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Aaron Hertzmann
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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 Göttingen 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.

In fact she proved two important theorems on this subject, but the easier one is more famous: Leon Lederman said it's "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean 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:

http://johncarlosbaez.wordpress.com/2014/05/03/noethers-theorem-quantum-vs-stochastic/

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 Göttingen.  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:

https://en.wikipedia.org/wiki/Emmy_Noether
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Got a Glass today and biked around SF, shooting pictures and video until its battery died. (Also applied YouTube's video stabilization.) #throughglass  
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I think I am going to throw up!
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Holy cow.
 
Learning to Execute and Neural Turing Machines

I'd like to draw your attention to two papers that have been posted in the last few days from some of my colleagues at Google that I think are pretty interesting and exciting:

  Learning to Execute: http://arxiv.org/abs/1410.4615

  Neural Turing Machines: http://arxiv.org/abs/1410.5401

The first paper, "Learning to Execute", by +Wojciech Zaremba and +Ilya Sutskever attacks the problem of trying to train a neural network to take in a small Python program, one character at a time, and to predict its output.  For example, as input, it might take:

"i=8827
c=(i-5347)
print((c+8704) if 2641<8500 else 5308)"

During training, the model is given that the desired output for this program is "12185".  During inference, though, the model is able to generalize to completely new programs and does a pretty good of learning a simple Python interpreter from examples.


The second paper, "Neural Turing Machines", by +alex graves, Greg Wayne, and +Ivo Danihelka from Google's DeepMind group in London, couples an external memory ("the tape") with a neural network in a way that the whole system, including the memory access, is differentiable from end-to-end.  This allows the system to be trained via gradient descent, and the system is able to learn a number of interesting algorithms, including copying, priority sorting, and associative recall.

Both of these are interesting steps along the way of having systems learn more complex behavior, such as learning entire algorithms, rather than being used for just learning functions.

(Edit: changed link to Learning to Execute paper to point to the top-level Arxiv HTML page, rather than to the PDF).
Abstract: We extend the capabilities of neural networks by coupling them to external memory resources, which they can interact with by attentional processes. The combined system is analogous to a Turing Machine or Von Neumann architecture but is differentiable end-to-end, allowing it to be ...
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Our new paper on recognizing image style: an algorithm for identify the artistic style of a photograph or painting, such as "Vintage," "Noir," "Depth-of-field," "Impressionist," and so on. (With +Sergey Karayev , +Matthew Trentacoste , Helen Han, +Aseem Agarwala , +Holger Winnemoeller , +Trevor Darrell )
Sergey Karayev is mostly a CS grad student.
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Our upcoming SIGGRAPH paper, introducing a better way to pick fonts! Check out the video; there's also a prototype of the interface online. With +Peter O'Donovan , +Jānis Lībeks , +Aseem Agarwala 
Abstract. This paper presents interfaces for exploring large collections of fonts for design tasks. Existing interfaces typically list fonts in a long, alphabetically-sorted menu that can be challenging and frustrating to explore. We instead propose three interfaces for font selection.
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I suppose this video was inevitable. I thought the lyrics were clever: they even slipped in a reference to the Google+ integration with YouTube! 
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Four and a half years in the making! This paper was just accepted to TOG. Michael Kass and I began the project during my sabbatical at Pixar in 2009. Over the next few years, we went through idea after idea that didn't work, with me writing lots of code in-between teaching and other stuff. Each idea flamed out but led to a new, more promising idea. Michael was a great collaborator. +Pierre BENARD took over the project during his post-doc at Toronto, and, through truly heroic efforts over more than a year, finally got great results. Whew! +
Abstract. This paper introduces a method for accurately computing the visible contours of a smooth 3D surface for stylization. This is a surprisingly difficult problem, and previous methods are prone to topological errors, such as gaps in the outline. Our approach is to generate, ...
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The most popular girl's name in the US
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New paper by +Peter O'Donovan, +Aseem Agarwala, and myself on analysis and synthesis of single-page graphic design.
Peter O'Donovan1 Aseem Agarwala 2 Aaron Hertzmann 1,2. 1University of Toronto 2Adobe Systems, Inc. Design Analysis. alignment, importance, segmentation. Alignment Detection, Importance Prediction, Segmentation. Design Synthesis. var1, var2, var3. Synthesized Example 1, Synthesized Example 2 ...
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This is very nice work!
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