Lisa and I visited Bern, where she has a colleague in the university. It's a great place — still sort of medieval in flavor. The name of this city sounds a bit like 'bear'. And indeed, there are statues and pictures of bears all over Bern.
Legend has it that, in 1191, Duke Berthold V of Zähringen vowed to name the city after the first animal he met in the forest that was to be chopped down for his new city. The story goes:
Then they caught a bear first, which is why the city was called Bern; and so the citizens had their coat and shield, which was a black bear in a white shield, going upright.
Others deny this, but everyone seems to agree that in 1513, when the Bernese returned home victorious from the Battle of Novara, they carried home a living bear... and for centuries, the city has kept bears in a pit called the Bärengraben.
This pit has been moved now and then, and in 1994 it was rebuilt to improve conditions for the bears. But keeping bears in a pit still seemed nasty to many people, so in 2000 the city built a special park for bears by the river Aar — the BärenPark — which is connected to the pit by a tunnel. So now the bears can visit the pit if they want, but they don't have to stay there. When I passed the pit, none were there. No surprise!
I'm in Erlangen, where the great German mathematician Emmy Noether was born in 1882. She was the daughter of the well-known mathematician Max Noether - but as a woman, she was only allowed to audit courses at the university here. Somehow she finished a PhD thesis in 1907. She then worked here without pay for 7 years, since women were excluded from academic jobs.
Her thesis advisor, Paul Gordan, specialized in doing complicated calculations to find all the polynomials that were unchanged by certain symmetries. Around this time David Hilbert proved a powerful general theorem that said all these polynomials could be gotten by adding, subtracting and multiplying a finite set of them, called 'generators'. But he didn't say how to find these generators! Gordan said "this is not mathematics; this is theology."
Noether did her thesis, On Complete Systems of Invariants for Ternary Biquadratic Forms, in the style of Gordan's work. It was well received, but she later said it was "crap". While working without pay, she learned Hilbert's ideas and started revolutionizing the subject of algebra.
In 1915 she was invited to the University of Gvttingen by David Hilbert and Felix Klein. Their attempt to recruit her was fought by the philologists and historians, who didn't want a woman on the faculty. Hilbert fought back, saying "After all, we are a university, not a bath house."
It took years for her to actually get paid, but she started working at Göttingen and soon proved the theorem physicists remember her for, relating symmetries and conservation laws. They call it Noether's Theorem.
Her theorem applies to classical mechanics and classical field theory, but there's also a quantum version, and more recently Brendan Fong and I proved a 'stochastic' version, which applies to random processes. The stochastic version is weirdly different from the quantum version, but Ville Bergholm has just written a nice article discussing this issue, and some results he discovered with Jacob Biamonte and Mauro Faccin:
Check it out!
Emmy Noether finally started getting a salary in 1923, sixteen years after finishing her thesis. If anyone asks why there are fewer famous women mathematicians than men, consider pointing this out!
Noether did extraordinary work until 1933, when the Nazis kicked her out of the University of Gvttingen. She wound up in Bryn Mawr College, a women's college near Philadelphia. She died of complications from surgery in 1935.
But here are some of the wonderful things she did:
In 1921 she stated the general definition of 'ring' and 'ideal', and proved that in a ring where every increasing sequence of ideals stops growing after finitely many steps, every ideal has finitely many generators. Such rings are now called Noetherian.
In 1927 she gave a massive generalization of the fundamental theorem of arithmetic, about unique factorization into primes. She characterized commutative rings in which the ideals have unique factorization into prime ideals as the integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields. Sorry - this sounds technical, and it is! But everyone who studies modern number theory takes this result as basic: such rings are now called Dedekind domains, but Noether discovered them.
Even more important than either of these massive results are the beautifully simple 'Noether isomorphism theorems' that everyone learns near the start of a course on group theory.
And perhaps even more important was her discovery of 'homology groups' while attending lectures by the famous topologists Alexandrov and Hopf. Other people would have made a whole career out of this discovery, which utterly revolutionized topology. But she only gave it a tiny mention in one of her works on group theory! She was truly a fountain of new ideas.
I now have an office in the Emmy-Noether-Zentrum für Algebra at the university in Erlangen.
For more, try:
The poet Martial was a kid when Caligula was emperor of Rome. Later he got support from the emperor Domitian. So Martial was an expert on decadence and depravity — and his work shows it.
He's famous for short, snappy, perfectly structured poems with surprise endings. People called them 'epigrams'.
But he's also infamous - because many of those epigrams are rude or even obscene. The Loeb Classical Library edition of his work says:
No account of the work of Martial would be complete without two features being touched upon which have darkened his fame, namely his indecency, and his adulation of Domitian. With regard to the first, however, of the 1171 epigrams in the first twelve books, those open to objection do not exceed a fourth, and if the 350 epigrams in Books XIII and XIV be included, the proportion is still smaller. On the other hand, of the objectionable epigrams the greater part are indescribably foul.Here's one that's not indescribably foul:
Praedia solus habes et solus, Candide, nummos,
aurea solus habes, murrina solus habes,
Massica solus habes et Opimi Caecuba solus,
et cor solus habes, solus et ingenium
omnia solus habes — hoc me puta velle negare! —
uxorem sed habes, Candide, cum populo.
You don't need to know Latin — I sure don't — to appreciate the tight structure. Almost every line has "solus habes" as its 2nd and 3rd words! This means "only you have" — and the poems is about possessiveness, and arrogance.
Here's a decent translation by A. S. Kline:
Only you have land, then, Candidus,A put-down, with a zinger at the end — typical Martial.
Gold plate, cash, and porcelain, only you,
Massic or Caecuban wine of famous vintage,
only you — judgement and wit, only you.
You have it all — well say I don't deny it —
But everyone has your wife, along with you.
Here's another, also translated by Kline:
Chloe, I could live without your face,
without your neck, and hands, and legs
without your breasts, and ass, and hips,
and Chloe, not to labour over details,
I could live without the whole of you.
But now maybe you want to read an "indescribably foul" one, to see how bad they get! Well, you're not getting it here. Try this:
This is how to get people to read poetry.
Personally I find many of Martial's poems annoying... but it's very interesting to see that art designed to shock is not new to the 20th century. Are we, like the Roman Empire of Martial's day (roughly 40-100 AD), a civilization that's become decadent?
On a brighter note, Martial was a jolly fellow, good to his friends, and he spent a lot of time living out in the countryside. This gives the flavor of it:
These, my dearest Martialis, are
the things that bring a happy life:
wealth left to you, not laboured for;
rich land, an ever-glowing hearth;
no law, light business, and a quiet mind;
a healthy body, gentlemanly powers;
a wise simplicity, friends not unlike;
good company, a table without art;
nights carefree, yet no drunkenness;
a bed that's modest, true, and yet not cold;
sleep that makes the hours of darkness brief:
the need to be yourself, and nothing more;
not fearing your last day, not wishing it.
Only the remark that inherited wealth makes for true happiness makes this outdated. Roman society, unlike ours, was openly aristocratic, with social equality not even a goal.
The photo above was taken by someone named Victor Manuel, who put it on WikiCommons. It shows a bronze bust of Martial created by the Spanish artist Juan Cruz Melero (1910-1986).
Over on Google+, Annarita Ruberto clarifies:
The novelty of Marco Valerio Marziale consists in the elimination of mythology, considered false and far-fetched. The aim of his poem is to totally take inspiration from reality.and then:(I studied Latin and Greek for five years before graduating in Physics;)).
With Marziale we have the affirmation of the epigram as a literary device: before him, the epigram, dating back to arcaic Greek age, was essentially a commemorative function and was used to positively remember a thing or a person (and in fact the word "epigram" comes from the greek and means "inscription" from epigraphein "to write on &mdas; inscribe"); but thanks to his work it, while retaining its brevity, deals with new issues such as parody, satire, politics and eroticism.From the stylistic point of view, Marziale opposes the mobility of the epigram both to the epic genre and to Greek tragedy, which through their famous and "heavy" themes kept away from everyday reality. Constant is in fact, in his verses, the literary controversy, often used to defend against those who considered poorly valid (from the artistic point of view) the epigrammatic kind, but also against those who accused him of being aggressive or obscene.
The language he uses is colloquial and everyday. His constant realism, however, allows him to develop a rich language by introducing in literature many terms and phrases that had never before found a place. He is able, finally, to demonstrate great flexibility in alternating elegant and sophisticated phrases with indecent and often vernacular sentences.
© 2014 John Baez
baez@math.removethis.ucr.andthis.edu