Death” based on a story by the great Japanese Program is a program of study of analogies and writer Yukio Mishima interconnections between four areas of science: and starred in it. Frenkel invented the plot and number theory, played the Mathematicianwho in both “Rites directed of Love the and film Math.” The Mathematician creates a formula for three on thecurves above over list are finite different fields, areasgeometry of love, but realizes that the formula can be used for mathematics.of Riemann surfaces, The modern and quantum formulation physics. of the The part both good and evil. To prevent it from falling into offirst the Langlands Program concerning these areas An invitation to math wrong hands, he hides the formula by tattooing was arrived at by the end of the 20th century. The it on the body of the woman he loves. The idea realization that the mathematics of the Langlands is that “a mathematical formula can be beautiful Program is intimately connected like a poem, a painting, or a piece of music” (page physics via mirror symmetry and electromagnetic Edward Frenkel’s to quantum 232). The rite of death plays an important role in duality came in the 21st century, around 2006- the Japanese culture. 2007. Frenkel personally participated in achieving review suggests that mathematics plays an important role the latter breakthrough. “Love & Math: in the world culture. HeThe describes title of Frenkel’s his motivation film hand account of the work done. His book presents a first- Why is the Langlands Program important? mathematicians are usually portrayed as weirdos Because it brings together several areas of the Heart of to create the film as follows. “In popular films,illness, science that cover a wide range of research, reinforcing the stereotype of mathematics as a boringand social and misfits cold subject, on the farverge removed of mental from reality. methods Hidden Reality” Who could want such a life for themselves, doing are sufficiently far apart, and use different been developing in its own way motivated by work that supposedly had nothing to do with and techniques. Each of the areas has anything?” (page 229). And here is his answer to important within it. Once an area is related to truth, the process of creating - a review results,another questions, one, the experts and conjectures in each of thesedeemed two new mathematics is a passionate pursuit, a deeply as methods personalhis own question. experience, “In just like creating art and area,areas which face questions are often and new ideas, and unexpected. as well The with the unknown and with oneself, which elicits andsituation techniques, is similar transplanted to a transfusion from theof fresh other blood strongmusic. Itemotions. requires And love the and formulas dedication, you a discover struggle to a person whose energy level has been stagnant do get under your skin, just like the tattooing in each of the areas involved. he eminent mathematician Edward Frenkel has written his Frenkel sheds light on the professional aspect for a while. The interconnections are beneficial to Mark Kleiner the film,” (page 233). The book discusses in detail precursors of the autobiography in order to convey to the general audience both of mathematics by introducing the reader to the Langlands Program. Tthe human and professional aspects of mathematics. Langlands Program, which he perceives as “a because One of them is the Shimura-Taniyama-Weil He explores the human aspect of mathematics by describing his it uncovers and brings into focus mysterious conjecture, which played an important role in work on several projects. He shows that, in addition to perseverance patternsGrand Unified shared Theory by different of Mathematics areas of mathematics the proof of Fermat’s Last Theorem by Andrew and thus points to deep, unexpected connections Wiles and Richard Taylor in 1994. The theorem, imagination and original ideas, and that beauty and elegance usually between them,” (page 70). Robert Langlands, a statement in number theory, had been the characterizeand hard work, important solving amathematical difficult mathematical results. The problem creative require processs of currently Professor Emeritus in the School of most famous open problem since 1637, when the a mathematician in many ways resembles that of an artist or musician Mathematics at the for Advanced Study French mathematician Pierre Fermat wrote about and generates the whole gamut of emotions, the most important of in Princeton, initiated the program in the late it on the margin of an old book he was reading. which is love. For it is a genuine love for mathematics that carries 1960s as a conjectureInstitute that hard problems of The reasons for the fame were, on one hand, the a mathematician through the often frustrating and emotionally number theory can be solved by using methods simplicity of the statement: there are no positive demanding work. The “Rites of Love of harmonic analysis. Since then, the scope of integers > 2 and , , satisfying and Math,” which was inspired by the short n x y z book describes the erotic film xn + yn = zn, film “Rite of Love and contributions of other mathematicians and Keywords: Mathematics, beauty, elegance, Langlands program, Shimura- ofthe physicists, program hasand significantly at this time thegrown Langlands due to Taniyama-Weil conjecture, Rosetta stone

66 At Right Angles | Vol. 4, No. 1, March 2015 At Right Angles | Vol. 4, No. 1, March 2015 67 Death” based on a story by the great Japanese Program is a program of study of analogies and writer Yukio Mishima interconnections between four areas of science: and starred in it. Frenkel invented the plot and number theory, played the Mathematicianwho in both “Rites directed of Love the and film Math.” The Mathematician creates a formula for three on thecurves above over list are finite different fields, areasgeometry of love, but realizes that the formula can be used for mathematics.of Riemann surfaces, The modern and quantum formulation physics. of the The part both good and evil. To prevent it from falling into offirst the Langlands Program concerning these areas An invitation to math wrong hands, he hides the formula by tattooing was arrived at by the end of the 20th century. The it on the body of the woman he loves. The idea realization that the mathematics of the Langlands is that “a mathematical formula can be beautiful Program is intimately connected like a poem, a painting, or a piece of music” (page physics via mirror symmetry and electromagnetic Edward Frenkel’s to quantum 232). The rite of death plays an important role in duality came in the 21st century, around 2006- the Japanese culture. 2007. Frenkel personally participated in achieving review suggests that mathematics plays an important role the latter breakthrough. “Love & Math: in the world culture. HeThe describes title of Frenkel’s his motivation film hand account of the work done. His book presents a first- Why is the Langlands Program important? mathematicians are usually portrayed as weirdos Because it brings together several areas of the Heart of to create the film as follows. “In popular films,illness, science that cover a wide range of research, reinforcing the stereotype of mathematics as a andboring social and misfits cold subject, on the farverge removed of mental from reality. methods Hidden Reality” Who could want such a life for themselves, doing are sufficiently far apart, and use different been developing in its own way motivated by work that supposedly had nothing to do with and techniques. Each of the areas has anything?” (page 229). And here is his answer to important within it. Once an area is related to truth, the process of creating - a review anotherresults, questions, one, the experts and conjectures in each of thesedeemed two new mathematics is a passionate pursuit, a deeply as methods personalhis own question. experience, “In just like creating art and area,areas which face questions are often and new ideas, and unexpected. as well The with the unknown and with oneself, which elicits andsituation techniques, is similar transplanted to a transfusion from theof fresh other blood strongmusic. Itemotions. requires And love the and formulas dedication, you a discover struggle to a person whose energy level has been stagnant do get under your skin, just like the tattooing in each of the areas involved. he eminent mathematician Edward Frenkel has written his Frenkel sheds light on the professional aspect for a while. The interconnections are beneficial to Mark Kleiner the film,” (page 233). The book discusses in detail precursors of the autobiography in order to convey to the general audience both of mathematics by introducing the reader to the Langlands Program. Tthe human and professional aspects of mathematics. Langlands Program, which he perceives as “a because One of them is the Shimura-Taniyama-Weil He explores the human aspect of mathematics by describing his it uncovers and brings into focus mysterious conjecture, which played an important role in work on several projects. He shows that, in addition to perseverance patternsGrand Unified shared Theory by different of Mathematics areas of mathematics the proof of Fermat’s Last Theorem by Andrew and thus points to deep, unexpected connections Wiles and Richard Taylor in 1994. The theorem, imagination and original ideas, and that beauty and elegance usually between them,” (page 70). Robert Langlands, a statement in number theory, had been the characterizeand hard work, important solving amathematical difficult mathematical results. The problem creative require processs of currently Professor Emeritus in the School of most famous open problem since 1637, when the a mathematician in many ways resembles that of an artist or musician Mathematics at the for Advanced Study French mathematician Pierre Fermat wrote about and generates the whole gamut of emotions, the most important of in Princeton, initiated the program in the late it on the margin of an old book he was reading. which is love. For it is a genuine love for mathematics that carries 1960s as a conjectureInstitute that hard problems of The reasons for the fame were, on one hand, the a mathematician through the often frustrating and emotionally number theory can be solved by using methods simplicity of the statement: there are no positive demanding work. The “Rites of Love of harmonic analysis. Since then, the scope of integers > 2 and , , satisfying and Math,” which was inspired by the short n x y z book describes the erotic film xn + yn = zn, film “Rite of Love and contributions of other mathematicians and Keywords: Mathematics, beauty, elegance, Langlands program, Shimura- ofthe physicists, program hasand significantly at this time thegrown Langlands due to Taniyama-Weil conjecture, Rosetta stone

66 At Right Angles | Vol. 4, No. 1, March 2015 At Right Angles | Vol. 4, No. 1, March 2015 67 and, on the other hand, the number of years the Frenkel mathematical The reader learns about the widespread anti- The reader gets a glimpse of the great problem had resisted the efforts of people to solve theories . . . : universality, objectivity, endurance, Semitism when Frenkel describes how he, the it. Familiarity with middle school mathematics is and relevanceformulates to the “four physical qualities world,” of (page son of a Russian mother, was denied admission to seminar at MGU, “an important mathematical understand the problem, 228), of which objectivity seems to be the Mekh-Mat of MGU (Moskovskiy Gosudarstvennyy mathematicianand social event, Israel which Gelfand had been and running his weekly for and many generations of people, both experts only controversial one in that it claims that Universitet), that is, the Department of Mechanics andmore amateurs, than sufficient had confused to the simplicity of mathematical concepts and ideas inhabit the and Mathematics at the Moscow State University, over the world,” (page 61). Many outstanding Platonic world of mathematics that exists because he had a Jewish father. He discusses mathematiciansmore than fifty years considered and was it anrenowned honor to all meet main step towards the proof was made in 1986 independently of the physical reality or the mental careers of other Jewish mathematicians, some of Gelfand and give a talk in the seminar. was the statement with the ease of proof. The first them truly exceptional, who were mistreated by open to both Gentiles and Jews. Since most of the Theorem follows from the Shimura-Taniyama- and physicists who support his point of view. the Soviet system. The reader also learns about with ItMGU, the Weilby Ken conjecture. Ribet who The showed second that main Fermat’s step was Last world. He quotes several famous mathematicians prominent Soviet mathematicians, both Gentile seminar provided “a safe haven” where they could The book is aimed at the reader who may have no the proof of the latter by Wiles and Taylor in and Jewish, who recognized Frenkel’s talent, latterdevelop were as mathematicians.not formally affiliated The core of Gelfand’s background or interest in mathematics or even mentored him, and seminar comprised the “Gelfand mathematical hates math. Using plain language and examples Weil conjecture, which establishes a connection mathematics. They did this of their own volition school,” of which Frenkel was a member. from everyday life and minimizing the use of sufficient generality. The Shimura-Taniyama- and without any remuneration.helped him find his place in formulas, Frenkel presents highly nontrivial The book also mentions correspondence schools, numbers and modular forms, relates number mathematical facts. At the same time, if the reader organized by famous Soviet mathematicians, between elliptic curves over the field of rationalcase of gets interested and decides to delve into the graduated from Mekh-Mat University Gelfand among them, that reached out to talented the Langlands Program. I cannot resist adding a personal touch here. I theory to harmonic analysis. It is a special described theories more deeply, the Notes contain in 1969, being one of the less than ten graduates high school students who lived outside of Another precursor of the Langlands Program with straight A’s. Out of the of graduating Kiev State class major cities and did not have access to special is the Rosetta stone proposed by the French heuristic arguments. The author also describes of 175, more than half were admitted to the contrast, very few mathematician Andre Weil in 1940. The name manynumerous formal applications definitions, of rigorous mathematics proofs, to andsupport State, but the door was prominent American mathematicians get directly Rosetta stone is a reference to the famous his point that an advanced society should be involvedmathematical with schools.teaching In high school students. ancient stele, on which essentially the same text mathematically literate. jobgraduate as a computer school at programmer, Kiev continued doing To summarize, the book is a lively account of the was written in three different languages and shutmathematics for me because in my spare of my time, Jewish and descent. defended I got my a The book discusses mathematics in the context life and work of a prominent mathematician. which was used to decipher ancient Egyptian Ph D thesis in 1972. An academic position was of the author’s mathematical career and, hieroglyphs. Weil’s Rosetta stone establishes an The issues raised are diverse enough to make therefore, places the reader in the Soviet Union analogy between number theory, the theory of States in 1979. That was possible due to the the book interesting to a wide range of readers, between years 1983, when Frenkel was in the algebraic curves over Jackson-Vanikbeyond my reach, amendment so I immigrated to the US to federalthe United law, from a person having little or no knowledge of ninth grade of high school, and 1989, when at surfaces. He visualized these three areas of which forced the Soviet government to lessen the mathematics to an experienced professional the age of 21 he became one of the mathematics as three finitecolumns fields, of Table and Riemann 1 (below). restrictions on the emigration of Jews. mathematician. recipients of the Harvard Prize Fellowship after Weil’s main idea was that the middle column is a his mathematical papers had been smuggledfirst four bridge between the left and right columns, i.e., the abroad and perestroika, which was initiated by results and arguments from the left column can General Secretary of the Communist Party Mikhail MARK KLEINER was born in the Soviet Union in 1946. He received his MS and PhD from Kiev State University. be translated to the right column and vice versa Gorbachev, had lifted the iron curtain. Frenkel has He worked later as a computer programmer in Kiev, but in his spare time he worked on representations via the middle column. Based on his Rosetta stone, lived in the United States since 1989. of algebras. He was a postdoctoral student at Brandeis University (USA), and has been a professor of mathematics at Syracuse University (USA) since 1981. He may be contacted at [email protected]. Weil formulated three conjectures, the proof of Although the focus of Frenkel’s narrative is his which “greatly stimulated the development of learning and doing mathematics, as well as his mathematics in the second half of the twentieth interaction with other mathematicians, by default century,” (page 104). Weil’s Rosetta stone is he paints a picture of life in the Soviet Union. another special case of the Langlands Program. Error Regretted Dear Readers, In the previous issue of At Right Angles, November 2014, we had missed acknowledging the contributor for the Number Theory Curves over Finite Fields Riemann Surfaces article “Folding a 450, 600, 750 Triangle from a square sheet of paper in 6 easy steps” on page 24. The article was contributed by Roopika Jayaram, (Sahyadri School (KFI), Chas Khaman Dam, Tiwai Hill, Rajgurunagar, Pune – 410513. E-mail: [email protected])

The error is regretted. Table 1

68 At Right Angles | Vol. 4, No. 1, March 2015 At Right Angles | Vol. 4, No. 1, March 2015 69 The reader learns about the widespread anti- The reader gets a glimpse of the great Semitism when Frenkel describes how he, the son of a Russian mother, was denied admission to seminar at MGU, “an important mathematical Mekh-Mat of MGU (Moskovskiy Gosudarstvennyy andmathematician social event, Israel which Gelfand had been and running his weekly for Universitet), that is, the Department of Mechanics and Mathematics at the Moscow State University, over the world,” (page 61). Many outstanding because he had a Jewish father. He discusses mathematiciansmore than fifty years considered and was it anrenowned honor to all meet careers of other Jewish mathematicians, some of Gelfand and give a talk in the seminar. was them truly exceptional, who were mistreated by open to both Gentiles and Jews. Since most of the the Soviet system. The reader also learns about with ItMGU, the prominent Soviet mathematicians, both Gentile seminar provided “a safe haven” where they could and Jewish, who recognized Frenkel’s talent, developlatter were as mathematicians.not formally affiliated The core of Gelfand’s mentored him, and seminar comprised the “Gelfand mathematical mathematics. They did this of their own volition school,” of which Frenkel was a member. and without any remuneration.helped him find his place in The book also mentions correspondence schools, organized by famous Soviet mathematicians, graduated from Mekh-Mat University Gelfand among them, that reached out to talented inI cannot 1969, resistbeing addingone of the a personal less than touch ten graduates here. I high school students who lived outside of with straight A’s. Out of the of graduating Kiev State class major cities and did not have access to special of 175, more than half were admitted to the contrast, very few State, but the door was prominent American mathematicians get directly involvedmathematical with schools.teaching In high school students. jobgraduate as a computer school at programmer, Kiev continued doing To summarize, the book is a lively account of the mathematicsshut for me because in my spare of my time, Jewish and descent. defended I got my a life and work of a prominent mathematician. Ph D thesis in 1972. An academic position was The issues raised are diverse enough to make States in 1979. That was possible due to the the book interesting to a wide range of readers, Jackson-Vanikbeyond my reach, amendment so I immigrated to the US to federalthe United law, from a person having little or no knowledge of which forced the Soviet government to lessen the mathematics to an experienced professional restrictions on the emigration of Jews. mathematician.

MARK KLEINER was born in the Soviet Union in 1946. He received his MS and PhD from Kiev State University. He worked later as a computer programmer in Kiev, but in his spare time he worked on representations of algebras. He was a postdoctoral student at Brandeis University (USA), and has been a professor of mathematics at Syracuse University (USA) since 1981. He may be contacted at [email protected].

Error Regretted Dear Readers, In the previous issue of At Right Angles, November 2014, we had missed acknowledging the contributor for the article “Folding a 450, 600, 750 Triangle from a square sheet of paper in 6 easy steps” on page 24.

The article was contributed by Roopika Jayaram, (Sahyadri School (KFI), Chas Khaman Dam, Tiwai Hill, Rajgurunagar, Pune – 410513. E-mail: [email protected])

The error is regretted.

At Right Angles | Vol. 4, No. 1, March 2015 69