Lifeboat Foundation

Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription.

Special thanks to Prof. Asaf Karagila for consultation on set theory and specific rewrites, to Prof. Alex Kontorovich for reviews of earlier drafts, Prof. Toby ‘Qubit’ Cubitt for the help with the spectral gap, to Henry Reich for the helpful feedback and comments on the video.

Continue reading “Math Has a Fatal Flaw” »

Neil deGrasse Tyson explains the early state of our Universe. At the beginning of the universe, ordinary space and time developed out of a primeval state, where all matter and energy of the entire visible universe was contained in a hot, dense point called a gravitational singularity. A billionth the size of a nuclear particle.

While we can not imagine the entirety of the visible universe being a billion times smaller than a nuclear particle, that shouldn’t deter us from wondering about the early state of our universe. However, dealing with such extreme scales is immensely counter-intuitive and our evolved brains and senses have no capacity to grasp the depths of reality in the beginning of cosmic time. Therefore, scientists develop mathematical frameworks to describe the early universe.

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In three towering papers, a team of mathematicians has worked out the details of Liouville quantum field theory, a two-dimensional model of quantum gravity.

Once researchers have done the hard work of translating a set of mathematical concepts into a proof assistant, the program generates a library of computer code that can be built on by other researchers and used to define higher-level mathematical objects. In this way, proof assistants can help to verify mathematical proofs that would otherwise be time-consuming and difficult, perhaps even practically impossible, for a human to check.

Proof assistants have long had their fans, but this is the first time that they have played a major role at the cutting edge of a field, says Kevin Buzzard, a mathematician at Imperial College London who was part of a collaboration that checked Scholze and Clausen’s result. “The big remaining question was: can they handle complex mathematics?” Says Buzzard. “We showed that they can.”

And it all happened much faster than anyone had imagined. Scholze laid out his challenge to proof-assistant experts in December 2020, and it was taken up by a group of volunteers led by Johan Commelin, a mathematician at the University of Freiburg in Germany. On 5 June — less than six months later — Scholze posted on Buzzard’s blog that the main part of the experiment had succeeded. “I find it absolutely insane that interactive proof assistants are now at the level that, within a very reasonable time span, they can formally verify difficult original research,” Scholze wrote.

Physics World

Quantum mechanics describes this frustration by suggesting that the orientation of the spins is not rigid. Instead, it constantly changes direction in a fluid-like way to produce an entangled ensemble of spin-ups and spin-downs. Thanks to this behaviour, a spin liquid will remain in a liquid state even at temperatures near absolute zero, where most materials usually freeze solid.

The holon and the spinon

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Glow is an iconic interesting research about deep neural networks that can generalize with small training sets.

Since the early days of machine learning, artificial intelligence scenarios have faced with two big challenges in order to experience mainstream adoption. First, we have the data efficiency problem that requires machine or deep learning models to be trained using large and accurate datasets which, as we know, are really expensive to build and maintain. Secondly, we have the generalization problem which AI agents face in order to build new knowledge that is different from the training data. Humans, by contrast, are incredibly efficient learning with minimum supervision and rapidly generalizing knowledge from a few data examples.

Generative models are one of the deep learning disciplines that focuses on addressing the two challenges mentioned above. Conceptually, generative models are focused on observing an initial dataset, like a set of pictures, and try to learn how the data was generated. Using more mathematical terms, generative models try to infer all dependencies within very high-dimensional input data, usually specified in the form of a full joint probability distribution. Entire deep learning areas such as speech synthesis or semi-supervised learning are based on generative models. Recently, generative models such as generative adversarial networks(GANs) have become extremely popular within the deep learning community. Recently, OpenAI experimented with a not-very well-known technique called Flow-Based Generative Models in order to improve over existing methods.

Continue reading “This Neural Networkfrom OpenAI can Learn from Small Datasets” »

The accelerating effort to understand the mathematics of quantum field theory will have profound consequences for both math and physics.

A Penn State scientist studying crystal structures has developed a new mathematical formula that may solve a decades-old problem in understanding spacetime, the fabric of the universe proposed in Einstein’s theories of relativity.

“Relativity tells us space and time can mix to form a single entity called spacetime, which is four-dimensional: three space-axes and one time-axis,” said Venkatraman Gopalan, professor of materials science and engineering and physics at Penn State. “However, something about the time-axis sticks out like sore thumb.”

For calculations to work within relativity, scientists must insert a negative sign on time values that they do not have to place on space values. Physicists have learned to work with the negative values, but it means that spacetime cannot be dealt with using traditional Euclidean geometry and instead must be viewed with the more complex hyperbolic geometry.

Quantum computing began in the early 1980s. It operates on principles of quantum physics rather than the limitations of circuits and electricity which is why it is capable of processing highly complex mathematical problems so efficiently. Quantum computing could one day achieve things that classical computing simply cannot. The evolution of quantum computers has been slow, but things are accelerating, thanks to the efforts of academic institutions such as Oxford, MIT, and the University of Waterloo, as well as companies like IBM, Microsoft, Google, and Honeywell.

IBM has held a leadership role in this innovation push and has named optimization as the most likely application for consumers and organizations alike.

Honeywell expects to release what it calls the “world’s most powerful quantum computer” for applications like fraud detection, optimization for trading strategies, security, machine learning, and chemistry and materials science.