers that are more accurate or that can be more applied. In the latter case, a correspondence principle will be required to recover the known result. Sometimes though, advances may proceed along different paths. For example, an correct theory may need some conceptual or factual revisions.
However, an exception to all the above is the wave–particle duality, a theory combining aspects of different, opposing models via the Bohr complementarity principle. Physical theories become accepted if they are able to make correct predictions and no incorrect ones; the theory should have, at least as a secondary objective, a certain economy and elegance, a notion sometimes called "Occam's razor" after the 13th-century English philosopher William of Occam, in which the simpler of two theories that describe the same matter just as adequately is preferred. They are more to be accepted if they connect a wide range of phenomena. Testing the consequences of a theory is part of the scientific method. Physical theories can be grouped into three categories: mainstream theories, proposed theories and fringe theories. Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy, continued by Plato and Aristotle, whose views held sway for a millennium. During the rise of medieval universities, the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar and rhetoric and of the Quadrivium like arithmetic, geometry and astronomy.
During the Middle Ages and Renaissance, the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon. As the Scientific Revolution gathered pace, the concepts of matter, space and causality began to acquire the form we know today, other sciences spun off from the rubric of natural philosophy, thus began the modern era of theory with the Copernican paradigm shift in astronomy, soon followed by Johannes Kepler's expressions for planetary orbits, which summarized the meticulous observations of Tycho Brahe.
In mathematics, a block matrix or a partitioned matrix is a matrix, interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned; this notion can be made more precise for an n by m matrix M by partitioning n into a collection r o w g r o u p s, partitioning m into a collection c o l g r o u p s. The original matrix is considered as the "total" of these groups, in the sense that the entry of the original matrix corresponds in a 1-to-1 way with some offset entry of some, where x ∈ r o w g r o u p s and y ∈ c o l g r o u p s. Block matrix algebra arises in general from biproducts in categories of matrices; the matrix P = can be partitioned into four 2×2 blocks P 11 =, P 12 =, P 21 =, P 22 =.
The partitioned matrix can be written as P =. It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors; the partitioning of the factors is not arbitrary and requires "conformable partitions" between two matrices A and B such that all submatrix products that will be used are defined. Given an matrix A with q row partitions and s column partitions A = and a matrix B with s row partitions and r column partitions B = [ B 11 B 12 ⋯ B 1
Pierre René, Viscount Deligne is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a complete proof in 1973, he is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, 1978 Fields Medal. Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles, writing a dissertation titled Théorème de Lefschetz et critères de dégénérescence de suites spectrales, he completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled Théorie de Hodge. Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques near Paris on the generalization within scheme theory of Zariski's main theorem. In 1968, he worked with Jean-Pierre Serre. Deligne's focused on topics in Hodge theory, he tested them on objects in complex geometry. He collaborated with David Mumford on a new description of the moduli spaces for curves.
Their work came to be seen as an introduction to one form of the theory of algebraic stacks, has been applied to questions arising from string theory. Deligne's most famous contribution was his proof of the third and last of the Weil conjectures; this proof completed a programme initiated and developed by Alexander Grothendieck. As a corollary he proved the celebrated Ramanujan–Petersson conjecture for modular forms of weight greater than one. Deligne's 1974 paper contains the first proof of the Weil conjectures, Deligne's contribution being to supply the estimate of the eigenvalues of the Frobenius endomorphism, considered the geometric analogue of the Riemann hypothesis. Deligne's 1980 paper contains a much more general version of the Riemann hypothesis. From 1970 until 1984, Deligne was a permanent member of the IHÉS staff. During this time he did much important work outside of his work on algebraic geometry. In joint work with George Lusztig, Deligne applied étale cohomology to construct representations of finite groups of Lie type.
He received a Fields Medal in 1978. In 1984, Deligne moved to the Institute for Advanced Study in Princeton. In terms of the completion of some of the underlying Grothendieck program of research, he defined absolute Hodge cycles, as a surrogate for the missing and still conjectural theory of motives; this idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. He reworked the Tannakian category theory in his 1990 paper for the Grothendieck Festschrift, employing Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate Weil cohomology. All this is part of the yoga of uniting Hodge theory and the l-adic Galois representations; the Shimura variety theory is related, by the idea that such varieties should parametrize not just good families of Hodge structures, but actual motives. This theory is not yet a finished product, more recent trends have used K-theory approaches, he was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, the Abel Prize in 2013.
In 2006 he was ennobled by the Belgian king as viscount. In 2009, Deligne was elected a foreign member of the Royal Swedish Academy of Sciences, he is a member of the Norwegian Academy of Letters. Deligne, Pierre. "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS. 43: 273–307. Doi:10.1007/bf02684373. Deligne, Pierre. "La conjecture de Weil: II". Publications Mathématiques de l'IHÉS. 52: 137–252. Doi:10.1007/BF02684780. Deligne, Pierre. "Catégories tannakiennes". Grothendieck Festschrift vol II. Progress in Mathematics. 87: 111–195. Deligne, Pierre. "Real homotopy theory of Kähler manifolds". Inventiones Mathematicae. 29: 245–274. Doi:10.1007/BF01389853. MR 0382702. Deligne, Pierre. Commensurabilities among Lattices in PU. Princeton, N. J.: Princeton University Press. ISBN 0-691-00096-4. Quantum fields and strings: a course for mathematicians. Vols. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C.
Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison and Edward Witten. American Mathematical Society, Providence, RI. Vol. 1: xxii+723 pp.. ISBN 0-8218-1198-3. Deligne wrote multiple hand-written letters to other mathematicians in the 1970s; these include "Deligne's letter to Piatetskii-Shapiro". Archived from the original on 7 December 2012. Retrieved 15 December 2012. "Deligne's letter to Jean-Pierre Serre". 2012-12-15. "Deligne's letter to Looijenga". Retrieved 15 December 2012; the following mathematical concepts are named after Deligne: Deligne–Lusztig theory Deligne–Mumford moduli space of curves Deligne–Mumford stacks Fourier–Deligne transform Deligne cohomology Deligne motive Deligne tensor product of abelian categories Langlands–Deligne local constantAdditionally, many different conjectures in mathematics have been called the De
Yuri Ivanovitch Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book "Computable and Uncomputable". Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich, he is now a Professor at the Max-Planck-Institut für Mathematik in Bonn, a professor emeritus at Northwestern University. Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, algebraic differential equations; the Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties. He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra.
He indicated the role of the Brauer group, via Grothendieck's theory of global Azumaya algebras, in accounting for obstructions to the Hasse principle, setting off a generation of further work. He formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties, he has further written on Yang–Mills theory, quantum information, mirror symmetry. Manin had over 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Arend Bayer and Victor Kolyvagin, as well as foreign students including Hà Huy Khoái, he was awarded the Brouwer Medal in 1987, the Nemmers Prize in Mathematics in 1994, the Schock Prize in 1999 and the Cantor Medal in 2002, the Bolyai Prize of the Hungarian Academy of Sciences in 2010. In 1990 he became foreign member of the Royal Netherlands Academy of Sciences. Manin: Selected works with commentary, World Scientific 1996 Manin: Mathematics as metaphor - selected essays, American Mathematical Society 2009 Manin: Rational points of algebraic curves over function fields.
AMS translations 1966 Manin: Algebraic topology of algebraic varieties. Russian Mathematical Surveys 1965 Manin: Modular forms and Number Theory. International Congress of Mathematicians, Helsinki 1978 Manin: Frobenius manifolds, quantum cohomology, moduli spaces, American Mathematical Society 1999 Manin: Quantum groups and non commutative geometry, Centre de Recherches Mathématiques, 1988 Manin: Topics in non-commutative geometry, Princeton University Press 1991 Manin: Gauge field theory and complex geometry. Springer 1988 Manin: Cubic forms - algebra, arithmetics, North Holland 1986 Manin: A course in mathematical logic, Springer 1977, second expanded edition with new chapters by the author and Boris Zilber, Springer 2010. Manin: The provable and the unprovable, Moscow 1979 Manin: Computable and Uncomputable, Moscow 1980 Manin: Mathematics and physics, Birkhäuser 1981 Manin: New dimensions in geometry. In Arbeitstagung Bonn 1984, Lectures Notes in Mathematics Vol. 1111, Springer Verlag Manin, Alexei Ivanovich Kostrikin: Linear algebra and geometry and Breach 1989 Manin, Sergei Gelfand: Homological algebra, Springer 1994.
Manin, Sergei Gelfand: Methods of Homological algebra, Springer 1996 Manin, Igor Kobzarev: Elementary Particles: mathematics and philosophy, Kluwer, 1989 Manin, Alexei A. Panchishkin: Introduction to Number theory, Springer Verlag 1995, 2nd edn. 2005 Manin Moduli, Mirrors, 3. European Congress Math. Barcelona 2000, Plenary talk Manin Classical computing, quantum computing and Shor´s factoring algorithm, Bourbaki Seminar 1999 Manin Von Zahlen und Figuren 2002 Manin, Mathilde Marcolli Holography principle and arithmetic of algebraic curves, 2002 Manin 3-dimensional hyperbolic geometry as infinite-adic Arakelov geometry, Inventiones Mathematicae 1991 Manin: Mathematik, Kunst und Zivilisation, e-enterprise, 2014 ADHM Arithmetic topology Gauss–Manin connection Manin–Drinfeld theorem Manin matrices Manin–Mumford conjecture Manin obstruction Manin triple Manin conjecture Némethi, A.. "Yuri Ivanovich Manin", Acta Mathematica Hungarica, April 2011, Volume 133, pp. 1–13. Jean-Paul Pier. Development of Mathematics 1950–2000.
Springer Science & Business Media. P. 1116. ISBN 978-3-7643-6280-5. Yuri Manin at the Mathematics Genealogy Project Manin's page at Max-Planck-Institut für Mathematik website Good Proofs are Proofs that Make us Wiser, interview by Martin Aigner and Vasco A. Schmidt Biography
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to