Thank you, Prof. Borcherds! It's wonderful how you concentrate on the interesting points and leave the boring details aside. An ideal math textbook would have the odd-numbered chapters written in such a style for motivation and overview and use the even-numbered chapters to clean up the technical details. Unfortunately, authors don't bother (or dare?) to do this.
For the exercise of finding number of norm 2 vectors in a 32 dimensional (self-dual, even) lattice with no norm sqrt(2) vector, I get 146880. Do you guys also get the same number?
If I may suggest a future topic, could you talk you have done on automorphic forms, Jacobi modular forms etcetera? It is just some suggestion or vague idea. For example, work from: Automorphic forms on Os+2,2(R) and infinite products.
Sorry I don't get the point why I should expect that the coefficients of a Theta series give me the number of certain vectors in a lattice. Could someone explain.
About the definition of Fourier transform. I agree that the definition you use is cleaner BUT all variables have periodicity 1, this is not what a physicist wants. It is better and clearer having x with periodicity 2\pi R or L also because it is dimensional. Dimensionality helps in checking the expressions and guess results, think of Fermi guesstimation. It is obviously possible to work with adimensional quantities and periodicity 1 but you loose a way of checking your results.
Amazing as always! Just a minor mistake: if M = R^n/L the eigenvalues of its Laplacian are the norms of elements of L' the dual of L. This is obviously irrelevant here as you consider only selfdual lattices. It would be very nice if you continue lecturing on the subject by saying a few things on the Niemeier lattices!
8 and 24 also come up in string theory. Bosonic strings live in 24+2 dimensions and superstrings live in 8+2 dimensions. (Where the world sheet of a string is 2 dimensional).
The 16 dimensional lattices also come up in string theory as the compactification geometry for the small dimensions of the bosonic half of the heterotic string. There are two different such lattices, which give rise to the E8xE8 and the SO(32) gauge groups from the bosonic half-strings twisting in the internal geometry.
Null vectors or light rays are self dual! Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference). Homo is dual to hetero, same is dual to different. The council of Nicaea in 325 AD was set up to answer the question is the Christ consciousness the same substance as God or a different substance? Christianity is based upon duality and not triality or trinity. Colours in physics are different frequencies of the same substance namely pure energy -- colours are therefore dual. Null vectors or light rays are perpendicular or dual to themselves from our perspective -- self duality, electro-magnetic energy is dual, electro is dual to magnetic. Yin is dual to yang. You can use this to create the 4th law of thermodynamics:- "Entropy is a measure of randomness" -- Roger Penrose. Syntropy is a measure of order. Randomness (entropy, uncertainty) is dual to order (syntropy, certainty). Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. From a converging, convex (lens) or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics. All observers have a syntropic perspective according to the 2nd law of thermodynamics. My syntropy is your entropy and your syntropy is my entropy -- duality! Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided lone. There are patterns of duality hardwired into physics, mathematics & philosophy. Schrodinger's cat is based upon Hegelian philosophy or metaphysics but the books do not tell you that. Bosons (symmetric wave functions, waves) are dual to Fermions (anti-symmetric wave functions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. You and your mind are built form atoms hence duality. Energy is duality, duality is energy. Energy is dual to mass -- Einstein. Dark matter is dual to dark energy. That is the good news. Good news is dual to bad news. The bad news is that main stream physics is currently dominated by materialists or teleophobia. Teleophilia is dual to teleophobia.
@@edcify8241 Your mind creates or synthesizes reality -- a syntropic process, teleological. Thesis is dual to anti-thesis creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Hence there is a 4th law of thermodynamics! Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. All numbers fall within the complex plane -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Mathematicians create new concepts or ideas all the time from their perceptions or measurements. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Your mind converts average information (entropy) into mutual or co-information (syntropy) -- information is dual. Analog is dual to digital -- information or data. AC (waves) is dual to DC -- signals or currents of information. Signals are dual to noise -- the signal to noise ratio in electronics. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or vectors are dual. "Always two there are" -- Yoda.
This is my favourite video series. I'm looking forward to every episode!
Thank you, Prof. Borcherds! It's wonderful how you concentrate on the interesting points and leave the boring details aside. An ideal math textbook would have the odd-numbered chapters written in such a style for motivation and overview and use the even-numbered chapters to clean up the technical details. Unfortunately, authors don't bother (or dare?) to do this.
the application to drum shapes was totally unexpected
For the exercise of finding number of norm 2 vectors in a 32 dimensional (self-dual, even) lattice with no norm sqrt(2) vector, I get 146880. Do you guys also get the same number?
What is your definition of the E_n lattices in general? Is the E_16 lattice the root lattice of D_16=so(32)?
Vous ici ? Quelle coïncidence ! ;)
@@MathsEtoile et réciproquement!
If I may suggest a future topic, could you talk you have done on automorphic forms, Jacobi modular forms etcetera? It is just some suggestion or vague idea.
For example, work from:
Automorphic forms on Os+2,2(R) and infinite products.
Sorry I don't get the point why I should expect that the coefficients of a Theta series give me the number of certain vectors in a lattice. Could someone explain.
About the definition of Fourier transform. I agree that the definition you use is cleaner BUT all variables have periodicity 1, this is not what a physicist wants. It is better and clearer having x with periodicity 2\pi R or L also because it is dimensional. Dimensionality helps in checking the expressions and guess results, think of Fermi guesstimation. It is obviously possible to work with adimensional quantities and periodicity 1 but you loose a way of checking your results.
Amazing as always! Just a minor mistake: if M = R^n/L the eigenvalues of its Laplacian are the norms of elements of L' the dual of L. This is obviously irrelevant here as you consider only selfdual lattices. It would be very nice if you continue lecturing on the subject by saying a few things on the Niemeier lattices!
oh yes
8 and 24 also come up in string theory. Bosonic strings live in 24+2 dimensions and superstrings live in 8+2 dimensions. (Where the world sheet of a string is 2 dimensional).
The 16 dimensional lattices also come up in string theory as the compactification geometry for the small dimensions of the bosonic half of the heterotic string. There are two different such lattices, which give rise to the E8xE8 and the SO(32) gauge groups from the bosonic half-strings twisting in the internal geometry.
yeee
196560 vectors, this is begging for an explanation :D
Null vectors or light rays are self dual!
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
The council of Nicaea in 325 AD was set up to answer the question is the Christ consciousness the same substance as God or a different substance?
Christianity is based upon duality and not triality or trinity.
Colours in physics are different frequencies of the same substance namely pure energy -- colours are therefore dual.
Null vectors or light rays are perpendicular or dual to themselves from our perspective -- self duality, electro-magnetic energy is dual, electro is dual to magnetic.
Yin is dual to yang.
You can use this to create the 4th law of thermodynamics:-
"Entropy is a measure of randomness" -- Roger Penrose.
Syntropy is a measure of order.
Randomness (entropy, uncertainty) is dual to order (syntropy, certainty).
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
From a converging, convex (lens) or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics.
All observers have a syntropic perspective according to the 2nd law of thermodynamics.
My syntropy is your entropy and your syntropy is my entropy -- duality!
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided lone.
There are patterns of duality hardwired into physics, mathematics & philosophy.
Schrodinger's cat is based upon Hegelian philosophy or metaphysics but the books do not tell you that.
Bosons (symmetric wave functions, waves) are dual to Fermions (anti-symmetric wave functions, particles) -- wave/particle or quantum duality.
Bosons are dual to Fermions -- atomic duality.
You and your mind are built form atoms hence duality.
Energy is duality, duality is energy.
Energy is dual to mass -- Einstein.
Dark matter is dual to dark energy.
That is the good news.
Good news is dual to bad news.
The bad news is that main stream physics is currently dominated by materialists or teleophobia.
Teleophilia is dual to teleophobia.
Is this some sort of copy paste?
@@edcify8241 Your mind creates or synthesizes reality -- a syntropic process, teleological.
Thesis is dual to anti-thesis creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic.
Hence there is a 4th law of thermodynamics!
Syntax is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
All numbers fall within the complex plane -- complex numbers are dual.
All numbers are dual.
The integers are self dual as they are their own conjugates.
Mathematicians create new concepts or ideas all the time from their perceptions or measurements.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Your mind converts average information (entropy) into mutual or co-information (syntropy) -- information is dual.
Analog is dual to digital -- information or data.
AC (waves) is dual to DC -- signals or currents of information.
Signals are dual to noise -- the signal to noise ratio in electronics.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or vectors are dual.
"Always two there are" -- Yoda.