I thank DARPA and the National Science Foundation for supporting some of my research described in this book. The book was completed while I was a Miller Professor at the Miller Institute for Basic Research in Science at UC Berkeley.
I thank my editor T.J. Kelleher and project editor Melissa Veronesi at Basic Books for their expert guidance.
While working on the book, I have benefitted from fruitful discussions with Sara Bershtel, Robert Brazell, David Eisenbud, Marc Gerald, Masako King, Susan Rabiner, Sasha Raskin, Philibert Schogt, Margit Schwab, Eric Weinstein, and David Yezzi.
I thank Alex Freedland, Ben Glass, Claude Levesque, Kayvan Mashayekh, and Corinne Trang for reading parts of the book at various stages and offering helpful advice. I am grateful to Andrea Young for taking the photos of the “cup trick” used in Chapter 15.
Special thanks are due to Thomas Farber for numerous insights and expert advice, and to Marie Levek for reading the manuscript and asking probing questions that helped me improve the presentation in many places. My father, Vladimir Frenkel, read the book’s many drafts and his feedback was invaluable.
My debts to my teachers, mentors, and others who helped me on my journey are, I hope, clear from the story I’ve told.
Above all, my gratitude is for my parents, Lidia and Vladimir Frenkel, whose love and support made possible all I have achieved. I dedicate this book to them.
Notes
Preface
1. Edward Frenkel, Don’t Let Economists and Politicians Hack Your Math, Slate, February 8, 2013, http://slate.me/128ygaM
Chapter 1. A Mysterious Beast
1. Image credit: Physics World, http://www.hk-phy.org/index2.html
2. Images credit: Arpad Horvath.
Chapter 2. The Essence of Symmetry
1. In this discussion we use the expression “symmetry of an object” as the term for a particular transformation preserving an object, such as a rotation of a table. We do not say “symmetry of an object” to express the property of an object to be symmetrical.
2. If we use clockwise rotation, we get the same set of rotations: clockwise rotation by 90 degrees is the same as the counterclockwise rotation by 270 degrees, etc. As a matter of convention, mathematicians usually consider counterclockwise rotations, but this is just a matter of choice.
3. This may seem superfluous, but we are not just being pedantic here. We must include it, if we are to be consistent. We said that a symmetry is any transformation that preserves our object, and the identity is such a transformation.
To avoid confusion, I want to stress that in this discussion we only care about the end result of a given symmetry. What we do to the object in the process does not matter; only the final positions of all points of the object do. For example, if we rotate the table by 360 degrees, then every point of the table ends up in the same position as it was initially. That’s why for us rotation by 360 degrees is the same symmetry as no rotation at all. For the same reason, rotation by 90 degrees counterclockwise is the same as rotation by 270 degrees clockwise. As another example, suppose that we slide the table on the floor ten feet in a certain direction and then slide it back ten feet, or that we move the table to another room and then bring it back. As long as it ends up in the same position and each of its points ends up in the same position as it was initially, this is considered the same symmetry as the identical symmetry.
4. There is an important property that the composition of symmetries satisfies called associativity: given three symmetries, S,S′, and S″, taking their composition in two different orders, and , gives the same result. This property is included in the formal definition of a group as an additional axiom. We do not mention it in the main body of the book because for the groups we consider, this property is obviously satisfied.
5. When we talked about the symmetries of a square table, we found it convenient to identify the four symmetries with the four corners of the table. However, such an identification depends on the choice of one of the corners – the one that represents the identity symmetry. Once this choice is made, we can indeed identify each symmetry with the corner to which the chosen corner is transformed by this symmetry. The drawback is that if we choose a different corner to represent the identity symmetry, we obtain a different identification. Hence it is better to make a distinction between the symmetries of the table and the points of the table.
6. See Sean Carroll, The Particle at the End of the Universe: How the Hunt for the Higgs Boson Leads Us to the Edge of a New World, Dutton, 2012.
7. Mathematician Felix Klein used the idea that shapes are determined by their symmetry properties as the point of departure for his highly influential Erlangen Program in 1872, in which he declared that salient features of any geometry are determined by a symmetry group. For example, in Euclidean geometry, the symmetry group consists of all transformations of the Euclidean space that preserve distances. These transformations are compositions of rotations and translations. Non-Euclidean geometries correspond to other symmetry groups. This allows us to classify possible geometries by classifying relevant symmetry groups.
8. This is not to say that no aspects of a mathematical statement are subject to interpretation; for instance, questions like how important a given statement is, how widely applicable, how consequential for the development of mathematics, and so on, could be subject to a debate. But the meaning of the statement – what exactly it says – is not open to interpretation if the statement is logically consistent. (The logical consistency of the statement is not subject to a debate, either, once we choose the system of axioms within which the statement is made.)
9. Note that each rotation also gives rise to a symmetry of any round object, such as a round table. Therefore, in principle, one could also speak of a representation of the group of rotations by symmetries of the round table rather than a plane. However, in mathematics the term “representation” is reserved specifically for the situation in which a given group gives rise to symmetries of an n-dimensional space. These symmetries are required to be what mathematicians call linear transformations, a concept explained in endnote 2 to Chapter 14.
10. For any element g of the group of rotations, denote the corresponding symmetry of the n-dimensional space by Sg. It has to be a linear transformation for any g, and the following properties must be satisfied: first, for any pair of elements of the group, g and h, the symmetry Sg·h must be equal to the composition of the symmetries Sg and Sh. And second, the symmetry corresponding to the identity element of the group must be the identity symmetry of the plane.
11. Later on, it was discovered that there are three more quarks, called “charm,” “top,” and “bottom,” and the corresponding anti-quarks.
Chapter 3. The Fifth Problem
1. There was also a small semi-official synagogue in Marina Rosha. The situation improved after perestroika as more synagogues and Jewish community centers opened in Moscow and other cities.
2. Mark Saul, Kerosinka: An episode in the history of Soviet mathematics, Notices of the American Mathematical Society, vol. 46, November 1999, pp. 1217–1220. Available online at http://www.ams.org/notices/199910/fea-saul.pdf
3. George G. Szpiro, Bella Abramovna Subbotovskaya and the “Jewish People’s University,” Notices of the American Mathematical Society, vol. 54, November 2007, pp. 1326–1330. Available online at http://www.ams.org/notices/200710/tx071001326p.pdf
4. Alexander Shen gives a list of some of the problems that were given to Jewish students at the MGU entrance exams in his article Entrance examinations to the Mekh-Mat, Mathematical Intelligencer, vol. 16, No. 4, 1994, pp. 6–10. This article is reprinted in the book M. Shifman (ed.), You Failed Your Math Test, Comrade Einstein, World Scientific, 2005 (available online at http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf). See also other articles about MGU admissions in this book, especially those by I. Vardi and A. Vershik.
Another list of problems is compiled in T. Khovanova and A. Radul, Jewish Problems, available at http://a
rxiv.org/abs/1110.1556
5. George G. Szpiro, ibid.
Chapter 4. Kerosinka
1. Mark Saul, ibid.
Chapter 5. Threads of the Solution
1. The story of the Jewish People’s University and the circumstances of Bella Muchnik Subbotovskaya’s death are recounted in the articles by D.B. Fuchs and others in M. Shifman (ed.), You Failed Your Math Test, Comrade Einstein, World Scientific, 2005.
See also George G. Szpiro, ibid.
2. If we put the identity braid on top of another braid and remove the middle plates, we will get back the original braid after shortening the threads. This means that the result of the addition of a braid b and the identity braid is the same braid b.
3. Here is what the addition of a braid and its mirror image looks like:
Now, in the braid shown on the right-hand side of the above picture, we pull to the right the thread starting and ending at the right-most “nail.” This gives us the left braid below. Then we do the same with the thread starting and ending at the third nail on this braid. This gives us the right braid below.
Next, we pull to the left the thread starting and ending on the second nail. In the resulting braid, there is a seeming overlap between the first and the second threads. But this is an illusion: by pulling the second thread to the right, we eliminate this overlap. These moves are shown on the next picture. The resulting braid, on the right-hand side of the picture below, is nothing but the identity braid that we saw above. More precisely, to get the identity braid, we need to straighten the threads, but this is allowed by our rules (we should also shorten the threads, so that our braid has the same height as the original braids). Note that at no step did we cut or sew the threads or allow one to go through the other.
4. This is a good opportunity to discuss the difference between “definition” and “theorem.” In Chapter 2, we gave the definition of a group. Namely, a group is a set endowed with an operation (variably called composition, addition, or multiplication, depending on the circumstances) that satisfies the following properties (or axioms): there is an identity element in the set (in the sense explained in Chapter 2); every element of the set has an inverse; and the operation satisfies the associativity property described in endnote 4 to Chapter 2. Once we have given this definition, the notion of a group is fixed once and for all. We are not allowed to make any changes to it.
Now, given a set, we can try to endow it with the structure of a group. This means constructing an operation on this set and proving that this operation satisfies all properties listed above. In this chapter, we take the set of all braids with n threads (we identify the braids that are obtained by tweaking the threads, as explained in the main text), and we construct the operation of addition of any two such braids by the rule described in the main text. Our theorem is then the statement that this operation satisfies all of the above properties. The proof of this theorem consists of a direct verification of these properties. We have checked the first two properties (see endnotes 2 and 3 above, respectively), and the last property (associativity) follows automatically from the construction of addition of two braids.
5. Because one of our rules is that a thread is not allowed to get entangled with itself, the sole thread we have has no place to go but straight down from the only nail on the top plate to the one on the bottom plate. Of course, it could go along a complicated path, like a winding mountain road or a meandering street, but by shortening it, if needed, we can make the thread go down vertically. In other words, the group B1 consists of only one element, which is the identity (it is also its own inverse and the result of addition with itself).
6. In mathematical jargon, we say “the braid group B2 is isomorphic to the group of integers.” This means that there is a one-to-one correspondence between the two groups – namely, we assign to each braid the number of overlaps – so that the addition of braids (in the sense described above) corresponds to the usual addition of integers. Indeed, putting two braids on top of each other, we are getting a new braid in which the number of overlaps is equal to the sum of these numbers assigned to the two original braids. Furthermore, the identity braid, in which no overlapping of braids occurs, corresponds to the integer 0, and taking the inverse braid corresponds to taking the negative of an integer.
7. See David Garber, Braid group cryptography, in Braids: Introductory Lectures on Braids, Configurations and Their Applications, eds. A. Jon Berrick, e.a., pp. 329–403, World Scientific 2010; available at http://arxiv.org/pdf/0711.3941v2.pdf
8. See, for example, Graham P. Collins, Computing with Quantum Knots, Scientific American, April 2006, pp. 57–63.
9. De Witt Sumners, Claus Ernst, Sylvia J. Spengler, and Nicholas R. Cozzarelli, Analysis of the mechanism of DNA recombination using tangles, Quarterly Reviews of Biophysics, vol. 28, August 1995, pp. 253–313.
Mariel Vazquez and De Witt Sumners, Tangle analysis of Gin recombination, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, 2004, pp. 565–582.
10. A more precise statement, which we will discuss in Chapter 9, is that the braid group Bn is the fundamental group of the space of n distinct unordered points on the plane. Here is a useful interpretation of the collections of n distinct unordered points on the plane in terms of polynomials of degree n. Consider a monic quadratic polynomial x2 + a1 x + a0, where a0 and a1 are complex numbers (“monic” means here that the coefficient in front of the term with the highest power of x, that is, x2, is equal to 1). It has two roots, which are complex numbers, and conversely, these roots uniquely determine a monic quadratic polynomial. Complex numbers may be represented as points on the plane (see Chapter 9), so a monic quadratic polynomial with two distinct roots is the same as a pair of distinct points on the plane.
Likewise, a monic polynomial of degree n, xn + an−1xn−1 + ... + a1x + a0, with n distinct complex roots is the same as a collection of n distinct points on the plane – its roots. Let’s fix one such polynomial: (x − 1)(x − 2)...(x − n), with the roots 1,2,3,...,n. A path in the space of all such polynomials, starting and ending at the polynomial (x − 1)(x − 2)... (x − n), may be visualized as a braid with n threads, each thread being the trajectory of a particular root. Hence we find that the braid group Bn is the fundamental group of the space of polynomials of degree n with distinct roots (see Chapter 14).
11. To each overlap between two threads, we assign +1 if the thread coming down from the left goes under the thread coming down from the right; we assign −1 if the opposite is true. Consider, for example, this braid:
When we sum up these numbers (+1 and −1) over all pairwise overlaps, we obtain the total overlap number of a given braid. If we tweak the threads, we will always add or eliminate the same number of +1 overlaps as the number of −1 overlaps, so the total overlap number will stay the same. This means that the total overlap number is well-defined: it does not change when we tweak the braid.
12. Note that the total overlap number of the braid obtained by the addition of two braids will be equal to the sum of the total overlap numbers of those two braids. Therefore, the addition of two braids having total overlap numbers 0 will again be a braid with the total overlap number 0. The commutator subgroup B′n consists of all such braids. In a certain precise sense, it is the maximal non-abelian part of the braid group Bn.
13. The concept of Betti numbers originated in topology, the mathematical study of the salient properties of geometric shapes. The Betti numbers of a given geometric shape, such as a circle or a sphere, form a sequence of numbers, b0,b1,b2,..., each of which could be either 0 or a natural number. For example, for a flat space, such as a line, a plane, etc., b0 = 1, and all other Betti numbers are equal to 0. In general, b0 is the number of connected components of the geometric shape. For the circle, b0 = 1, b1 = 1, and the rest of the Betti numbers are 0. The fact that b1, the first Betti number, is equal to 1 reflects the presence of a non-trivial one-dimensional piece. For the sphere, b0 = 0, b1 = 0, b2 = 1, and the othe
r Betti numbers are all equal to 0. Here b2 reflects the presence of a non-trivial two-dimensional piece.
The Betti numbers of the braid group Bn are defined as the Betti numbers of the space of monic polynomials of degree n with n distinct roots. The Betti numbers of the commutator subgroup B′n are the Betti numbers of a closely related space. It consists of all monic polynomials of degree n with n distinct roots and with the additional property that their discriminant (the square of the product of the differences between all pairs of roots) takes a fixed non-zero value (for instance, we can say that this value is 1). For example, the discriminant of the polynomial x2 + a1 x + a0 is equal to a21 – 4a0, and there is a similar formula for all n.
It follows from the definition that the discriminant of a polynomial is equal to zero if and only if it has multiple roots. Therefore, the discriminant gives us a map from the space of all monic polynomials of degree n with n distinct roots to the complex plane without the point 0. Thus, we obtain a “fibration” of this space over the complex plane without the origin. The Betti numbers of B′n reflect the topology of any of these fibers (topologically, they are the same), while the Betti numbers of Bn reflect the topology of the entire space. The desire to understand the topology of the fibers was what motivated Varchenko to suggest this problem to me in the first place. For more on the Betti numbers and the related concepts of homology and cohomology, you may consult the following introductory textbooks:
William Fulton, Algebraic Topology: A First Course, Springer, 1995;
Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001.
Chapter 6. Apprentice Mathematician
1. Some have speculated that Fermat may have bluffed when he left that note on the margin. I don’t think so; I think he made an honest mistake. Anyway, we have to be grateful to him – his little note on the margin has definitely had a positive effect on the development of mathematics.
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