Angel "Java" Lopez en Blog

Publicado el 7 de Marzo, 2021, 20:13

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The Napkin Ring Problem
https://www.youtube.com/watch?v=J51ncHP_BrY

MATHCOUNTS Practice Competition #4 Live Solve by Po-Shen Loh SAT
https://www.youtube.com/watch?v=vgWV-6l3ouM&feature=emb_logo

Martin"s doileys
https://www.johndcook.com/blog/2021/02/17/martins-doileys/

In 1918, Albert Einstein wrote to Felix Klein decrying the great injustice that brilliant mathematician Emmy Noether, because of being a woman, was denied the right to teach in Germany as a university professor:
https://twitter.com/phalpern/status/1361745091526864903
https://einsteinpapers.press.princeton.edu/vol8-trans/742

Nina Holden just won the 2021 Maryam Mirzakhani New Frontiers Prize for her work on random surfaces and the mathematics of quantum gravity
https://twitter.com/johncarlosbaez/status/1304450533051097090

El día que el arte me ayudó a escribir un artículo de matemáticas
https://culturacientifica.com/2021/03/07/el-dia-que-el-arte-me-ayudo-a-escribir-un-articulo-de-matematicas/

Groups in the form of objects and morphisms
https://twitter.com/SchmiegSophie/status/1367308613153222656

When Math Gets Impossibly Hard
https://www.quantamagazine.org/when-math-gets-impossibly-hard-20200914/

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 6 de Marzo, 2021, 15:12

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Imaginary Numbers May Be Essential for Describing Reality
https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/

Good math problems for high school students
https://twitter.com/AbellsTheorem/status/1362268697482596352

Grover"s Algorithm
https://en.wikipedia.org/wiki/Grover's_algorithm

World's hardest easy geometry problem
https://twitter.com/pickover/status/1361453165187633153
https://mathsmed.co.uk/hardest/taskD.html

The Coach Who Led the U.S. Math Team Back to the Top
https://www.quantamagazine.org/po-shen-loh-led-the-u-s-math-team-back-to-first-place-20210216/

Fast Factoring Integers by SVP Algorithms
https://eprint.iacr.org/2021/232

Lattice problem
https://en.wikipedia.org/wiki/Lattice_problem

Isadore Singer Transcended Mathematical Boundaries
https://www.quantamagazine.org/isadore-singer-bridged-math-and-physics-20210304/

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 28 de Febrero, 2021, 16:50

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A quantum algorithm for the direct estimation of the steady state of open quantum systems
https://quantum-journal.org/papers/q-2021-02-22-399/

On Computable Numbers, with an Application to the Entscheidungsproblem
https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

The Halting Problem - An Impossible Problem to Solve
https://www.youtube.com/watch?v=t37GQgUPa6k&t=1s

Turing Machines - The Accidental Birth of Computer Science
https://www.youtube.com/watch?v=PLVCscCY4xI&t=1s

Una integral muy «dorada»
https://www.gaussianos.com/una-integral-muy-dorada/

Tübinger Dreiecke
https://commons.wikimedia.org/wiki/File:T%C3%BCbinger_Dreiecke.gif

Which great mathematicians had great political commitments?
https://mathoverflow.net/questions/384230/which-great-mathematicians-had-great-political-commitments

Is there any integral for the Golden Ratio?
https://twitter.com/pickover/status/1362458565198237696
https://math.stackexchange.com/questions/1653979/is-there-any-integral-for-the-golden-ratio

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 27 de Febrero, 2021, 6:55

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The larger square from the seventh simple nonperiodic tiling using pattern blocks
https://ar.pinterest.com/pin/121949102394724563/

Hay una transición de fase en el modelo de Erdös-Rényi
https://twitter.com/pgroisma/status/1365271264542261248

Hausdorff and Box dimensions of continuous functions and lineability
https://www.tandfonline.com/doi/abs/10.1080/03081087.2019.1612832?journalCode=glma20#.YDeBCccVH0E.twitter

How (1-z)/(1+z) transforms the unit disk
https://twitter.com/AnalysisFact/status/1364984115376902147

Mathematicians Set Numbers in Motion to Unlock Their Secrets
https://www.quantamagazine.org/with-arithmetic-dynamics-mathematicians-unlock-new-insights-20210222/

42 métodos de demostración en Matemáticas
https://www.gaussianos.com/42-metodos-de-demostracion-en-matematicas/

La infinitud de los números primos y Fermat
https://www.gaussianos.com/la-infinitud-de-los-numeros-primos-y-fermat/

Descenso infinito: un método de demostración poco conocido
https://www.gaussianos.com/descenso-infinito-un-metodo-de-demostracion-poco-conocido/

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 21 de Febrero, 2021, 20:17

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Ludwig Boltzmann
https://mathshistory.st-andrews.ac.uk/Biographies/Boltzmann/

Classical Mechanics and Category Theory
https://twitter.com/johncarlosbaez/status/1291048030515339264

0019: Article 9 (More Pi Formulas)
https://sites.google.com/site/tpiezas/0019

Mathematicians Open a New Front on an Ancient Number Problem
https://www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/

Seven Sketches in Compositionality: An Invitation to Applied Category Theory
https://arxiv.org/abs/1803.05316

At the Math Olympiad, Computers Prepare to Go for the Gold
https://www.quantamagazine.org/at-the-international-mathematical-olympiad-artificial-intelligence-prepares-to-go-for-the-gold-20200921/

Legendre"s Constant
https://en.wikipedia.org/wiki/Legendre%27s_constant

The Simple Math Problem We Still Can"t Solve
https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 16 de Febrero, 2021, 9:21

Llegó la hora de escribir las resoluciones de febrero, algo tarde. Primero, un repaso de las de enero:

- Escribir sobre Matemáticas [pendiente]
- Escribir sobre Física [pendiente]
- Escribir sobre Historia de las Matemáticas [pendiente]
- Escribir sobre Historia de la Ciencia [pendiente]
- Estudiar blues en guitarra [completo]

Le he dedicado bastante tiempo estas semanas pasadas a leer sobre matemáticas, física y su historia: temas muy interesantes, que quisiera comentar. Pero no he tenido el tiempo de pasar en limpio por escrito en posts. Espero que pueda retomar esa actividad en las semanas que siguen.

Mis resoluciones del nuevo mes:

- Escribir sobre Matemáticas
- Escribir sobre Física
- Escribir sobre Historia de las Matemáticas
- Escribir sobre Historia de la Ciencia
- Estudiar blues en guitarra

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Publicado el 15 de Febrero, 2021, 17:39

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How I Learned to Love and Fear the Riemann Hypothesis
https://www.quantamagazine.org/how-i-learned-to-love-and-fear-the-riemann-hypothesis-20210104/

The NASA Engineer Who"s a Mathematician at Heart
https://www.quantamagazine.org/the-nasa-engineer-whos-a-mathematician-at-heart-20210119/

The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12?
https://medium.com/cantors-paradise/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793

Pi Day
http://mathmisery.com/comic/03062015/piday.png

La Teoría de Conjuntos de Georg Cantor
https://www.youtube.com/watch?v=YLiS_flCFt4&feature=youtu.be

How Gödel"s Proof Works
https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/

Landmark Math Proof Clears Hurdle in Top Erdős Conjecture
https://www.quantamagazine.org/landmark-math-proof-clears-hurdle-in-top-erdos-conjecture-20200803/

Knuth's advice on writing mathematical stuff
https://twitter.com/ben_golub/status/1290327297531080705

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 14 de Febrero, 2021, 14:14

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Premio al mejor post de la Edición 9.4: «Regla y compás» del Carnaval de Matemáticas
https://www.gaussianos.com/premio-al-mejor-post-de-la-edicion-9-4-regla-y-compas-del-carnaval-de-matematicas/

11 Life Lessons From History"s Most Underrated Genius
https://medium.com/the-mission/10-000-hours-with-claude-shannon-12-lessons-on-life-and-learning-from-a-genius-e8b9297bee8f

Soy matemático
http://eliatron.blogspot.com/2012/05/soy-matematico.html

La comprobación por encima de la muerte
https://www.gaussianos.com/la-comprobacion-por-encima-de-la-muerte/

What Is A Complexity Class?
https://rjlipton.wordpress.com/2010/11/07/what-is-a-complexity-class/

Balthasar van der Pol
https://mathshistory.st-andrews.ac.uk/Biographies/Van_der_Pol/

Power Fractional Parts
https://mathworld.wolfram.com/PowerFractionalParts.html

Sofia Vasilyevna Kovalevskaya
https://mathshistory.st-andrews.ac.uk/Biographies/Kovalevskaya/

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 13 de Febrero, 2021, 15:43

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Combinatoire
https://www.youtube.com/watch?v=lLsCdEDaNoo&feature=emb_logo

Pi Calculus
https://twitter.com/lostinio/status/1359874839784296455
https://flylib.com/books/en/1.165.1.31/1/

Ruggero Giuseppe Boscovich
https://mathshistory.st-andrews.ac.uk/Biographies/Boscovich/

Web Equation, de mano alzada a código LaTeX
https://www.gaussianos.com/web-equation-de-mano-alzada-a-codigo-latex/

Topology 101: The Hole Truth
https://www.quantamagazine.org/topology-101-how-mathematicians-study-holes-20210126/

Claude Louis Marie Henri Navier
https://mathshistory.st-andrews.ac.uk/Biographies/Navier/

Meet a Mathematician! - Tai-Danae Bradley
https://www.youtube.com/watch?v=dd3a9LaF3UY&feature=emb_logo

arctan 1 + arctan 2 + arctan 3 = π
https://twitter.com/AlgebraFact/status/1351975491381682183
Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 7 de Febrero, 2021, 12:28

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Guillaume François Antoine Marquis de L'Hôpital
https://mathshistory.st-andrews.ac.uk/Biographies/De_LHopital/

Inductive Graphs and Functional Graph Algorithms
https://web.engr.oregonstate.edu/~erwig/papers/InductiveGraphs_JFP01.pdf

Tetrahedron Solutions Finally Proved Decades After Computer Search
https://www.quantamagazine.org/mathematicians-finally-prove-rational-tetrahedron-solutions-20210202/

Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time
https://twitter.com/OxUniMaths/status/1356724980894072833
https://www.maths.ox.ac.uk/node/38304

Dynamic Rectangles are rectangles generated from the unit square
https://twitter.com/fermatslibrary/status/1129747875523645441

Renaissance Science – II
https://thonyc.wordpress.com/2021/01/27/renaissance-science-ii/

The Future of Mathematics
https://www.microsoft.com/en-us/research/video/the-future-of-mathematics/

Lean Prover Tutorial
https://leanprover.github.io/tutorial/index.html

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 6 de Febrero, 2021, 11:17

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Look-and-say y la constante de Conway
https://www.gaussianos.com/look-and-say-y-la-constante-de-conway/

Look-and-say sequence
https://en.wikipedia.org/wiki/Look-and-say_sequence

Elliptic Curve PKC and Key Sizes
https://medium.com/iovlabs-innovation-stories/elliptic-curve-pkc-and-key-sizes-941db80040b6

Jost Bürgi
https://mathshistory.st-andrews.ac.uk/Biographies/Burgi/

Carl Friedrich Gauss: el príncipe de las matemáticas
https://www.gaussianos.com/carl-friedrich-gauss-el-principe-de-las-matematicas/

246B, Notes 3: Elliptic functions and modular forms
https://terrytao.wordpress.com/2021/02/02/246b-notes-3-elliptic-functions-and-modular-forms/

Mark Kac y su nacionalidad compleja
https://www.gaussianos.com/mark-kac-y-su-nacionalidad-compleja/

The Cistercian monks invented a numbering system in the 13th century
https://twitter.com/MathematicsUCL/status/1356558846093914114

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 31 de Enero, 2021, 15:02

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Descubren un sistema de seis planetas que orbitan en asombrosa resonancia
https://www.vozpopuli.com/altavoz/next/Descubre-sistema-planetas-orbital-resonancia_0_1431757949.html

Cuando decimos que “cada quien tiene derecho a pensar lo que quiera”, ¿de qué “derecho” estamos hablando realmente?
https://twitter.com/ClaudioJavierCM/status/1353764838493839364

CK Vulpeculae: Something Went Boom but We Don't Know What
https://www.syfy.com/syfywire/ck-vulpeculae-something-went-boom-but-we-dont-know-what

A Six-Planet System Dances in Time to the Tune of Gravity
https://www.syfy.com/syfywire/a-six-planet-system-dances-in-time-to-the-tune-of-gravity

Physicists Study How Universes Might Bubble Up and Collide
https://www.quantamagazine.org/physicists-study-how-our-universe-might-have-bubbled-up-in-the-multiverse-20210125/

Podcast CB SyR 304: Estrella de Tabby, colores de las estrellas, el cuásar más lejano y otras noticias
https://francis.naukas.com/2021/01/29/podcast-cb-syr-304-estrella-de-tabby-colores-de-las-estrellas-el-cuasar-mas-lejano-y-otras-noticias/

The Hard Lessons of Modeling the Coronavirus Pandemic
https://www.quantamagazine.org/the-hard-lessons-of-modeling-the-coronavirus-pandemic-20210128/

Elementary Particles Cards
https://scoollab.web.cern.ch/sites/scoollab.web.cern.ch/files/documents/particle_cards_instructions_english.pdf

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia

Publicado el 24 de Enero, 2021, 13:11

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New Fish Data Reveal How Evolutionary Bursts Create Species
https://www.quantamagazine.org/new-fish-data-reveal-how-evolutionary-bursts-create-species-20201201/

Las variantes del SARS-CoV-2 que reducen la eficacia de las vacunas contra la COVID-19
https://francis.naukas.com/2021/01/24/las-variantes-del-sars-cov-2-que-reducen-la-eficacia-de-las-vacunas-contra-la-covid-19/

NASA: decenas de civilizaciones alienígenas habrían habitado la Vía Láctea
https://www.clarin.com/internacional/nasa-decenas-civilizaciones-alienigenas-habitado-via-lactea_0_H11OM557M.html

El Ovillo y la Espada
https://rafabravo.blog/2021/01/19/el-ovillo-y-la-espada/

Have We Already Been Visited by Aliens?
https://www.newyorker.com/magazine/2021/01/25/have-we-already-been-visited-by-aliens

We need hard science, not software, to power our post-pandemic recovery
https://phys.org/news/2021-01-hard-science-software-power-post-pandemic.html

La Nueva Era de la Sinrazón
http://revistaanfibia.com/ensayo/la-nueva-la-sinrazon/

Crab Nebula
https://www.astrobin.com/327338/0/
https://twitter.com/AdirondackAstro/status/1350916239099625476

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia

Publicado el 23 de Enero, 2021, 16:09

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Return of the Lost Evolutionary Psychology Interviews
https://www.psychologytoday.com/us/blog/sex-murder-and-the-meaning-life/201906/return-the-lost-evolutionary-psychology-interviews

Ask Ethan: Why Do Gravitational Waves Travel Exactly At The Speed Of Light?
https://www.forbes.com/sites/startswithabang/2019/07/06/ask-ethan-why-do-gravitational-waves-travel-exactly-at-the-speed-of-light/#5a6da3e232dc

Plant Cells of Different Species Can Swap Organelles
https://www.quantamagazine.org/plant-cells-of-different-species-can-swap-organelles-20210120/

The Search for Dark Matter Is Dramatically Expanding
https://www.quantamagazine.org/physicists-are-expanding-the-search-for-dark-matter-20201123/

Machine Learning Finds More Gravitational Lenses Than All Astronomers Combined
https://www.forbes.com/sites/startswithabang/2021/01/18/machine-learning-finds-more-gravitational-lenses-than-all-astronomers-combined/

How mRNA went from a scientific backwater to a pandemic crusher
https://www.wired.co.uk/article/mrna-coronavirus-vaccine-pfizer-biontech

The Curious Strength of a Sea Sponge"s Glass Skeleton
https://www.quantamagazine.org/the-curious-strength-of-a-sea-sponges-glass-skeleton-20210111/

Secret Ingredient Found to Power Supernovas
https://www.quantamagazine.org/supercomputer-simulations-reveal-the-power-inside-a-supernova-20210121/

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia

Publicado el 17 de Enero, 2021, 15:36

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Some Fixed Point Theorems for Partially Ordered Sets
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7BE2EFF0FF71A8CBFC34FF973BED5AA/S0008414X00030704a.pdf/some_fixed_point_theorems_for_partially_ordered_sets.pdf

Theoretical Physics
http://www.tug.org/texshowcase/cheat.pdf

New Geometric Perspective Cracks Old Problem About Rectangles
https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/

Enlaces útiles para la clase de Matemáticas
https://docs.google.com/document/d/174DY7xK8rLVvVwqYCweQDZ0tO4Zluf8RiG2k-cTWPDw/edit
https://twitter.com/lolamenting/status/1285510978445225989

Applied Category Theory 2020
https://twitter.com/johncarlosbaez/status/1277643847363772416

The Mathematical Structure of Particle Collisions Comes Into View
https://www.quantamagazine.org/new-particle-collision-math-may-offer-quantum-clues-20200820/

The Crooked Geometry of Round Trips
https://www.quantamagazine.org/the-crooked-geometry-of-round-trips-20210113/

Mathematicians Probe Unsolved Hilbert Polynomial Problem
https://www.quantamagazine.org/mathematicians-probe-unsolved-hilbert-polynomial-problem-20210114/

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez

Publicado el 16 de Enero, 2021, 14:52

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Scientists rise up against statistical significance
https://www.nature.com/articles/d41586-019-00857-9

New videos show RNA as it's never been seen
https://phys.org/news/2021-01-videos-rna.html

Dark matter local density determination: recent observations and future prospects
https://arxiv.org/abs/2012.11477

The Mystery of Mistletoe"s Missing Genes
https://www.quantamagazine.org/the-mystery-of-mistletoes-missing-genes-20201221/

¿Quién demonios es ese cangrejo y qué tiene que ver con la vacuna del COVID?
https://twitter.com/AzucenaMS/status/1331331376633819144

Scattering amplitudes and positive Grassmannian by Jaroslav Trnka
https://www.youtube.com/watch?v=M-4Iomz0Ce0

The Parke-Taylor Amplitudes: Why Quantum Field Theory Might Not Be So Hard, After All
https://4gravitons.com/2013/11/29/the-parke-taylor-amplitudes-why-quantum-field-theory-might-not-be-so-hard-after-all/

New Hubble data explains missing dark matter
https://phys.org/news/2020-11-hubble-dark.html

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia

Publicado el 14 de Enero, 2021, 18:31

Este comienzo de año me toma ocupado, así que tardé algunos días adicionales en publicar este post. Primero, revision de las resoluciones del mes pasado;

- Escribir sobre Matemáticas [completo] ver post
- Escribir sobre Física [pendiente]
- Escribir sobre Historia de las Matemáticas [pendiente]
- Escribir sobre Historia de la Ciencia [pendiente]
- Estudiar blues en guitarra [completo]

Estuve estudiando mucho en estos días no laborables que llegaron a fin de año, así que será hora de pasar al por escrito de estos temas. Tambien volví a escribir Objective Key Results personales, pero todavía los estoy revisando, a compartir espero algunos en el próximo mes.

Resoluciones del nuevo mes, sigo insistiendo con:

- Escribir sobre Matemáticas
- Escribir sobre Física
- Escribir sobre Historia de las Matemáticas
- Escribir sobre Historia de la Ciencia
- Estudiar blues en guitarra

Nos leemos!

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: General

Publicado el 10 de Enero, 2021, 19:08

El año pasado (2020) leí un artículo de Quanta Magazine que me hizo conocer el trabajo de Emily Riehl en teoría de categorías (el artículo abarca más que esa actividad), Quiero comentar hoy brevemente los primeros párrafos de su excelente libro Category Theory in Context. Serán comentarios livianos, el tema da para mucho más, sirva esto como una leve introducción.

Teoría de categorías es un tema que goza de la fama de abstracto, y en gran parte es así. Tambien va surgiendo en este siglo una rama aplicada de la teoría, pero en general, aún muchos matemáticos profesionales ven a la teoría como abstracta. Algo que no colabora para su diffusion, es que, como muchas otras ramas, si uno comienza a estudiarla se encuentra con muchas definciones y conceptos, que si bien son interesantes, NO PARECEN tener motivación que justifique su desarrollo. Es por eso que es bienvenido este libro de Riehl (la pueden segurr en su twitter @emilyriehl donde se describe como working mathematician, supongo que un guiño a uno de los más conocidos libros de teoría de categorás) que pone en contexto, como dice su título, las ideas que se desarrollan en el pensamiento de categorías. Leo al comienzo del libro:

Atiyah described mathematics as the "science of analogy." In this vein, the purview of category theory is mathematical analogy. Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another. The category-theoretic perspective can function as a simplifying1 abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline.

Suenore recuerdo el comentario de Bourbaki sobre Gauss, que en su Disquisitione Mathematica probaba una y otra vez una propiedad que quedaba evidente dentro de la teoría de grupos, pero debía hacerlo así porque esa teoría no estaba desarrollada. Las matemáticas como que extraen lo común a varias situaciones y trabajan sobre ello. Por ejemplo, las propiedades de un anillo, luego de haber establecido los axiomas a cumplir por todo anillo. O los espacios topológicos, dados las propiedades de sus conjuntos abiertos. La teoría de categorías va un paso (yo diría más de un paso), en esa dirección.

A subtle shift in perspective enables mathematical content to be described in language that is relatively indifferent to the variety of objects being considered. Rather than characterize the objects directly, the categorical approach emphasizes the transformations between objects of the same general type. A fundamental lemma in category theory implies that any athematical object can be characterized by its universal property—loosely by a representation of the morphisms to or from other objects of a similar form. For example, tensor products, "free" constructions, and localizations are characterized by universal properties in appropriate categories, or mathematical contexts. A universal property typically expresses one of the mathematical roles played by the object in question. For instance, one universal property associated to the unit interval identifies self-homeomorphisms of this space with re-parameterizations of paths. Another highlights the operation of gluing two intervals end to end to obtain a new interval, the construction used to define composition of paths.

El punto importante, que exige un esfuerzo intellectual, es justamente ver el objeto a través de sus relaciones con otros objetos parecidos. El tema de propiedad universal es fundamental en teoría de categorías, pero también cuesta aprehenderlo porque en general lo manejos en dominios particulares, como anillos o campos.

Certain classes of universal properties define blueprints which specify how a new object may be built out of a collection of existing ones. A great variety of mathematical constructions fit into this paradigm: products, kernels, completions, free products, "gluing" constructions, and quotients are all special cases of the general category-theoretic notion of limits or colimits, a characterization that makes it easy to define transformations to or from the objects so- defined. The input data for these constructions are commutative diagrams, which are themselves a vehicle for mathematical definitions, e.g., of rings or algebras, representations of a group, or chain complexes.

Es por todo esto, que este libro es para matemáticos: RIehl pone en contexto las ideas de teoría de categorías, pero apelando a varios conocimientos, desde el algebra conmutativa y no conmutativa hasta topología general.

Important technical differences between particular varieties of mathematical objects can be described by the distinctive properties of their categories: that rings have all limits and colimits while fields have few, that a continuous bijection defines an isomorphism of compact Hausdorff spaces but not of generic topological spaces. Constructions that convert mathematical objects of one type into objects of another type often define transformations between categories, called functors. Many of the basic objects of study in modern algebraic topology and algebraic geometry involve functors and would be impossible to define without
category-theoretic language.

De alguna forma, teoría de categorías expone lo que muchos matemáticos hacen: encontrar analogías entre distintos objetos y sacar provecho de ello. Sí, Atiyah tenia razón: es la ciencia de la analogia.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Publicado el 9 de Enero, 2021, 7:26

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Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia

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Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
https://twitter.com/ajlopez