Love and Math: The Heart of Hidden Reality

Frenkel's book tackles a difficult challenge, that of writing a mathematics book for a popular audience while still actually writing about mathematics. In my mind Frenkel succeeds. He does so through unbridled passion and the telling of engaging autobiographical stories. Not all readers will be able to understand all of the mathematics on the first try but you will still come away with curiosity and excitement about mathematics. Aside from the popularization of mathematics, I found the book valuable because it describes in comprehensible terms Langlands program, a very exciting ongoing research effort. An understandable overview from an expert in a specialized field is a rare thing.

Anyone who reads Love and Math will be richer for it.

The observation to which this book is a response is the fact that intelligent, educated, and sophisticated readers who take pride in the breadth of their knowledge, reveal with pride that they hate math. Those of us who use advanced mathematics integrally and intimately in our work have all heard this refrain. Why would someone take pride in their ignorance? Why would someone who would never say they hate art take pleasure in affirming their aversion to mathematics, which is doubtless and indisputably one of the most stunning achievements of human intellect?

Frenkel's answer is that students are not taught math properly in school. Students learn how to perform calculations, but are never exposed to the deep principles of symmetry and dynamics that embody the true beauty of mathematics. His response is to show how human an enterprise modern mathematical research can be. I read this book from cover to cover with some pleasure, but I do not think he has made his case. Frenkel tell us early of the insights of Galois and even explains with some effectiveness exactly what an incredible tour de force Galois theory is. Moreover, Galois proved something with his detour through group theory that is truly remarkable: there are no general solutions to polynomial equations of degree greater than four using the operations of arithmetic operating on integers and their various integral roots. Frenkel explains that the Langlands Program is a sophisticated and ambitious extension of Galois' project, but he fails in two ways. First, he introduces a plethora of mathematical constructs to which he gives a best vague descriptions, and gives no intuition as to why they are related to one another in interesting ways. Second, he gives us no concrete examples of how all of this high-level math solves some real-world problem of the sort that Galois theory addresses. Why all the huffing and puffing? All we are told is that some of the theory Is useful in banking and cryptography. How and why? Who knows?

In fact, probably most people who become mathematicians or use mathematics extensively actually like to solve rote problems. I know I did. I would spend hours and hours solving problems, deriving trigonometric identities, and integrating funny-looking rational functions. Moreover, I treat mathematics as very hard and exhausting---like running laps and lifting weights. The rewards are surely there, but it is simply hard work most of the time. Art and music are fun, not work. Of course, we must learn to appreciate art and music, but once the brain is prepared, it is all fun thereafter. Math is not like that. Math is sweat and pain all the way. Struggling through a three page paper full of symbols is excruciating effort. The reward is surely there at the end---understanding something of ineffable majesty.

I still think that the beauty of mathematics can be explained to humanists, I'm not sure how, though.

A boy who grew up in a town closer to Moscow became interested in quantum physics. Plenty of popular science books were easy for him to find to read. His parents were engineers and were quick to recognize boy's talents. A mathematics professor of the only college in the town who is also a friend of the family took the boy under his wings. The professor asked him if he knew about the group SU(3). This conversation made the boy realize that he had to learn math to answer many questions he had by reading popular science books. Thus began the journey of this boy becoming one of the premier mathematicians of today. He was introduced to great mathemati- cians from the beginning. Mathematicians gave him problems to solve. He solved his first problem on Braid groups as a freshman. He attended Israel Gelfand’s legendary seminars at Moscow State University. He was offered a visiting professorship at Harvard even before he received his bachelor’s degree. To get a permanent job he needed a degree. He enrolled in the Harvard PhD program and received his PhD in one year. He first heard about the Langlands Program at Harvard. Since then until today he has been working on the Lang- lands Program. Langlands program is like a Rosetta stone revealing similarities in three apparently unrelated fields; namely, Number the- ory & curves/finite fields, Riemann surfaces and Quantum Physics. The book "Love and Math" is Edward Frenkel’s autobiography. He has done a masterful job of trying to bring hard mathematics that he has been working on to a level of an average reader by inter-vowing it with his fascinating life experiences. It is hard not to get excited about the math that you know practically nothing about. "Love and Math" makes you wanting to read more and get to know more about this fascinating mathematics. Frenkel has included many references for an interested reader to follow through as notes at the end of the book. "Love and Math" is a must read for young and old alike.

I recommend that every high school student, who wants to know what mathematics is and what research mathematicians do, read this book. It is not an easy read so read it very carefully with pencil and paper in hand, again, and again, and again; until you have mastered it. You will be richly rewarded for your efforts.

Frenkel has obviously devoted an considerable amount of his time and energy (which he could have used to pursue his cutting-edge research) to write this book, and the mathematics community should be grateful for his efforts, for there is too much misconception among laypersons and,in particular,among too many high school mathematics teachers, concerning what mathematics is and what research mathematicians do. Frenkel has made a very significant contribution to rectifying this unfortunate situation.

C. J. Mozzochi, Ph.D. Author of The Fermat Diary and Author of The Fermat Proof.

I gave this book the full five stars because I read it twice and enjoyed it even more the second time.

However, I must warn non-mathematics-trained readers that although some of the author's explanations will be understandable by you, most of them will be over your head. Don't be discouraged. Take the Guide for the Reader on p.8 to heart. You don't have to comprehend every detail to enjoy the personal story told by Frenkel and the passionate way he conveys the beauty of major discoveries and programs in mathematics. It is a treat to have someone like Frenkel, at the highest level of research, invite the rest of us into his world to share in his interactions with other greats, and especially to share in the "highs" he experiences when he solves difficult problems and opens new vistas to human understanding.

I am trained in certain branches of mathematics but not in physics. So I did not understand the details of the quantum field theories and string theories Frenkel discussed in Chapters 16 and 17. But I got the flavor of what those researchers were trying to accomplish, and I enjoyed his description of what it was like to collaborate with the great Edward Witten of the Institute for Advanced Study in Princeton.

This is basically a very personal memoir. It traces his rise from being a Jewish student in the anti-semitic Soviet Union, barred from admission to the leading university in Moscow but able to secretly collaborate in research with professors there who recognized his talent, to becoming a Full Professor at U.C. Berkeley, then co-leading a research project about the Langlands Program that attracted the very best minds in the subject, funded by a multi-million dollar grant from the U.S. government.

Note: There is a large time-range (1991 - 2003) in which his journey is not discussed - probably because doing so would have made this book too lengthy. In end-note 1, p.278, he alludes to important events in his life that occurred during that time, saying he'll save the details of that part of his journey for another book. I hope he writes it.

Okay I am a math teacher and have never thought about all the history and relationships of math. This book brought a new thought about how I teach and why people don't like math

A review of the book: “Love & Math: The Heart of Hidden Reality”, Basic Books, hardcover 2013, by Edward Frenkel. (Review by Ján Mináč.)

Caresses of tenderness, burning passion, feelings of closeness and hope; alternating with utter desperation and blind alleys, crushed with heaviness; then darkness slowly breaking into daylight, singing birds, sunshine, and soft beautiful music in the distance.

These are the images and emotions associated with love, longing, dreaming, and creating.

The texts one usually reads on automorphic representations of a Lie group and representations of a Langlands dual group, or almost any text on mathematics, is seldom associated with these feelings; at least among the general public. (Perhaps an exception is the image of being “crushed with heaviness”.)

But Edward Frenkel’s book is not a usual text on mathematics. It is a unique, brilliant book about discovering the secret, beautiful, and magical world of mathematics.

If you did not know that this world existed, now you will know. If you did not long to enter this world, now you may long to do so, and if you already knew about this special world, then you will appreciate it even more.

This is a deeply personal and dramatic book which will make you feel absorbed, curious and delighted; but also angry.

How could you not be angry and offended, and experience many and varied emotions, while reading about young Edward’s mathematical oral examination on July 13, 1984 at Moscow State University? Perhaps your fists will clench, perhaps you will feel ashamed for those who participated in this hypocritical play with the sole purpose of crushing the dreams of a sixteen-year-old dedicated and wonderfully talented young man, to study pure mathematics in the only place in Moscow designed for this purpose.

Why did this happen to Edward? Because Frenkel is a Jewish name, and Frenkel’s father is Jewish.

But no later than five years after this event at Moscow State University, Frenkel was a visiting Professor at Harvard, and now he is one of the leading experts and researchers working on the Langlands program at the University of California at Berkeley. As Frenkel explains: “The Langlands program is one of the deepest and most exciting mathematical theories which has emerged in the last fifty years.”

Frenkel is able to give the reader a glimpse into the extraordinary links between representations of the Galois groups in the Langlands dual of Lie groups, and in automorphic representations of Lie groups. He introduces the reader to quantum duality, and to Weil’s “work on deciphering a triangular text”, where the languages of number theory, curves over finite fields, and Riemann surfaces are used, and one learns about many more exciting and genuinely surprising mathematical discoveries.

The reader is not expected to understand everything, and in fact the reader on his/her first reading is advised to skip through some parts of the book here and there which contain some of the more challenging material.

This entire book is a breathtaking story of the quest of Edward Frenkel. You will feel with him the urge, the determination, the joy and the triumph; along with some anxious and sometimes painful moments. You will feel privileged and happy, as Edward will allow you to enter Professor Vladimir Drinfeld’s office at Harvard, and to witness some real discussions where technicalities have been replaced by the relatively simple ways of how frequently mathematicians can, and in fact must, think about their beloved objects.

The reader will see a number of amazingly creative mathematicians thinking and creating together with Edward, as his mentors, friends, colleagues, or students. Mathematics and creativity come alive.

Then there is a surprising and fitting finale dedicated to the search for the formula of love. Here, as in the real life of Edward Frenkel, it all comes together: Mathematics, writing, and film. How Edward’s film, “Rites of Love and Math”, a film about truth; was inspired, created, and so successfully shown at a number of film festivals around the world.

After one completes the reading of this book, one finds a moving epilogue describing Edward Frenkel’s prestigious AMS Colloquium lectures in 2012 in Boston, his first home in the U. S. A., and the home of his parents.

Finally as an appendix, there are some very valuable notes explaining some of the more technical details, some nice references, and some interesting further comments which one may consult, and which one is referred to while reading the main text. Still, making life even easier for the reader, this book finishes with a useful and friendly glossary of terms and an index.

During the time of the Cold War, we in the East often wondered how life was beyond the Iron Curtain. How did people in “free” countries live? What did they do? We did not believe the propaganda that they were all taking drugs, were all “corrupt capitalists”, or were all the exploited proletariat or other types of victims; but otherwise we did not really know nor understand the truth about life beyond the Iron Curtain.

It may be that the perception of mathematics among the general public is comparable to this confusing picture we once had about the West.

In his book, Edward Frenkel has succeeded in unveiling the beauty, the richness, and the vitality and magic of mathematics. He lifts the “iron curtain” between the general public and mathematics; and invites crowds, as in times past; crowds who traveled from the East – crowds who poured into Austria, West Germany, Greece, Israel, Italy; and further into the West; people who discovered that life in those “corrupt capitalist countries” could actually be great fun and very uplifting. This move and this new life was an amazing change which allowed people to be free, to live, to laugh, and to enjoy things which they could not even dream about before.

This book is a great success, a wonderful eye opener, and pure delight. This book is a “must-read” for everyone.

I am not embarrassed to share the fact that I LOVE MATH (I don't mean to "yell" but I would in order to get this point across). In, fact, I'd guess that I love math more than most of the friends and acquaintances I've made over the years. So, I am not exactly the target audience for this book, which the author hopes will (p 3), "...unlock the beauty and power of mathematics, and enable you to enter this magical world the way I did, even if you are the sort of person who has never used the words "math" and "love" in the same sentence." But, in spite of the fact that I am an extreme fan of math, mathematical concepts, data analysis and more, I did not enjoy the math parts of this book. In fairness, though, I did love the "life" (of the author) parts.

Reading Love & Math went, for me, like this:

(1) Enthralled interest in an intriguing story about Edward Frenkel's love of math, for example, his experience during his oral exams (during which he was denied entry into an elite school because of his Jewish surname), or interaction with other super smart, famous math person, etc.

followed by

(2) Extreme confusion while reading a very simplistic (supposedly, at least from the author's view) long, complicated explanation of a mathematical concept almost entirely foreign to me complete with some sort of visual (this helped, though not much) during which my eyes began to glaze over, though, fortunately, not enough to preclude me from becoming interested in the next example .
followed by

(1), then (2), repeatedly through the end of the book.

Although I did understand (and enjoy) a few of the explanations (the bit about his work with a doctor involved in kidney transplantation in order to help design an algorithm to determine whether or not to remove the transplanted organ, for example), for the most part I was lost but felt compelled to read on in order to learn everything I could about his interesting life as a mathematician.

In summary, in spite of my inability to appreciate the story in its entirety because the author explained math concepts at a level several notches above my ability to comprehend them, I enjoyed his anecdotes and other recollections about his life as a mathematician, while wondering how the author's over-my-head explanations of complicated math concepts could possibly lead non-mathlovers to suddenly appreciate the subject. Better: A Beautiful Mind by Sylvia Nasar, Rocket Boys by Homer Hickam and Help Your Kids with Math by Barry Lewis.

My wife bought this book for me, and handed it to me with some apprehension, because I am a serious scientist with a very strong background in mathematics, physics and computing, now a retired professor of biomedical informatics. She knows I have little patience for popularizations of math, physics and computing. This book is very different. It has something for everyone. If you have no math background, you won't get everything, but so what? If you go to an ice hockey game and have never played you will not see or appreciate everything that is happening. Still, you can appreciate the flow, the beauty, the greatness of the game. We don't get to watch mathematicians at work like watching ice hockey, but Frenkel's book gives you a view. The personal stories, like the interviews of ice hockey players, help give a personal perspective to what it is to be a mathematician. This book will help you appreciate the beauty and greatness of mathematics. It is an honest exposition. The explanations are the real thing, not silly metaphors that don't really convey the truth. I learned real math (theory of groups, topology, symmetry, transfinite numbers, etc.) from books like Courant and Robbins's classic, "What is Mathematics", and yes, in school I also had to learn arithmetic, algebra, calculus, etc., but the fun stuff is not much taught in schools. It should be. I hope this book will stir people to make some changes in the way math is taught. Math is for everyone, not so they can become professionals, but so that they can enjoy it, and feel more comfortable in a world that is and always has been mathematical to the core, now more than ever. Frenkel's book belongs with Courant and Robbins and likely also will become a classic. Mathematics is a mind boggling adventure, as exciting and spectacular as mountain climbing, but a lot safer. So play safe, take the advice of the ancient Greek philosopher Plato, study mathematics and enjoy the trip of a lifetime.

The early parts of this autobiography are interesting, but the later explanations of the math research the author did are fairly opaque. He tries to defend the Langlands program without generating much enthusiasm for the average reader, and the transcripts of dialog among participants don't seem to further any purpose.