“Quantum field theory is a very rich subject for mathematics as well as physics. But its development in the last seventy years has been mainly by physicists, and it is still largely out of reach as a rigorous mathematical theory despite important efforts in constructive field theory. So most of its impact on mathematics has not yet been felt. Yet in many active areas of mathematics, problems are studied that actually have their most natural setting in quantum field theory. Examples include Donaldson theory of four-manifolds, the Jones polynomial of knots and its generalizations, mirror symmetry of complex manifolds, elliptic cohomology, and many aspects of the study of affine Lie algebras.
To a certain extent these problems are studied piecemeal, with difficulty in understanding the relations among them, because their natural home in quantum field theory is not now part of the mathematical theory. To make a rough analogy (Figure1), one has here a vast mountain range, most of which is still covered with fog. Only the loftiest peaks, which reach above the clouds, are seen in the mathematical theories of today, and these splendid peaks are studied in isolation, because above the clouds they are isolated from one another. Still lost in the mist is the body of the range, with its quantum field theory bedrock and the great bulk of the mathematical treasures.”
By Lukáš Lánský If you want to help me create some nicer or more accurate version of this image, see Python code that generates it. I welcome pull requests. – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=19134117
Dear Sangha Members and Friends of the Padmasambhava Buddhist Center,
I have the great privilege to share with you one of the most beautiful announcements that I can make: the longtime wishes and prayers of all the PBC members and devoted disciples of Ven. Khenchen Palden Sherab Rinpoche have been answered.
Recently, His Eminence Terton Namkha Drimed Rinpoche discovered the reincarnation of Ven. Khenchen Palden Sherab Rinpoche on the eighth day of this month, just two days ago. He was given the name Palden Yonten Thaye Lodro Chokyi Gyaltsen!
At Padma Samye Ling, the North American Monastery of PBC, we enjoyed a great celebration for the 10th Day of Guru Rinpoche, and all the gathered Sangha rejoiced in this announcement with tremendous joy and excitement.
Now that we have discovered the reincarnation of Ven. Khenchen Palden Sherab Rinpoche, H.E. Terton Namkha Drimed Rinpoche is requesting that everyone recite many Dusum Sangye prayers of Guru Padmasambhava. This is my wish as well. We pray for the continued success of all the Dharma activities of Khenchen Yangsi Rinpoche, that there are no obstacles and everything unfolds very smoothly, for the fulfillment of all of his wishes, and that he will benefit the Teachings and all sentient beings for aeons.
Yours in the Dharma,
Khenpo Tsewang Dongyal
Padma Samye Ling
10th Day of Guru Rinpoche
April 16, 2016
Ramanujan’s manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. The equation expressing the near counter examples to Fermat’s last theorem appears further up: α3 + β3 = γ3 + (-1)n. Image courtesy Trinity College library.
The animations above illustrate the typical four-dimensional structure of gluon-field configurations averaged over in describing the vacuum properties of QCD [ Quantum Chromodynamics ]. The volume of the box is 2.4 by 2.4 by 3.6 fm, big enough to hold a couple of protons. Contrary to the concept of an empty vacuum, QCD induces chromo-electric and chromo-magnetic fields throughout space-time in its lowest energy state. After a few sweeps of smoothing the gluon field (50 sweeps of APE smearing), a lumpy structure reminiscent of a lava lamp is revealed. This is the QCD Lava Lamp. The action density (left) and the topological charge density (right) are displayed. The former is similar to an energy density while the latter is a measure of the winding of the gluon field lines in the QCD vacuum.
Doctor Lloyd Bannerman [ 1915 – 2014 ]. He was the first to teach me what it meant to “grasp the essence” or what is known in the Japanese Zen tradition as Kenshō [ (見性) Ken meaning “seeing” and shō meaning “nature or essence”].
Dark energy density
Baryon mass per photon ρb / nγ
Cold dark matter mass per photon ρc / nγ
Neutrino mass per photon ρν / nγ = 3⁄11 Σ mνi
Scalar fluctuation amplitude δH on horizon
Scalar spectral index
If reality has finite information content, space has finite fidelity. The quantum wave function that encodes spatial relationships maybe limited to information that can be transmitted in a “Planck broadcast”, with a bandwidth given by the inverse of the Planck time, about 2×1043 bits per second. Such a quantum system can resemble classical space-time on large scales, but locality emerges only gradually and imperfectly. Massive bodies are never perfectly at rest, but very slightly and slowly fluctuate in transverse position, with aspectrum of variation given by the Planck time. This distinctive new kind of noise associated with quantum geometry would not have been noticed up to now, but may be detectable in a new kind of experiment.