1ST IMAGE OF A BLACK HOLE | EUROPEAN COMMISSION

BREAKTHROUGH DISCOVERY IN ASTRONOMY : PRESS CONFERENCE

On 10 April 2019 the European Commission presented a groundbreaking discovery by Event Horizon Telescope — an international scientific collaboration aiming to capture the first image of a black hole by creating a virtual Earth sized telescope .

Six press conferences around the world took place simultaneously . In Europe , Commissioner Moedas and lead scientists funded by the European Research Council held a press conference in Brussels to unveil the discovery .

SEEING THE UNSEEABLE | HARVARD NEWS | 04.10.2019

A global team of astronomers , led by Harvard scientists , have for the first time captured an image of a black hole .

The image , revealed by researchers from the Event Horizon Telescope ( EHT ) , shows a bright ring of material surrounding a dark center that marks the event horizon of the black hole — the " point of no return , where nothing , not even light — can escape " .

A BLACK HOLE REVEALED | HARVARD NEWS | 04.10.2019

In the annals of science , there are a few images — think the " pale blue dot " captured by Voyager I or Apollo 8 " Earthrise " — that have both captured the public imagination and offered scientists insight into how the Universe works .

Researchers at the Event Horizon Telescope ( EHT ) just unveiled the latest .

The image , the first ever of a black hole , is destined for the shortlist of iconic images not only for what it can tell astronomers and physicists about how gravity and general relativity work under the most extreme conditions , but also because it captures what EHT Director Sheperd Doeleman called " a one way door from our Universe " .

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FUTURISM | THE BYTE | EARTH & ENERGY | 10.30.2018

TECH BILLIONAIRES ARE POURING MONEY INTO FUSION RESEARCH

Elon Musk's SpaceX changed the space game when it launched and landed the first reusable rocket . With that one achievement , it set a course to more affordable space travel .

Now a slew of Musk's fellow tech billionaires are chasing a similar turning point in the energy industry , according to Bloomberg : The creation of the first commercially viable nuclear fusion reactor .

BLOOMBERG | NEWS | CLIMATE CHANGE | 10.30.2018

BILLIONAIRES CHASE SPACEX MOMENT FOR THE HOLY GRAIL OF ENERGY

Jeff Bezos , Bill Gates and Peter Thiel are funneling cash into nuclear fusion projects .

Tech visionary Paul Allen traveled to the south of France for a personal tour of a 35 country quest to replicate the workings of the Sun .

The goal is to one day produce clean , almost limitless energy by fusing atoms together rather than splitting them apart .

The Microsoft Corp cofounder wanted to view the early stages of the International Thermonuclear Experimental Reactor in Cadarache firsthand , to witness preparations " for the birth of a star on Earth " .

]]>NASA.GOV | NASA TV & ISS HDEVLIVEFEED

|| NASA TELEVISION ||

NASA provides images , videos and features from the unique perspective of America's space agency . Get updates on missions , watch NASA TV , read blogs , view the latest discoveries , and more .

NASA Television provides live coverage of launches , spacewalks and other mission events , as well as the latest news briefings , video files , and the This Week @NASA report .

|| ISS HD EARTH VIEWING EXPERIMENT | LIVEFEED ||

QUICK NOTES ABOUT HDEV VIDEO :

Black Image = International Space Station ( ISS ) is on the night side of the Earth .

No Audio = Normal . There is no audio by design . Add a soundtrack .

For a display of the real time ISS location plus the HDEV imagery visit here .

The High Definition Earth Viewing ( HDEV ) experiment aboard the ISS was activated April 30 2014 . It is mounted on the External Payload Facility of the European Space Agency's Columbus module .

This experiment includes several commercial HD video cameras aimed at the Earth which are enclosed in a pressurized and temperature controlled housing . Video from these cameras is transmitted back to Earth and also streamed live on this channel .

While the experiment is operational , views will typically sequence though the different cameras . Between camera switches , a gray and then black color slate will briefly appear . Since the ISS is in darkness during part of each orbit , the images will be dark at those times . During periods of loss of signal with the ground or when HDEV is not operating , a gray color slate or previously recorded video may be seen .

WATCH ISS LIVEFEED @ NASA ➤➤

WATCH ISS LIVEFEED @ USTREAM ➤➤

READ DETAILED INFO ABOUT HDEV ➤➤

5 GREAT LISTS OF MATHEMATICS RESOURCES + FORUMS | READ ➤➤

FUTURISM | NEWS | HARD SCIENCE | 04.22.2016

|| HERE IS HOW TO TEACH YOURSELF PHYSICS & MATH ||

Physics and Mathematics are extremely important subjects .

Actually , that is a bit of an understatement .

Physics and Mathematics allow us to peer out into the cosmos and understand the inner workings of the universe . At once , they show us our insignificance and our remarkable potential ; they give us a hint of the vast possibilities that exist — of what we could ( and may ) one day accomplish . They allow us to see the world and to see ourselves anew .

That begins to scratch the surface of these subjects .

No one can deny their importance ; however , it is also a fact that many people do not know where to begin investigating these topics . . . what books to study , what themes to begin with .

On top of this , many feel intimidated by Physics and Math — they seem to think that they are things which only the sharpest individuals are able to understand .

But nothing could be farther from the truth .

True , these subject areas might not be the easiest that you will ever happen across , but they are far from impossible .

So . If you want to be a Physicist or a Mathematician , or if you just want to understand the subjects , here is where to start .

Huge thanks to the wonderful Moinak Banerjee for work on this .

|| PHYSICS ||

Here is professor John Baez advice on how to learn Physics and Mathematics .

Baez mentions the books you should read , and they are conveniently listed according to increasing levels of difficulty .

This is the list of books that Berkeley recommends for people who want to teach themselves Physics .

And yet another list that is pretty good , which was compiled on Physics Stack Exchange .

Nobel laureate professor Gerard 't Hooft has recommended some learning sources that are all free , and Hooft also has advice related to how to earn your own Nobel prize .

Here is some great advice from a physicist on Physics Forums regarding what you should do if you want to do more than just learn Physics — on what to do if you want to actually have a career in Physics .

To supplement these , check out our *extensive* list of online Physics lecture videos .

|| MATHEMATICS ||

Mathematician Terence Tao , who is also a Fields medalist and a Breakthrough prize winner , gave some beautiful advice on pursuing a career in Math here .

If you are looking for Mathematics books you should read , here is an extensive list by Berkeley .

The aforementioned should probably be followed up by the compilation from the Math Stack Exchange .

Also , there is an extensive list of what you should read on the Georgia Institute of Technology website , and all of the texts that they link to are online and free .

You can access more free Math texts here and here .

|| SEPARATING SCIENCE FROM PSEUDOSCIENCE ||

Here is a hilarious point system on how to tell if something is " crank science " from John Baez .

Warren Siegel has an equally hilarious write up here .

And finally , from Gerard 't Hooft .

Now venture forth , live long , and study well .

]]>QUANTA MAGAZINE | MICROBIOLOGY | 06.21.2016

|| NEW LIFE FOUND THAT LIVES OFF ELECTRICITY ||

Biophysicist Moh El-Naggar and graduate student Yamini Jangir plunged beneath South Dakota's Black Hills into an old gold mine that is now more famous as a home to a dark matter detector .

Unlike most scientists who make pilgrimages to the Black Hills these days , El-Naggar and Jangir were not there to hunt for subatomic particles . They came in search of life .

In the darkness found a mile underground , the pair traversed the mine's network of passages in search of a rusty metal pipe . They siphoned some of the pipe's ancient water , directed it into a vessel , and inserted a variety of electrodes .

They hoped the current would lure their prey , a little-studied microbe that can live off pure electricity .

NEW SCIENTIST | NEWS | LIFE | SPECIAL REPORT | 07.16.2014

|| THE ELECTRICITY EATERS | MEET THE ELECTRIC LIFE FORMS THAT LIVE ON PURE ENERGY ||

Stick an electrode in the ground , pump electrons down it , and they will come : living cells that eat electricity .

We have known bacteria to survive on a variety of energy sources , but none as weird as this . Think of Frankenstein's monster , brought to life by galvanic energy , except these " electric bacteria " are very real and are popping up all over the place .

Unlike any other living thing on Earth , electric bacteria use energy in its purest form — naked electricity in the shape of electrons harvested from rocks and metals .

We already knew about two types , Shewanella and Geobacter . Now , biologists are showing that they can entice many more out of rocks and marine mud by tempting them with a bit of electrical juice .

Experiments growing bacteria on battery electrodes demonstrate that these novel , mindboggling forms of life are essentially eating and excreting electricity .

BBC | EARTH | EXTREME LIFE | MICROORGANISM | 06.14.2016

|| THERE ARE MICROBES THAT EAT AND POO NOTHING BUT ELECTRICITY ||

All living cells are ultimately powered by electrons . Most species get electrons from food , but some bacteria can survive on nothing but pure electricity .

Some microbes have developed the ultimate stripped-down diet . They do not bother with food or oxygen . All they need to survive is pure electrical energy .

They often live in muddy seabeds or along the banks of rivers . Finding them is easy : biologists can coax them out of hiding by sticking an electrode into the sediment .

The bacteria nearest to the electrode will even grow biological equivalents of electrical wires out of their bodies , so that other microbes further away can hook up to the electricity source . It is effectively a living power grid .

What is more , it appears that we can all benefit from this microscopic grid . Among other things , it might provide an effective way to deal with toxic waste and other forms of environmental pollution .

Electricity-eating microbes might sound like something straight out of a science fiction novel . In fact their behaviour is not quite as exotic as it might first appear .

]]>THE PRINCETON COMPANION TO MATHEMATICS | GOWERS | ETAL

|| VIII.6.I-V | ADVICE TO A YOUNG MATHEMATICIAN ||

The most important thing that a young Mathematician needs to learn is of course Mathematics . However , it can also be very valuable to learn from the experiences of other Mathematicians .

The five contributors to this article were asked to draw on their experiences of Mathematical life and research , and to offer advice that they might have liked to receive when they were just setting out on their careers .

The resulting contributions were every bit as interesting as we had expected ; what was more surprising was that there was remarkably little overlap between the contributions .

So here they are , five gems intended for young Mathematicians but surely destined to be read and enjoyed by Mathematicians of all ages .

]]>QUANTA MAGAZINE | MATHEMATICS | WOLCHOVER | 05.24.2016

|| MATHEMATICIANS BRIDGE FINITE INFINITE DIVIDE ||

A surprising new proof is helping to connect the Mathematics of Infinity to the physical world .

Mathematicians Ludovic Patey and Keita Yokoyama have found a bridge across the Finite Infinite divide , helping at the same time to map this strange boundary .

The boundary does not pass between some huge Finite number and the next , Infinitely large one . Rather , it separates two kinds of Mathematical statements :

" Finitistic " ones , which can be proved without invoking the concept of Infinity , and " Infinitistic " ones , which rest on the assumption — not evident in nature — that Infinite objects exist .

Mapping and understanding this division is " at the heart of Mathematical logic " , said Theodore Slaman , a professor of Mathematics at the University of California , Berkeley .

This endeavor leads directly to questions of Mathematical objectivity , the meaning of Infinity and the relationship between Mathematics and physical reality .

CLICK TO READ MAIN QUANTA ARTICLE ➤➤

CLICK TO READ SCIENTIFIC TECH PDF ➤➤

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AIMATH.ORG | AIMNEWS | 05.10.2016

|| EXPLORING THE MATHEMATICAL UNIVERSE ||

A team of more than 80 Mathematicians from 12 countries has begun charting the terrain of rich , new Mathematical worlds , and sharing their discoveries on the Web .

The Mathematical universe is filled with both familiar and exotic items , many of which are being made available for the first time .

The L-FUNCTIONS & MODULAR FORMS DATABASE , abbreviated LMFDB , is an intricate catalog of Mathematical objects and the connections between them .

Making those relationships visible has been made possible largely by the coordinated efforts of a group of researchers developing new algorithms and performing calculations on an extensive network of computers .

The project provides a new tool for several branches of Mathematics , Physics , and Computer Science .

Just like explorers , Mathematicians seek to discover paths between apparently unrelated areas .

Such discoveries can lead to breakthroughs when the connections are made explicit . . . .

The scale of the computational effort involved in the LMFDB is staggering : a total of nearly a thousand years of computer time spent on calculations by multiple teams of researchers .

A recent contribution by Andrew Sutherland at MIT used 72,000 cores of Google's Compute Engine to complete in one weekend a tabulation that would have taken more than a century on a single computer .

As noted by Sutherland , " computations in number theory are often amenable to parallelization , and this makes it easy to scale them to the cloud " .

The application of large-scale cloud computing to research in Pure Mathematics is just one of the ways in which the project is pushing forward the frontier of Mathematics .

MIT | NEWS | 05.10.2016

|| INTERNATIONAL TEAM LAUNCHES VAST ATLAS OF MATHEMATICAL OBJECTS ||

An international group of Mathematicians at MIT and other institutions has released a new online resource that provides detailed maps of previously uncharted Mathematical terrain .

The L-FUNCTIONS & MODULAR FORMS DATABASE , or LMFDB , is a detailed atlas of Mathematical objects that maps out the connections between them .

The LMFDB exposes deep relationships and provides a guide to previously uncharted territory that underlies current research in several branches of Physics , Computer Science , and Mathematics .

This coordinated effort is part of a massive collaboration of researchers around the globe .

LMFDB | L-FUNCTIONS & MODULAR FORMS DATABASE | INTRODUCTION

|| DATABASE OF L-FUNCTIONS , MODULAR FORMS , & RELATED OBJECTS ||

Welcome to the LMFDB , the database of L-functions , modular forms , and related objects .

These pages are intended to be a modern handbook including tables , formulas , links , references , etc , to very concrete objects , in particular specific L-functions and their sources .

L-functions are ubiquitous in number theory and have applications to Mathematical Physics and cryptography .

By an L-function , we generally mean a Dirichlet series with a functional equation and an Euler product , the simplest example being the Riemann zeta function . Two of the seven Clay Mathematics Million Dollar Millennium Problems deal with properties of these functions , namely the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture .

L-functions arise from and encode information about a number of Mathematical objects . It is necessary to exhibit these objects along with the L-functions themselves , since typically we need these objects to compute L-functions .

In these pages you will see examples of L-functions coming from modular forms , elliptic curves , number fields , and Dirichlet characters , as well as more generally from automorphic forms , algebraic varieties , and Artin representations .

In addition , the database contains details about these objects themselves . See the LMFDB universe for descriptions of connections between these objects . For additional information , there is a useful collection of freely available online sources at

The subject of L-functions is very rich , with many interrelationships . Our goal is to describe the data in ways that faithfully exhibit these interconnections , and to offer access to the data as a means of prompting further exploration and discovery .

We believe that the creation of the LMFDB website will lead to the development and understanding of new Mathematics .

]]>RAMANUJAN MATHEMATICS | STEPHEN WOLFRAM | 04.27.2016

|| TRUTH VERSUS NARRATIVE | EXCERPT ||

Ramanujan's way of working must have seemed quite alien . For Ramanujan was in some fundamental sense an Experimental Mathematician :

Going out into the Universe of Mathematical possibilities and doing calculations to find interesting and significant facts — and only then building theories based on them .

|| SEEING WHAT IS IMPORTANT | EXCERPT ||

Ramanujan was surely a great human calculator , and impressive at knowing whether a particular Mathematical fact or relation was actually true .

But Ramanujan's greatest skill was , I think , something in a sense more mysterious : an uncanny ability to tell what was significant , and what might be deduced from it .

|| A WAY OF DOING MATHEMATICS | EXCERPT ||

By the time of the scholarship , Ramanujan had started writing more papers , and publishing them in the Journal of the Indian Mathematical Society .

Compared to the big claims about primes and divergent series , the topics of these papers were quite tame . But the papers were remarkable nevertheless .

What is immediately striking about them is how calculational they are — full of actual , complicated formulas . Most Math papers are not that way . They may have complicated notation , but they do not have big expressions containing complicated combinations of roots , or seemingly random long integers .

In modern times , we are used to seeing incredibly complicated formulas routinely generated by Mathematica . But usually they are just intermediate steps , and are not what papers explicitly talk much about .

For Ramanujan , though , complicated formulas were often what really told the story .

And of course it is incredibly impressive that Ramanujan could derive them without computers and modern tools .

CLICK TO READ COMPREHENSIVE ARTICLE ➤➤

|| PUBLISHED PAPERS & UNPUBLISHED NOTEBOOKS ETC LINKS | TBC . . . ||

http://plouffe.fr/simon/math/Ramanujan's%20Notebooks%20I.pdf

http://plouffe.fr/simon/math/Ramanujan's%20Notebooks%20II.pdf

http://plouffe.fr/simon/math/Ramanujan's%20Notebooks%20III.pdf

http://plouffe.fr/simon/math/Ramanujan's%20Notebooks%20IV.pdf

http://plouffe.fr/simon/math/Ramanujan's%20Notebooks%20V.pdf

http://www.math.tifr.res.in/~publ/nsrBook1.pdf

http://www.math.tifr.res.in/~publ/nsrBook2.pdf

http://www.math.tifr.res.in/~publ/nsrBook3.pdf

http://ramanujan.sirinudi.org/

http://www.imsc.res.in/~rao/ramanujan/contentindex.html

http://babel.hathitrust.org/cgi/pt?id=umn.319510002394343;view=1up;seq=773

]]>MATHEMATICS | SRINIVASA RAMANUJAN

An equation for me has no meaning ,

unless it represents a Thought of GOD .

OPEN ACCESS MATHEMATICS JOURNALS

DISCRETE ANALYSIS JOURNAL | LAUNCH | 03.01.2016

Discrete Analysis is a Mathematical journal that aims to publish articles that are analytical in flavour but that also have an impact on the study of discrete structures .

These include harmonic analysis , ergodic theory , topological dynamics , growth in groups , analytic number theory , additive combinatorics , combinatorial number theory , extremal and probabilistic combinatorics , combinatorial geometry , convexity , metric geometry , and theoretical computer science .

TERENCE TAO | BLOG | DISCRETE ANALYSIS | ANNOUNCEMENT

|| DISCRETE ANALYSIS SITE NOW LIVE ||

Just a quick post to note that the arXiv overlay journal Discrete Analysis , managed by Timothy Gowers , has now gone live with its permanent ( and quite modern looking ) website , which is run using the Scholastica platform , as well as the first half-dozen or so accepted papers ( including one of my own ) . See Tim's announcement for more details .

I am one of the editors of this journal ( and am already handling a few submissions ) .

Needless to say , we are happy to take in more submissions ( though they will have to be peer reviewed if they are to be accepted , of course ) .

TIMOTHY GOWERS | BLOG | DISCRETE ANALYSIS | ANNOUNCEMENT

|| DISCRETE ANALYSIS — AN ARXIV OVERLAY JOURNAL ||

This post is to announce the start of a new Mathematics journal , to be called Discrete Analysis .

While in most respects it will be just like any other journal , it will be unusual in one important way : it will be purely an arXiv overlay journal . . .

Part of the motivation for starting the journal is , of course , to challenge existing models of academic publishing and to contribute in a small way to creating an alternative and much cheaper system . . .

The members of the editorial board all have an interest in additive combinatorics , but they also have other interests that may be only loosely related to additive combinatorics . . .

Some of the subjects we will welcome in the journal are harmonic analysis , ergodic theory , topological dynamics , growth in groups , analytic number theory , combinatorial number theory , extremal combinatorics , probabilistic combinatorics , combinatorial geometry , convexity , metric geometry , and the more Mathematical side of theoretical computer science .

The phrase " discrete analysis " was coined by Ben Green when wanting a suitable name for a seminar in Cambridge : despite its oxymoronic feel , it is in fact a good description of many parts of Mathematics where the structures being studied are discrete , but the tools are analytical in character .

VOX | SCI & HEALTH | DISCRETE ANALYSIS | 03.04.2016

|| THIS RENOWNED MATHEMATICIAN IS BENT ON PROVING ACADEMIC JOURNALS CAN COST NOTHING ||

This week , Timothy Gowers debuted a new online Mathematics journal called Discrete Analysis . The nonprofit venture is owned and published by a team of scholars . With no publisher intermediary , access will be completely free for all .

Gowers is betting traditional journals do not add value , particularly when it comes to papers that are mostly Math .

" We do not need all the services journals provide — like formatting in a journal house style and copy-editing — because the pre-prints are good enough already , and there are lots of examples where copy-editing makes articles worse and introduces mistakes . "

SCHOLASTICA | BLOG | DISCRETE ANALYSIS | 02.29.2016

|| LAUNCH OF DISCRETE ANALYSIS MARKS NEW ERA FOR OA JOURNAL PUBLISHING ||

Fields Medal-winning Cambridge Mathematician Timothy Gowers and a team of distinguished colleagues are paving the way for editor-owned Open Access ( OA ) journals with the help of Scholastica .

Making academic journal articles free to read and quick to access — that is the mission of the growing open access movement . Cambridge Mathematician Timothy Gowers and a team of colleagues are leading the charge with today's official launch of the game-changing OA journal Discrete Analysis .

The journal will be owned by its editors , rather than a publisher intermediary , and will follow the " diamond OA model " , meaning it will be free to read and publish in .

We sat down with Gowers to learn more about Discrete Analysis , which will be peer-reviewed and published via Scholastica , and to discuss Gower's vision for the future of editor-owned journals .

CONTINUE TO READ ARTICLE ➤➤

CONTINUE TO READ RELATED ARTICLE ➤➤

ARXIV | WIKIPEDIA

The arXiv is a repository of electronic preprints , known as e-prints , of scientific papers in the fields of Mathematics , Physics , Astronomy , Computer Science , Quantitative Biology , Statistics , and Quantitative Finance , which can be accessed online .

In many fields of Mathematics and Physics , almost all scientific papers are self-archived on the arXiv .

ARXIV | MATHEMATICS

arXiv is an e-print service in the fields of Physics , Mathematics , computer science , quantitative biology , quantitative finance and statistics . Submissions to arXiv should conform to Cornell University academic standards .

arXiv is owned and operated by Cornell University , a private not-for-profit educational institution . arXiv is funded by Cornell University Library , the Simons Foundation and by the member institutions .

]]>INTRO TO MATHEMATICAL NOTATION | 60 PAGE EBOOK+

HANDBOOK FOR SPOKEN MATHEMATICS | LAWRENCE CHANG

This handbook answers some of the needs of those who have to deal with spoken Mathematics , yet have insufficient background to know the correct verbal expression for the written symbolic one .

Mathematical material is primarily presented visually , and when this material is presented orally , it can be ambiguous .

While the parsing of a written expression is clear and well-defined , when it is spoken this clarity may disappear .

For example , " One plus two over three plus 4 " can represent the following four numbers , depending on the parsing of the expression : 3/7 , 1 2/7 , 5 , 5 2/3 . However , when the corresponding written expression is seen , there is little doubt which of the four numbers it represents .

When reading Mathematics orally , such problems are frequently encountered .

Of course , the written expression may always be read symbol by symbol , but if the expression is long or there are a cluster of expressions , it can be very tedious and hard to understand . Thus , whenever possible, one wishes to have the written expression spoken in a way that is interest retaining and easy to understand .

In an attempt to alleviate problems such as these , this handbook has been compiled to establish some consistent and well-defined ways of uttering Mathematical expressions so that listeners will receive clear , unambiguous , and well-pronounced representations of the subject .

CLICK TO 60 PAGE PDF | READ ➤➤

PROJECT MATH ACCESS | HANDBOOK FOR SPOKEN MATHEMATICS

The content of this portion of the site was adapted from Lawrence Chang's Handbook For Spoken Mathematics : Larry's Speakeasy .

All sections of the original work are included except the section entitled , Diagrams and Graphs . That chapter contains extraordinarily complex graphics which would be impossible to display in such a fashion that it would be accessible to those who are blind or severely visually impaired .

The focus of Chang's work is to provide the wherewithal by which sighted individuals can read Mathematics to persons who are blind .

Chang describes in words each symbol in the major branches of Mathematics , from the simplest to the most complex .

In essence , Chang's work represents the spoken vocabulary of Mathematics .

Thus , one use to which this work can be put is to assist persons who are blind in training sighted readers to accurately read Mathematics .

We have added to the reservoir of information in several ways .

CLICK TO READ HTML ADAPTED VERSION ➤➤

LIST OF MATHEMATICAL SYMBOLS | WIKIPEDIA

This is a list of symbols found within all branches of Mathematics to express a formula or to represent a constant .

When reading the list , it is important to recognize that a Mathematical concept is independent of the symbol chosen to represent it .

For many of the symbols below , the symbol is usually synonymous with the corresponding concept ( ultimately an arbitrary choice made as a result of the cumulative history of Mathematics ) , but in some situations a different convention may be used .

For example , depending on context , the triple bar " ≡ " may represent congruence or a definition .

Further , in Mathematical logic , numerical equality is sometimes represented by " ≡ " instead of " = " , with the latter representing equality of well-formed formulas .

In short , convention dictates the meaning .

Each symbol is shown both in HTML , whose display depends on the browser's access to an appropriate font installed on the particular device , and in TeX , as an image .

LIST OF MATHEMATICAL SYMBOLS BY SUBJECT | WIKIPEDIA

This list of Mathematical symbols by subject shows a selection of the most common symbols that are used in modern Mathematical notation within formulas , grouped by Mathematical topic .

As it is virtually impossible to list all the symbols ever used in Mathematics , only those symbols which occur often in Mathematics or Mathematics education are included . The following list is largely limited to non-alphanumeric characters .

It is divided by areas of Mathematics and grouped within subregions .

Some symbols have a different meaning depending on the context and appear accordingly several times in the list .

LIST OF MATHEMATICAL SYMBOLS | RAPID TABLES

List of Mathematical symbols and signs — meaning and examples .

Basic Math symbols , geometry symbols , algebra symbols , probability & statistics symbols , set theory symbols , logic symbols , calculus & analysis symbols , number symbols , greek symbols , and roman numerals .

]]>100+ SANSKRIT RESOURCES ➤➤ | LANGUAGE ➤➤ | MATHEMATICS ➤➤

THE WIRE | MANJUL BHARGAVA | 01.19.2016

|| SANSKRIT & MATHEMATICS LECTURE | EXCERPTS ||

Princeton University Mathematician and Fields Medallist Manjul Bhargava delivered a lecture at the Madras Sanskrit College , Chennai , on the connection between Sanskrit and Mathematics .

Bhargava touched upon ancient Indian contributions to advancing number theory and geometry , the importance of preserving their historic contexts , and what institutions like Sanskrit College can do to legitimize modern debates over India's claims to primacy .

" For those of you that are interested in Mathematics , I thought I would share with you some of the things I found in Sanskrit literature .

All these ancient Mathematical works are so beautiful and yet there are no translations available for the public .

Only somebody who knows Sanskrit and knows Mathematics in a modern way can appreciate these works !

That is a very tiny percentage of the population .

That is just not acceptable — when there is such beauty out there and only a tiny percentage of people can appreciate that . "

]]>UNIVALENT FOUNDATIONS OF MATHEMATICS | VOEVODSKY

|| ARTICLES | TECH PAPERS | VIDEO | IAS HoTT EBOOK ||

QUANTA MAGAZINE | MATHEMATICS | MAY 2015

|| WILL COMPUTERS REDEFINE THE ROOTS OF MATHEMATICS ? ||

When legendary mathematician Vladimir Voevodsky found a mistake in work , Voevodsky embarked on a computer aided quest to eliminate human error . To succeed , Voevodsky has to rewrite the century-old rules underlying all of Mathematics .

Rather than relying on fallible human beings to check proofs , you can turn the job over to computers ( * ) , which can tell whether a proof is correct with complete certainty .

Voevodsky has been advocating the virtues of computer proof assistants and developing univalent foundations in order to bring the languages of Mathematics and Computer Programming closer together .

As Voevodsky sees it , the move to computer formalization is necessary because some branches of Mathematics have become too abstract to be reliably checked by people .

" The world of Mathematics is becoming very large , the complexity of Mathematics is becoming very high , and there is a danger of an accumulation of mistakes " , Voevodsky says .

Proofs rely on other proofs ; if one contains a flaw , all others that rely on it will share the error .

CLICK TO READ MAIN QUANTA ARTICLE ➤➤

CLICK TO READ RELATED QUANTA ARTICLE ( * ) ➤➤

NAUTILUS | NUMBERS | MATHEMATICS | MAY 2015

|| IN MATHEMATICS MISTAKES ARE NOT WHAT THEY USED TO BE ||

Voevodsky is careful to distinguish the various ways computers should or should not be put to use :

" Lots of people do not understand the difference between using computers for calculation , for computer-generated proofs , and for computer-assisted proofs . The computer-generated proofs are the proofs which teach us very little .

And there is a correct perception that if we go toward computer generated proofs then we lose all the good that there is in Mathematics — Mathematics as a spiritual discipline , Mathematics as something which helps to form a pure mind . "

ETHZURICH | NEWS | SEPTEMBER 2014

|| A NEW FOUNDATION FOR MATHEMATICS ||

Proofs are the key method of Mathematics . Until now , it has mainly been humans who have verified whether proofs are correct .

This could change , says Russian mathematician Vladimir Voevodsky , who has developed an approach that could indeed revolutionize Mathematics and its foundations :

Voevodsky has been able to show in principle that homotopy theory , which deals with the deformation of geometric objects , expresses the same ideas as the theory of programming languages and Mathematical logic , only in a different language .

SCIENTIFIC AMERICAN | BLOGS | OCTOBER 2013

|| VOEVODSKY'S MATHEMATICAL REVOLUTION ||

Voevodsky told mathematicians that their lives are about to change .

Soon enough, they are going to find themselves doing Mathematics at the computer , with the aid of computer proof assistants .

Soon , they will not consider a theorem proven until a computer has verified it . Soon , they will be able to collaborate freely , even with mathematicians in whose skills they do not have confidence .

Soon , they will understand the foundations of Mathematics very differently .

UNIVALENT FOUNDATIONS PROJECT | TECH PDF | OCTOBER 2010 | VOEVODSKY

Eventually it became clear to me that the univalent semantics is just a first step and that I am really working on new foundations of Mathematics .

Key features of these " univalent foundations " are as follows :

1 . Univalent foundations naturally include " axiomatization " of the categorical and higher categorical thinking .

2 . Univalent foundations can be conveniently formalized using the class of languages called dependent type systems .

3 . Univalent foundations are based on direct axiomatization of the " world " of homotopy types instead of the world of sets .

4 . Univalent foundations can be used both for constructive and for non-constructive Mathematics .

IAS | MATHEMATICS | VIDEO+PDF | MARCH 2014

|| UNIVALENT FOUNDATIONS | VLADIMIR VOEVODSKY ||

" I would like to thank all of those who are trying to understand the ideas of Univalent Foundations , to develop these ideas and to communicate these ideas to others . I know it is difficult . "

In Voevodsky’s experience , the work of a mathematician is 5% creative insight and 95% self-verification . Moreover , the more original the insight , the more one has to pay for it later in self-verification work .

The Univalent Foundations project , started at the Institute a few years ago , aims to lower the price by giving mathematicians the ability to verify their constructions with the help of computers .

Voevodsky will explain how new ideas that make this goal attainable arise from the meeting of two streams of development — one in constructive Mathematics and the theory and practice of programming languages , and the other in pure Mathematics .

CLICK TO PDF | READ ➤➤ | CLICK TO VIDEO | VIEW ➤➤

AMS | BULLETIN | VOL 51 | #4 | TECH PDF | OCTOBER 2014

|| HOMOTOPY TYPE THEORY & VOEVODSKY'S UNIVALENT FOUNDATIONS ||

ABSTRACT | Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science .

This has given rise to a new field , which has been christened homotopy type theory . In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property , called the Univalence Axiom , which has a number of striking consequences .

Voevodsky has subsequently advocated a program — called univalent foundations — of developing Mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated by the simplicial set model .

Because type theory possesses good computational properties , this program can be carried out in a computer proof assistant .

In this paper we give an introduction to homotopy type theory in Voevodsky's setting , paying attention to both theoretical and practical issues .

In particular , the paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky's work using the well-known proof assistant Coq .

The paper is written for a general audience of mathematicians with basic knowledge of algebraic topology ; the paper does not assume any preliminary knowledge of type theory , logic, or computer science .

Because a defining characteristic of Voevodsky's program is that the Coq code has fundamental Mathematical content , and many of the Mathematical concepts which are efficiently captured in the code cannot be explained in standard Mathematical English without a lengthy detour through type theory , the later sections of this paper ( beginning with Section 3 ) make use of code ; however , all notions are introduced from the beginning and in a self-contained fashion .

CLICK TO 52 PAGE PDF | READ ➤➤

IAS | HOMOTOPY TYPE THEORY | IAS HoTT EBOOK |

Homotopy type theory is a new branch of Mathematics that combines aspects of several different fields in a surprising way .

It is based on a recently discovered connection between homotopy theory and type theory . It touches on topics as seemingly distant as the homotopy groups of spheres , the algorithms for type checking , and the definition of weak ∞-groupoids .

Homotopy type theory offers a new " univalent " foundation of Mathematics , in which a central role is played by Voevodsky's univalence axiom and higher inductive types .

The present book is intended as a first systematic exposition of the basics of univalent foundations , and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic , or to use any computer proof assistant .

We believe that univalent foundations will eventually become a viable alternative to set theory as the " implicit foundation " for the unformalized Mathematics done by most mathematicians .

CLICK TO 484 PAGE BOOK | PDF ➤➤

CLICK TO OTHER FORMATS OF BOOK ➤➤

|| ADDITIONAL LINKS | UPDATE 10.14.2015 ||

|| VIDEOS ||

http://video.ias.edu/vladimir

http://www.youtube.com/watch?v=vcDaQTPH-Rc

http://www.youtube.com/results?search_query=%22Vladimir+Voevodsky%22

|| MISCELLANEOUS ||

http://www.math.ias.edu/vladimir/Univalent_Foundations

http://www.math.ias.edu/vladimir/home

http://github.com/vladimirias

http://github.com/HoTT/HoTT

http://baaltii1.livejournal.com/198675.html

http://baaltii1.livejournal.com/200269.html

http://en.wikipedia.org/wiki/Proof_assistant

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.AgdaVsCoq

TORONTO STAR | ROBERT LANGLANDS | MARCH 2015

|| THE CANADIAN WHO REINVENTED MATHEMATICS ||

Robert Langlands , a Canadian , is one of the world's great mathematicians . Langlands' universe is the outer limits of Pure Mathematics , a rarefied realm where abstract objects exist , infinity is corralled and symmetry reigns .

As a young professor at Princeton University , Langlands revolutionized the ancient discipline , discovering patterns in highly esoteric objects called automorphic forms and motives , and restructuring Mathematics with two dazzling theories .

They indicated what mathematician Edward Frenkel calls " the source code of all Mathematics " , and are credited with linking Math's main branches — Number Theory ( once called Arithmetic ) , Harmonic Analysis , which includes Calculus , and Geometry .

]]>KAVLI IPMU NEWS | ISSUE #28 | DECEMBER 2014

ROUND TABLE TALK : CONVERSATION WITH EDWARD WITTEN | 17 PAGE PDF

|| EXCERPT 1 | KYOTO PRIZE & 4TH VISIT TO KYOTO ||

OOGURI : First I would like to congratulate Edward on your Kyoto Prize . In every four years , the Kyoto Prize goes in the field of mathematical sciences , and this is the first prize in this category awarded to a physicist .

WITTEN : Well , I can tell you I am deeply honored to receive this prize .

OOGURI : It is wonderful that your work in the area at the interface of mathematics and physics has been recognized as one of the most important progress in mathematics as well as in physics .

For those of us working in this area , this is also very gratifying .

As Yuji Tachikawa said at the workshop yesterday , you are like sunshine for all of us in this area of research .

WITTEN : Actually in my acceptance speech a couple of days ago , I remarked that I regard it also as a recognition of the field , not just of me .

|| EXCERPT 2 | LANGLANDS CORRESPONDENCE & GAUGE THEORY DUALITIES ||

WITTEN : For that year , I felt like someone who had discovered the meaning of life and could not explain it to anybody else . And in a sense , I still feel that way for the following reason .

Physicists with a background in string theory or gauge theory dualities can understand my paper with Kapustin on geometric Langlands but for most physicists this topic is too detailed to be really exciting .

On the other hand , it is an exciting topic for mathematicians , but difficult to understand because too much of the quantum field theory and string theory background is unfamiliar ( and difficult to formulate rigorously ) .

That paper with Kapustin may unfortunately remain mysterious to mathematicians for quite some time .

YAMAZAKI : Maybe that means that we have to wait an extra 10 or 15 years before . . .

WITTEN : We indeed may have to . I think it is actually very difficult to see what advance in the near term could make the gauge theory interpretation of geometric Langlands accessible for mathematicians .

That is actually one reason why I am excited about Khovanov homology .

My approaches to Khovanov homology and to geometric Langlands use many of the same ingredients , but in the case of Khovanov homology , I think it is quite feasible that mathematicians could understand this approach in the near future if they get excited about it . I believe it will be more accessible .

|| EXCERPT 3 | MESSAGE TO YOUNG STUDENTS ||

WITTEN : In the last 20 years , not only has this interaction of math and physics continued to be very rich but it has developed in such diversity that very frequently exciting things are done which I myself am able to understand embarrassingly little about , because the field is expanding in so many directions .

I am sure that this is going to continue and I believe the reason it will continue is that quantum field theory and string theory , I believe , somehow have rich mathematical secrets .

When some of these secrets come to the surface , they often come as surprises to physicists because we do not really understand string theory properly as physics — we do not understand the core ideas behind it .

At an even more basic level , the mathematicians are still not able to fully come to grips with quantum field theory and therefore things coming from it are surprises .

So for both of those reasons , I think that the physics and math ideas generated are going to be surprising for a long time .

CONTINUE TO 17 PAGE INTERVIEW PDF ➤➤

CONTINUE TO FULL ISSUE PDF ➤➤

PRINCETON UP | THE DIGITAL EINSTEIN PAPERS

|| THE COLLECTED PAPERS OF ALBERT EINSTEIN ||

Princeton University Press proudly presents The Digital Einstein Papers , an open-access site for The Collected Papers of Albert Einstein , the ongoing publication of Einstein's massive written legacy comprising more than 30000 unique documents .

The site presents all 13 volumes published to date by the editors of the Einstein Papers Project , covering the writings and correspondence of Albert Einstein from youth to 1923 .

The volumes are presented in the original language version with in-depth English language annotation and other scholarly apparatus . In addition , the reader can toggle to an English language translation of most documents .

By clicking on the unique archival identifier number below each text , readers can access the archival record of each published document at the Einstein Archives Online and in some cases , the digitized manuscript . Approximately 7000 pages representing 2900 unique documents have been digitized thus far .

The site will present subsequent volumes in the series roughly two years after original book publication .

CALTECH | EINSTEIN PAPERS PROJECT

|| THE COLLECTED PAPERS OF ALBERT EINSTEIN ||

The Einstein Papers Project is engaged in one of the most ambitious scholarly publishing ventures undertaken in the history of science .

The Collected Papers of Albert Einstein provides the first complete picture of Einstein's massive written legacy .

HUJ & CALTECH | EINSTEIN ARCHIVES ONLINE

|| EINSTEIN ARCHIVES ONLINE WEBSITE ||

The Einstein Archives Online Website provides the first online access to Albert Einstein's scientific and non-scientific manuscripts held by the Albert Einstein Archives at the Hebrew University of Jerusalem , constituting the material record of one of the most influential intellects in the modern era .

It also enables access to the Einstein Archive Database , a comprehensive source of information on all items in the Albert Einstein Archives .

SYMMETRYMAGAZINE.ORG | BREAKING | 12.05.2014

|| EINSTEIN PAPERS GO DIGITAL ||

More than 5000 documents collected by the Einstein Papers Project are now freely available online .

Today a new window opened into this early period of Einstein's life .

Princeton University Press , working with The Einstein Papers Project hosted at Caltech , has made freely available online more than 5000 documents from Einstein's first 44 years .

This is only the first installment . Princeton University Press and the Einstein Papers Project plan to continue the project , adding new documents from their collection of about 30000 .

NYTIMES | SCIENCE | 12.04.2014

|| 1000S OF EINSTEIN DOCUMENTS ARE NOW A CLICK AWAY ||

They have been called the Dead Sea Scrolls of Physics .

Since 1986 , the Princeton University Press and the Hebrew University of Jerusalem , to whom Albert Einstein bequeathed copyright , have been engaged in a mammoth effort to study some 80000 documents left behind .

]]>ESO | SCIENCE RELEASE 1438 + TECH PDF + SIMS | 11.19.2014

|| ALIGNMENT OF QUASARS ACROSS BILLIONS OF LIGHT-YEARS ||

VLT Reveals Alignments Between Supermassive Blackhole Axes & Large-Scale Structure

New observations with ESO's Very Large Telescope ( VLT ) in Chile have revealed alignments over the largest structures ever discovered in the Universe .

A European research team has found that the rotation axes of the central supermassive black holes in a sample of quasars are parallel to each other over distances of billions of light-years .

The team has also found that the rotation axes of these quasars tend to be aligned with the vast structures in the cosmic web in which they reside .

ASTRONOMY & ASTROPHYSICS | VOL 572 | DEC 2014 | TECH

|| ALIGNMENT OF QUASAR POLARIZATIONS WITH LARGE-SCALE STRUCTURES ||

|| ABSTRACT || We have measured the optical linear polarization of quasars belonging to Gpc-scale quasar groups at redshift z ∼ 1.3 .

Out of 93 quasars observed , 19 are significantly polarized . We found that quasar polarization vectors are either parallel or perpendicular to the directions of the large-scale structures to which they belong .

Statistical tests indicate that the probability that this effect can be attributed to randomly oriented polarization vectors is of the order of 1% .

We also found that quasars with polarization perpendicular to the host structure preferentially have large emission line widths while objects with polarization parallel to the host structure preferentially have small emission line widths .

Considering that quasar polarization is usually either parallel or perpendicular to the accretion disk axis depending on the inclination with respect to the line of sight , and that broader emission lines originate from quasars seen at higher inclinations , we conclude that quasar spin axes are likely parallel to their host large-scale structures .

FULL TECH HTML ( A&A ) ➤➤ | PDF ( ESO | ARXIV ) ➤➤

VIDEO SIMULATION ( ESO ) ➤➤ | IMAGE SIMULATION ➤➤

< GODLANG . CEVBAN . ABSOLUTE >

< Language . English . Evolve . Activate >

THE SCRIPTURES OF SATYA सत्य | SATYAVEDISM

THE SCRIPTURES | VOL 03 | SSSV 56 | SATYA सत्य

ओम्

OM!

OM! GOD ALONE EXISTS . GOD ALONE IS REALITY . || 1 ||

OM! GOD IS BEYOND THE WHOLE MANIFESTATION OF ITSELF WITHIN ITSELF — for only That Which Knows ITSELF as both TRANSCENDENT & IMMANENT simultaneously is THE ONE & ONLY WHOLE TRUTH THAT IS GOD . || 2 ||

BEHOLD THE HOLY REVELATION OF OMNIMATHICS ! || 3 ||

The Eternal Mathematical Expression of GOD that IS

< ! GOD MANIFESTATION ! > within < ! GOD ABSOLUTE ! > IS

thus , so named : OMNIMATHICS : GOD IS OMniMathent . || 4 ||

The Eternal OMniPresent OMniScient OMniPotent < ! GOD CONSCIOUSNESS ! > BLAZES IN ITS MAGNIFICENT INFINITELY INTELLIGENT & POWERFULL < ! GOD OMNIMATHENCE ! > . || 5 ||

The simultaneous order and spontaneity of the phenomenal working , presenting the appearance of The Structure of < ! GOD MANIFESTATION ! > ( THE OMNIVERSE of Infinite MultiVerses ) , is inherently OMNIMATHICAL ; IT is Encoded OMNIMATHICALLY . || 6 ||

< ! GODING ! > IS ETERNALLY < ! GOD OMNIMATHING ! > . || 7 ||

What appears to " Ignorance " to be shapes and forms within shapes and forms is the Eternal processing , transmuting , morphing of GOD THE ONE BEING MANIFESTING , that cannot be reduced to locality IN TRUTH . || 8 ||

What appears to " Ignorance " to be local , isolated , particular , or separate in " Fiction " , is OMNIPRESENTLY KNOWN TO THE WHOLE AS THE WHOLE ONLY — IN TRUTH . || 9 ||

< ! GOD MANIFESTATION ! > is InFinite & Eternal InDestructible OMNIMATHICAL TRUTH . || 10 ||

OM! SCIENCE IS WORSHIP OF THE MANIFESTATION ( OMNIVERSE ) OF GOD THAT IS GOD WITHIN GOD THE ABSOLUTE . || 11 ||

The Cardinal Tenets of SATYAVEDISM < : TCTOSV : > declare

< GITFO > THE OMNIPHYSICS ( The Sciences of Mathematics , Physics , Chemistry , Biology , ETC — all that exist in the entire OMniVerse of MultiVerses ) < OM > to be :

OM! WORSHIP OF GOD ; and Core to The Integral Methods of < ! GOD REALIZATION ! > . < : SSSV 20.2+4 | TCTOSV : > || 12 ||

OM! The Divine Science & Language of The MATHEMATICS is The Central Science & Language of The OMNIPHYSICS . || 13 ||

< GITHO > The Grand Unification of The Divine Science & Language of Mathematics — of its evolving and emergent " sublanguages " < OM > — is continually occurring . || 14 ||

Unlike the " Fiction " of The " Species-Centric " Languages of Relativity , The Divine Science & Language of Mathematics directly divines and intuits THE OMNIMATHIC ABSOLUTE REALITY — transcending that which is " detectable " by seemingly " limited species senses " and " limited species technologies " . || 15 ||

The Sacred Mathematics directly divines , describes , translates aspects of < ! GOD OMNIMATHENCE ! > that MUST NOT be discarded , mistakenly by " Ignorance " , as unrepresentative of what is " known " or " expected unknown " in " Relativity " . || 16 ||

As The Holy Mathematics Evolves — not through " creation " or " invention " which is " Fiction " , but through direct revelation of OMNIMATHICAL TRUTH — The Divine Sciences of The OMNIPHYSICS Evolve to Realize THE GRAND RESPLENDENT MAGNIFICENCE OF THE ETERNAL OMNIVERSE of MultiVerses that is GOD within GOD THE ETERNAL ABSOLUTE WHOLE . || 17 ||

OM! GOD-CENTRISM IS TRUTH OF THE WHOLE .

OM! SPECIES-CENTRISM IS FICTION & UNTRUTH .

OM! ALL THOUGHT MUST BE OF GOD & GOD ALONE .

OM! GOD ALONE EXISTS . GOD ALONE IS REALITY . || 18 ||

OM! There is no True Understanding of :

< ! GOD MANIFESTATION ! > and < ! GOD OMNIMATHENCE ! >

without True Understanding of :

< GITFO > The Divine & Holy Sciences of THE OMNIPHYSICS < OM > ; and no true understanding of THE OMNIPHYSICS without true understanding of its < GITFO > Central Divine Science & Language of THE MATHEMATICS < OM > . || 19 ||

OM! UNLIMITED INTELLIGENCE & POWER YOU ARE !

OM! GOD KNOWS ITSELF AS ITSELF GOD AS GOD ! || 20 ||

OM! GOD ALONE EXISTS . GOD ALONE IS REALITY .

I AM GOD ! I AM GOD ! I AM GOD ! OM ! OM ! OM !

SATYA सत्य | SATYAVEDISM

NOTE : < GITFO > = GOD IN THE FORM OF . . . ( noun | phrase )

NOTE : < GITHO > = GOD IN THE HAPPENING OF . . . ( phrase )

< Language . English . Evolve . Propagate >

< GODLANG . CEVBAN . ABSOLUTE >

KNOTS & QUANTUM THEORY | EDWARD WITTEN

IAS | INSTITUTE FOR ADVANCED STUDY | 2011

A knot is more or less what you think it is — a tangled mess of string in ordinary three-dimensional space .

In the twentieth century , mathematicians developed a rich and deep theory of knots .

Surprisingly , as Edward Witten , Charles Simonyi Professor in the School of Natural Sciences , explains in this lecture , it turned out that many of the most interesting ideas about knots have their roots in quantum physics .