Is the Abstract Mathematics of Topology Applicable to the Real World?
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- Опубліковано 1 лип 2015
- Robert D. MacPherson; Randall D. Kamien; Raúl Rabadán
Hermann Weyl Professor, School of Mathematics; University of Pennsylvania; Columbia University
April 14, 2015
Topology is the only major branch of modern mathematics that wasn't anticipated by the ancient mathematicians. Throughout most of its history, topology has been regarded as strictly abstract mathematics, without applications. However, illustrating Wigner's principle of "the unreasonable effectiveness of mathematics in the natural sciences", topology is now beginning to come up in our understanding of many different real world phenomena.
In this minisymposium, Robert MacPherson speaks on "What is Topology?", Randall Kamien discusses topology and liquid crystals (like those in your computer display), and Raul Rabadan describes how topology modifies our understanding of the evolutionary "Tree of Life". Following the presentation, Robbert Dijkgraaf moderates a panel discussion on topology.
Video also available here: video.ias.edu
5:41 What is topology
21:51 Topology of Evolution
41:15 Liquid Crystals
54:23 Panel Discussion
On topic, resetting Magical (logarithmic projection-drawing coordination) Thinking Labelling systems is a "variation on the theme" of Mathematical Conjecture or Abstraction, so the self-defining measuring operation of Singularity-point positioning, 0-1-2-ness superposition identification of interference picture-planetopology of cross-sectional compositions.., which is as real, in stages, as it gets.
The Calculus of Superspin Superposition-point, the relative-timing frequency amplitudes of harmonic spacing timing modulation, is a pov, an abstract axial-tangential orthogonality freeze-framing that is "what you see" by orientation or orthogonal-normal cross-sectional interference, eg collinear with a z-axis, perpendicular as Euclid saw the sand pit, is equivalent to looking up or down the eye of an infinite Tornado reflecting at resonancevanishing-into-no-thing congruence horizons, proximal and/or distal depending on prime-cofactor frequency density-intensity alignment-integration.., perspective-picture.
(String Theoretical Vibration reasoning applies, and so on)
Well topology , general topology, deals with the relationships of sets of objects and their properties, if you can describe these relationships, whatever they are, in terms of connectedness, compactness, separation and continuity then it seems reasonable it applies to everything.
Not necessarily. Note that the real world is not a continuum, it's discrete.
@@santafucker1945 depends on how coarse or fine you look at your topology. Continuity in this sense is a way of evaluating a topology not a requirement to be a topology.
@@santafucker1945 That is still a big open problem
The only reason the eta-prime meson is heavier than the pions is the topological properties of the strong interaction gauge field. This is because the gauge group is SU(3), and the gauge holonomy can 'wrap around' the 3-dimensional sphere of space at infinity'. This is true for all simple nonabelian gauge groups. The simplest twisted configuration of lowest total action in 4d Euclidean space is called an 'instanton', and in normal space-time, in our space, it represents a tunnelling process where the strong field acquires a topological twist that spreads outward at the speed of light. The topological twisting means that the low-energy description of the strong interaction is by pions with the eta-prime more massive. This is historically the first serious application of topology to the real world, and needs a mention. The instanton is due to Polyakov, the explanation of the mass of the eta-prime is due to Gerard 't Hooft and is partly responsible for his Nobel prize.
First Murry Gel Man and next Furey and others showed me the magic of abstract algebra. I often wanted to dig deep into this beautiful world, but I am old (71) and gave up (I can understand Witten's procrastination when he was trying to master topology and dig deep into the world of 'homology'. Today mathematics is indispensable in acquiring the algorithm that enables us to design a thinking machine. Turing new it was impossible. QM/QC and 'complexity' leads us to what is called 'cosmic consciousness' (Anthropic Principle) and gives the insight how the universe is a QC function (conjectured by Maldacena) and leads to 'determinism' by eliminating randomness/chance/uncertainty (ID).
Life is also a QC function (SMNH).
Allen Hatcher's textbook on Algebraic Topology is freely available : pi.math.cornell.edu/~hatcher/AT/AT.pdf
This requires a small background on topology, and I think it can be filled in by reading Wikipedia articles for the following:
Metric space, topological space, basis, product topology, continuity, limit points, compactness
@@uzulim9234 this is a terrible textbook for beginners as it gives almost no intuition and it is mostly a compendium of proofs.
Weird ... being qualitative is exactly what makes topology interesting for science, because it gives the luxury of explanation ...
The audio is way too quiet .
grayxy Copy and paste the url into VLC, then increase the volume to above 100%.
Ah, thanks!
WTF it's a valid comment. The audio is low. I should have like normalized it or something before uploading but I'm too lazy.
David Lloyd-Jones Enjoy your ban!
I just read this list of comments and it makes no sense. How do people manage to argue over something like this?
wonderful!
we are on revolution
How do people in non-mathmatics academic areas come to discover that methods used in Topology might benefit their research? Topology is largely restricted to the high-priests of mathematics. It is not a subject that is normally recommended for students in the non-math subjects. And it is not a trivial thing to learn and understand. It would be interesting to get some idea about how the ideas of topology is transferred to those in non-math subjects.
Michael Edward Zeidler
well people in hugher physics know topology
I don't agree that it's only high priests (especially not after Rob Ghrist and Justin Curry). There are undergrad treatments and the topic is on math.SX, not mathoverflow. I think an average person could look at Hatcher and understand some of it.
That's a misguided impression. Anyone who thinks about deformations and how things are connected is doing topology. For example, traffic networks, social graphs, organization structure, disease spread, and I could reel off thousands of similar everyday examples. What I think you might be referring to is the difficulty of formalizing topological intuitions to make them into abstract mathematics. That is where the "high priests of mathematics" come into play.
I think measurement has been the greatest aberration in mathematic history ...
That betti number stuff makes no sense. For the torus, why did you need to drill a hole if the 2 cuts with sissors were sufficient? Actually, why would you need a second cut if the 1st cut was allowed to meet the beginning of that cut along any equator on the torus? Unless we're talking about breaking confined 2-d areas. But then shouldn't we also look at confined 1-d segments. Unless the drill is supposed to make/collapse a 3-d solid into a 2-d surface...which a real drill doesn't do.
You can't start cutting into a manifold with scissors without creating a cusp. Scissors need an edge or a hole to begin cutting.
Drills and scissors gives the insight into the geometry of the topology and homomorphism.
#Мозгу не было дано определение,в контексте его приёма Информац.ЭП кода Материи изменения синтеза....Поэтому говорить об абстракции не только неуместно,но и губительно,в том смысле,что вы определяете мозг,как нечто второстепенное от сложившихся постулатов определения среды,как объяснение и оправдание ,,смысла" переключения...И вот почему: математика не занимается построением объяснений систем,ибо она является системой сама,где формула решения смысла оперирует не заданием решения вычисления,а обнаружением сложности этого решения в синтезе самой Материи установки,как звена обнаружения идентичности кодовых характеристик мозга соответствовать расщеплению ступеней данного решения,в общей системе передачи вимбрац.волновых параметров продолжения синтеза цепи....Т.е.построение основы проекцируется не предметом математики обучения структуры владения,а МОЗГОМ ИНФОРМАЦИОННОГО ПОДТВЕРЖДЕНИЯ ВОЛНОВОЙ МАТЕМАТИКИ КОДА СИММЕТРИИ ДО НАЧАЛА ДАННОГО РЕШЕНИЯ,Т.Е.МЫ ГОВОРИМ О МАТЕМАТИКИ КОДА,ГДЕ САМ ВХОД ЦИФРЫ ОПЕРИРУЕТ И ПРОИЗВОДИТ РЕШЕНИЕ,В ОБЪЕКТИВНОСТИ ПРОСТРАНСТВЕННЫХ,ОРГАНИЧНЫХ ПЕРЕМЕЩЕНИЙ ПРИЕМА ЦИФРЫ,КАК ИМПУЛЬС ЭЛЕКТРОННОГО ВВОДА В СИНТЕЗ ОБЪЯСНЕНИЯ,ЛИБО ОТТОРЖЕНИЯ,ПО ПРИЧИНЕ ВИДИМОГО СБОЯ ДО РЕШЕНИЯ...Т.Е.ВОЛНОВАЯ МАТЕМАТИКА-ЭТО ИНСТРУМЕНТ ОРГАНИЗАЦИИ ПАРОЛЯ МОЗГА ЧЕЛОВЕКА,А НЕ РЕШЕНИЕ ВИДИМЫХ УСТРОЙСТВ КОМФОРТА ДАННОЙ ТЕОРИИ ПОСТРОЕНИЯ ВОВЛЕЧЕНИЯ МАСС,КАК Т.Н.ПРЕДМЕТА СИНТЕЗИРУЮЩЕГО РЕШЕНИЯ ТЕОРИЙ!!!.КВАНТ.МАТЕМАТ.СИНТЕЗ НЕЛЬЗЯ ПРОИЗВОДИТЬ В ЛЮБОЙ ФОРМЕ,ЕСЛИ НЕТ СУММЫ ТОЧНЫХ ПОДТВЕРЖДЕНИЙ РАВНОВЕСИЯ,НО ЕГО ТАКЖЕ НЕЛЬЗЯ ПРОИЗВОДИТЬ В АБСОЛЮТНОМ НЕЗНАНИИ УСТРОЙСТВА И ПРЕДНАЗНАЧЕНИЕ МОЗГА,ПРОСТРАНСТВА,СВЯЗИ,ЗАКОНОВ!!!.ПОСЕМУ ОПРЕДЕЛЯЯ МАТЕМАТИКУ,ВЫ ОПРЕДЕЛЯЕТЕ УСТРОЙСТВО СИНТЕЗА СМЫСЛА МОЗГА В ПРОСТРАНСТВЕ ПОЛНОЦЕННОСТИ,НЕ РЕШЕНИЙ,А ОСНОВЫ ИНФОРМАЦИОННОЙ ОБЪЕКТИВНОСТИ СУЩЕСТВОВАНИЯ СУТИ КОНТЕКСТА СВЯЗИ МЫСЛИ В РЕШЕНИИ ВОЛНОВОЙ,КЛЕТОЧНОЙ ПОЛНОЦЕННОСТИ СИММЕТРИИ!!!.ВОЗЬМИТЕ РУЧКУ.....,Х-0,1,2,3,4,5,6,7,8,9......ПОСТАВЬТЕ ЛЮБУЮ ПОСЛЕДОВАТЕЛЬНОСТЬ ЦИФР,КАК ВЫ СЧИТАЕТЕ НУЖНЫМ,ПОСЛЕ ИКС..ПОСТАВИЛИ???!ЧТО ВЫ ПОДРАЗУМЕВАЕТЕ ПОД ДАННЫМИ ЦИФРАМИ???!КАКОЙ СМЫСЛ И ВОЗМОЖНЫЕ РЕШЕНИЯ ОПРЕДЕЛЯЮТСЯ ВАМИ ,В ОТНОШЕНИИ СВЯЗИ СЛЕДСТВИЙ???!И Я ПОСТАВЛЮ...Х-01,04,140,03,230,410...ТАК ВОТ,РАДОСТИ МОИ,МАТЕМАТИКА ОПРЕДЕЛЯЕТ НЕ СМЫСЛ КОДА МАНИПУЛЯЦИЙ С ПОСТРОЕНИЕМ ЦИФР,ФОРМУЛ,ТЕОРИЙ,СООТНОШЕНИЙ В ЗАДАЧЕ ЦЕЛИ...МАТЕМАТИКА ОПРЕДЕЛЯЕТ ЦИФРУ,КАК ПРОХОЖДЕНИЕ УСТРОЙСТВА ЯДЕРНО/МАГНИТНОЙ СВЯЗИ СИНТЕЗА В ОБЪЕМЕ СОУЧАСТИЯ ВОЛНОВОЙ ПРИЧАСТНОСТИ ОПЕРЕЖЕНИЯ РЕШЕНИЯ СВЯЗИ СООТВЕТСТВИЯ!!!.#zZz#.
Шизофазия налицо.
Is this a video on math or evolutionary biology.... Seriously that guy went way over with his biology.
The main obstacle that I often encounter in trying to understand topology and algebraic geometry is the complexity of the apparatus used to describe everything. Open any of those books and find yourself, after some interesting examples, in a desert of abstraction and symbolism, with fewer and fewer examples of actual geometry. Too many fancy compound nouns, too much word salad, and too few examples that go beyond the trivial. It seems sometimes to me that mathematicians refuse to draw pictures, as if it would reveille that the esoteric stuff they write down might actually much simpler and less awe inspiring then they want to believe or admit. It’s like no one is allowed to draw an image of the prophet.
You are right about 'complexity. Modern physics of quantum computation involve a finite universe, so we should be able to count all the photons and Ramanujan's 'partition function, derived from q-series is a beautiful tool, leading QM to frame the theory leading to 'determinism' (in QM we can measure energy, momentum etc exactly, but the problem starts with 'spin' [Bohm's monogram on QM]. When spin can take any of the infinite values after you measure a quantum process leading to 'complexity'. There is no algorithm that provide 'determinism'.
Yes, this is a problem. The way to fix it is to actually compute the homology group of a simplicial complex explicitly with a presentation. To do this, represent a collection of triangles by "abc" "cbd" and so on, with each vertex a letter, and same-letter implying same point, same pair-of-letters means shared line. This allows you to represent a triangulated surface. Then you can imagine 'adding' triangles, this defines the abelian group of triangles, then you can find the homology groups by using the boundary operator (which is quite intuitive). This is how Poincare defined it.
The modern tools are trying to streamline the operations which allow you to compute homotopy groups. There is no greatest algorithm for that, you have to piddle around with various substandard tools. So they try to streamline the presentation.
Vaughan Jones wore a rugby shirt to pick up his Fields medal.
These chumps wear suits to a nano-symposium?
welcome to princeton nj
Yes, and don't they look very handsome, especially the one doing the introduction!