Love and Math

by Frenkel, Edward

  3. This citation appears in the book of Duchamp’s notes entitled À l’Infinitif, as quoted in Gerald Holton, Henri Poincaré, Marcel Duchamp and innovation in science and art, Leonardo, vol. 34, 2001, p. 130.

  4. Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, MIT Press, 2013, p. 493.

  5. Gerald Holton, ibid., p. 134.

  6. Charles Darwin, Autobiographies, Penguin Classics, 2002, p. 30.

  7. For more details, see for example, Shing-Tung Yau and Steve Nadis, The Shape of Inner Space, Basic Books, 2010.

  8. It turns out that the dimension of this group is equal to n(n−1)/2. In other words, to describe an element of this group we need n(n−1)/2 independent coordinates (in the case n = 3 we need 3(3−1)/2 = 3 coordinates, as we have seen in the main text).

  9. Mathematically, each loop may be viewed as the image of a particular “map” from the circle to the three-dimensional space, that is, a rule that assigns to each point φ on the circle a point f(φ) in the three-dimensional space. We only consider “smooth” maps. Roughly speaking, this means that the loop does not have any sharp angles or corners, and so it looks like the one shown on the picture in the main text.

  More generally, a map from a manifold S to a manifold M is a rule that assigns to each point s in S a point in M, called the image of s.

  10. See, for example, Brian Greene, The Elegant Universe, Vintage Books, 2003.

  11. More precisely, a loop in SO(3) is a collection {f (φ)} of elements of SO(3), parametrized by the angle φ (which is a coordinate on the circle). Given a second loop, which is a collection {g(φ)}, let’s compose the two rotations, f (φ) º g(φ) for each φ. Then we get a new collection {f (φ) º g(φ)}, which is another loop in SO(3). Thus, for each pair of loops in SO(3) we produce a third loop. This is the rule of multiplication in the loop group. The identity element of the loop group is the loop concentrated at the identity of SO(3), that is, f (φ) is the identity element of SO(3) for all φ. The inverse loop to the loop {f (φ)} is the loop {f (φ)−1}. It is easy to check that all axioms of the group hold. Hence, the loop space of SO(3) is indeed a group.

  12. To see this, let’s consider a simpler example: the loop space of the plane. The plane has two coordinates, x and y. Therefore a loop on the plane is the same as a collection of points on the plane with coordinates x(φ) and y(φ), one for each angle φ between 0 and 360 degrees. (For example, the formulas x(φ) = cos(φ), y(φ) = sin(φ) describe a particular loop: the circle of radius 1 centered at the origin.) Therefore to specify such a loop we need to specify an infinite collection of pairs of numbers, (x(φ), y(φ)), one pair for each angle φ. That’s why the loop space of the plane is infinite-dimensional. For the same reason, the loop space of any finite-dimensional manifold is also infinite-dimensional.

  13. Quoted in R. E. Langer, René Descartes, The American Mathematical Monthly, vol. 44, No. 8, October 1937, p. 508.

  14. The tangent plane is the closest plane to the sphere among all planes passing through this point. It only touches the sphere at this one point, whereas if we move this plane even slightly (so that it still passes through the same fixed point on the sphere), we obtain a plane that intersects the sphere at more points.

  15. By definition, the Lie algebra of a given Lie group is the flat space (such as a line, a plane, and so on) that is the closest to this Lie group among all other flat spaces passing through the point in the Lie group corresponding to the identity.

  16. A general circle does not have a special point. But the circle group does: it is the identity element of this group, which is a special point of the circle. It must be specified to make a circle into a group.

  17. Here is a more precise definition of a vector space:

  Once we choose a coordinate system in an n-dimensional flat space, we identify the points of this space with n-tuples of real numbers, (x1, x2,..., xn), the numbers xi being the coordinates of a point. In particular, there is a special point (0,0,...,0), at which all coordinates are equal to 0. This is the origin.

  Now fix a point (x1,x2,...,xn) in this space. We define a symmetry of our space, which sends any other point (z1,z2,...,zn) to (z1 + x1,z2 + x2,...,zn + xn). Geometrically, we can think of this symmetry as the shift of our n-dimensional space in the direction of the pointed interval connecting the origin and the point (x1,x2,...,xn). This symmetry is called a vector, and it is usually represented by this pointed interval. Let’s denote this vector by 〈x1,x2,...,xn〉. There is a one-to-one correspondence between points of the n-dimensional flat space and vectors. For this reason, the flat space with a fixed coordinate system may be viewed as the space of vectors. Hence, we call it vector space.

  The advantage of thinking in terms of vectors rather than points is that we have two natural operations on vectors. The first is the operation of addition of vectors, which makes a vector space into a group. As explained in Chapter 2, symmetries can be composed, and therefore they form a group. The composition of the shift symmetries described in the previous paragraph gives us the following rule of addition of vectors:

  The identity element in the group of vectors is the vector 〈0,0,...,0〉. The additive inverse of the vector 〈x1,x2,...,xn〉 is the vector 〈−x1,−x2,...,−xn〉.

  The second is the operation of multiplication of vectors by real numbers. The result of multiplication of a vector 〈x1, x2,..., xn〉 by a real number k is the vector 〈kx1, kx2,..., kxn〉.

  Thus, a vector space carries two structures: addition, satisfying the properties of a group, and multiplication by numbers. These structures must satisfy natural properties.

  Now, any tangent space is a vector space, and therefore any Lie algebra is a vector space.

  What is described above is the notion of a vector space over real numbers. Indeed, the coordinates of vectors are real numbers and so we can multiply vectors by real numbers. If we replace real numbers by complex numbers in this description, we obtain the notion of a vector space over complex numbers.

  18. The operation on a Lie algebra is usually denoted by square brackets, so if and denote two vectors in a Lie algebra (which is a vector space, as explained in the previous endnote), then the result of this operation on them is denoted by . It satisfies the following properties: for any number k, and the so-called Jacobi identity:

  19. The cross-product of two vectors in the three-dimensional space, and , is the vector, denoted by , which is perpendicular to the plane containing and , has length equal to the product of the lengths of and and the sine of the angle between them, and such that the triple of vectors , , and × has positive orientation (this may be expressed by the so-called right-hand rule).

  20. For example, the Lie algebra of the Lie group SO(3) is the three-dimensional vector space. Therefore the Lie algebra of the loop group of SO(3) consists of all loops in this three-dimensional space. The cross-product in the three-dimensional space gives a Lie algebra structure on these loops. Thus, given two loops, we produce a third, even though it’s not easy to describe what it is in words.

  21. More precisely, a Kac–Moody algebra is an extension of the Lie algebra of a loop group by a one-dimensional space. For more details, see Victor Kac, Infinite-dimensional Lie Algebras, Third Edition, Cambridge University Press, 1990.

  22. The models with Virasoro algebra symmetry are called conformal field theories, first introduced by the Russian physicists Alexander Belavin, Alexander Polyakov, and Alexander Zamolodchikov in 1984. Their seminal work relied on the results obtained by Feigin and Fuchs, as well as Victor Kac.

  23. The most well-known of these are the Wess–Zumino–Witten models. For more details, see Edward Frenkel and David Ben-Zvi, Vertex Algebras, Second Edition, American Mathematical Society, 2004.

  24. These “quantum fields” have nothing to do with “number fields” or “finite fields” that we discussed in the earlier chapters. This is another example of confusing mathematical terminology, though in other languages there is no confusion: the
French, for example, use the word “champs” for quantum fields and “corps” for number fields and finite fields.

  Chapter 11. Conquering the Summit

  1. Here is a precise construction: suppose we have an element of the loop group of SO(3), which is a collection {g(φ)} of elements of SO(3) parametrized by the angle φ (the coordinate on the circle). On the other hand, an element of the loop space of the sphere is a collection {f (φ)} of points of the sphere parametrized by φ. Given such {g(φ)} and {f (φ)}, we construct another element of the loop space of the sphere as the collection {g(φ)(f(φ))}. This means that we apply the rotation g(φ) to the point f (φ) of the sphere, independently for each φ. Thus, we see that each element of the loop group of SO(3) gives rise to a symmetry of the loop space of the sphere.

  2. A point of a flag manifold is a collection: a line in a fixed n-dimensional space, a plane that contains this line, the three-dimensional space that contains the plane, and so on, up to an (n−1)-dimensional hyperplane containing all of them.

  Contrast this with the projective spaces, which I had studied at first: a point of the projective space is just a line in the n-dimensional space, nothing else.

  In the simplest case, n = 2, our fixed space is two-dimensional, and so the only choice we have is that of a line (there is only one plane, the space itself). Therefore, in this case the flag manifold is the same as the projective space, and it turns out to coincide with the sphere. It is important to note that here we consider lines, planes, and so forth in a complex space (not a real space), and only those that pass through the origin of our fixed n-dimensional space.

  The next example is n = 3, so we have a three-dimensional space. In this case the projective space consists of all lines in this three-dimensional space, but the flag manifold consists of pairs: a line and a plane containing it (there is only one three-dimensional space). Therefore, in this case there is a difference between the projective space and the flag manifold. We can think of the line as the pole of a flag and the plane as the banner of the flag. Hence the name “flag manifold.”

  3. Boris Feigin and Edward Frenkel, A family of representations of affine Lie algebras, Russian Mathematical Surveys, vol. 43, No. 5, 1988, pp. 221–222.

  Chapter 12. Tree of Knowledge

  1. Mark Saul, Kerosinka: An episode in the history of Soviet mathematics, Notices of the American Mathematical Society, vol. 46, November 1999, pp. 1217–1220.

  2. I learned later that Gelfand, who collaborated with heart doctors (for the same reason as Yakov Isaevich collaborated with urologists), also successfully used this approach to medical research.

  Chapter 14. Tying the Sheaves of Wisdom

  1. A precise definition of a vector space was given in endnote 17 to Chapter 10.

  2. In the case of the category of vector spaces, morphisms from a vector space V1 to a vector space V2 are the so-called linear transformations from V1 to V2. These are maps f from V1 to V2 such that for any two vectors and in V1, and for any vector in V1 and number k. In particular, the morphisms from a given vector space V to itself are the linear transformations from V to itself. The symmetry group of V consists of those morphisms that have an inverse.

  3. See, for example, Benjamin C. Pierce, Basic Category Theory for Computer Scientists, MIT Press, 1991.

  Joseph Goguen, A categorical manifesto, Mathematical Structures in Computer Science, vol. 1, 1991, 49–67.

  Steve Awodey, Category Theory, Oxford University Press, 2010.

  4. See, for example, http://www.haskell.org/haskellwiki/Category_theory and references therein.

  5. See, for example, Masaki Kashiwara and Pierre Schapira, Sheaves on Manifolds, Springer-Verlag, 2010.

  6. This surprising property of arithmetic modulo primes has a simple explanation if we view it from the point of view of group theory. Consider the non-zero elements of the finite field: 1,2,..., p−1. They form a group with respect to multiplication. Indeed, the identity element with respect to multiplication is the number 1: if we multiply any element a by 1, we get back a. And each element has an inverse, as explained in endnote 8 to Chapter 8: for any a in {1,2,..., p−1}, there is an element b such that a·b = 1 modulo p.

  This group has p−1 elements. There is a general fact that holds for any finite group G with N elements: the Nth power of each element a of this group is equal to the identity element (which we will denote by 1),

  To prove this, consider the following elements in the group G: 1,a,a2,... Because the group G is finite, these elements cannot all be distinct. There have to be repetitions. Let k be the smallest natural number such that ak is equal to 1 or aj for some j = 1,...,k−1. Suppose that the latter is the case. Let a−1 denote the inverse of a, so that a·a−1 = 1 and take its jth power (a−1)j. Multiply both sides of the equation ak = aj with (a−1)j on the right. Each time we encounter a · a−1 we replace it by 1. Multiplying by 1 does not change the result, so we can always remove 1 from the product. We see then that each a−1 will cancel one of the a’s. Hence, the left-hand side will be equal to ak−j, and the right-hand side will be equal to 1. We obtain that ak−j = 1. But k−j is smaller than k, and this contradicts our choice of k. Therefore, the first repetition on our list will necessarily have the form ak = 1, so that the elements 1,a,a2,...,ak−1 are all distinct. This means that they form a group of k elements: {1,a,a2,...,ak−1}. It is a subgroup of our original group G of N elements, in the sense that it is a subset of elements of G such that the result of multiplication of any two elements of this subset is again an element of the subset, this subset contains the identity element of G, and this subset contains the inverse of each of its elements.

  Now, it is known that the number of elements of any subgroup always divides the number of elements of the group. This statement is called the Lagrange theorem. I’ll leave it for you to prove (or you may just Google it).

  Applying the Lagrange theorem to the subgroup {1,a,a2,...,ak−1}, which has k elements, we find that k must divide N, the number of elements of the group G. Thus, N = km for some natural number m. But since ak = 1, we obtain that

  which is what we wanted to prove.

  Let’s go back to the group {1,2,..., p−1} with respect to multiplication. It has p−1 elements. This is our group G, so our N is equal to p−1. Applying the general result in this case, we find that ap−1 = 1 modulo p for all a in {1,2,..., p−1}. But then

  It is easy to see that the last formula actually holds for any integer a, if we stipulate that

  whenever x − y = rp for some integer r.

  This is the statement of Fermat’s little theorem. Fermat first stated it in a letter to his friend. “I would send you a proof,” he wrote, “but I am afraid it’s too long.”

  7. Up to now, we have considered the arithmetic modulo a prime number p. However, it turns out that there is a statement analogous to Fermat’s little theorem in the arithmetic modulo any natural number n. To explain what it is, I need to recall the Euler function φ, which we discussed in conjunction with braid groups in Chapter 6. (In my braid group project, I had found that the Betti numbers of braid groups are expressed in terms of this function.) I recall that φ(n) is the number of natural numbers between 1 and n −1 that are relatively prime with n; that is, do not have common divisors with n (other than 1). For instance, if n is a prime, then all numbers between 1 and n−1 are relatively prime to n, and so φ(n) = n −1.

  Now, the analogue of the formula ap−1 = 1 modulo p that we proved in the previous endnote is the formula

  It holds for any natural number n and any natural number a that is relatively prime to n. It is proved in exactly the same way as before: we take the set of all natural numbers between 1 and n −1 that are relatively prime to n. There are φ(n) of them. It is easy to see that they form a group with respect to the operation of multiplication. Hence, by the Lagrange theorem, for any element of this group, its φ(n)th power is equal to the identity element.

  Consider, for example, the case that n is the product
of two prime numbers. That is, n = pq, where p and q are two distinct prime numbers. In this case, the numbers that are not relatively prime to n are either divisible by p or by q. The former have the form pi, where i = 1,..., q−1 (there are q −1 of those), and the latter have the form qj, where j = 1,..., p −1 (there are p −1 of those). Hence we find that

  Therefore we have

  for any number a that is not divisible by p and q. And it is easy to see that the formula

  is true for any natural number a and any integer m.

  This equation is the basis of one of the most widely used encryption algorithms, called RSA algorithm (after Ron Rivest, Adi Shamir, and Leonard Adleman, who described it in 1977). The idea is that we pick two primes p and q (there are various algorithms for generating them) and let n be the product pq. Number n is made public, but the primes p and q are not. Next, we pick a number e that is relatively prime to (p −1)(q −1). This number is also made public.

  The encryption process converts any number a (such as a credit card number) to ae modulo n:

  It turns put that there is an efficient way to reconstruct a from ae. Namely, we find a number d between 1 and (p −1)(q −1) such that

  In other words,

  for some natural number m. Then

  according to the formula above.

  Therefore, given b = ae, we can recover the original number a as follows:

  Let’s summarize: we make the numbers n and e public, but keep d secret. The encryption is given by the formula

  Anyone can do it because e and n are publicly available.