GATEQUESTIONS

THEORY OF COMPUTATION

Q1. Choose the correct statement. The set of all strings over an alphabet S ={0,1} with the concatenation operator for strings a) does not form a group b) forms a noncommutative group c) does not have a right identity d) forms a group if the empty string is removed from S * Q2. Consider the set of all strings S * over an alphabet S ={a,b} with the concatenation operator for strings, and a) the set does forms semigroup b) the set does not form a group c) the set has a left and right identity d) the set forms a monoid Q3. Consider the set of all strings S * over the alphabet S ={a,b,c,d,e} with the concatenation operator for strings. a. the set has a right identity and forms a semigroup b. the set has a left identity and forms a monoid c. the set does not form a commutative group but has an identity d. the set does not form a semigroup with identity Q4. Nobody knows yet if P = NP. Consider the language L defined as follows: L=()+1)* if P = NP And L=j otherwise Which of the following statements is true? a) L is recursive b) L is recursively enumerable but not recursive c) L is not recursively enumerable d) Whether L is recursive or not will be known after we find out if P = NP Q5. Consider the language defined as follows L= {a^n b^n|n>=1} if P=NP And L={ww|w in (a+b)+} otherwise Which of the following statements is true? a) L is recursive but not context sensitive b) L is context sensitive but not context free c) L is context sensitive d) What L is will be known after we resolve the P=NP question Q6. Consider the language defined as follows L=(0+1)* if the CSLs are closed under complement And L=(0*1)*0* if P=NP And L=(10*)1* if P is not the same as NP Which of the following statements is true? a) L is always a regular set b) L does not exist c) L is recursive but not a regular set d) What L is will be known after the two open problems P = NP and the closure of CSLs under complement are resolved Q7. Consider the language defined as follows L=(0+1)* if man goes to Mars by 2020AD And L=0* if man never goes to the Mars Which of the following is true? a. L is context free language but not recursive b. L is recursive c. Whether L is recursive or not will be known in 2020AD d. L is a r.e. set that is not regular Q8. Given an arbitrary context free grammar G, we define L as below. L=(0+1)* if G is ambiguous And L=j if G is not ambiguous a. L is a context-free language b. L is recursive but not r.e. c. What L is depends on whether we can determine if G is ambiguous or not d. What L is is undecidable Q9. Given an arbitrary turing machine M and a string w we define L as below. L=(0*1)*0* if M halts on w And L=(0*1*)* if M does not halt on w a. The type of L is undecidable because of the halting problem b. L is a context-sensitive language c. L is recursively enumerable and not context-free d. L is context sensitive and not regular Q10. Consider the language L defined below L=(0+1)* if P=NP And L=(a^nb^n|n>=1} otherwise a. Whether L is a regular set that is not context-free will be known after we resolve the P=NP question. b. Whether L is context-free but not regular will be known after we resolve the P=NP question c. L is context-sensitive d. L is not recursive Q11. It is undecidable if two cfls L1 and L2 are equivalent. Consider two cfls L1 and L2 and a language defined as follows L={a^nb^nc^n|n>=234} if L1=L2 And L={a^nb^nc^nd^n|n>=678} otherwise a. L is recursive b. L is context-free c. We can never say anything about L as it is undecidable d. L is regular Q12. At present it is not known if NP is closed under complementation. Consider L defined as below L={w wR w| w in (0+1+2)* and wR is the reverse of w} if NP is closed under complement And L = {a^nb^nc^nd^ne^n|n>=34567} otherwise a) L is recursive b) L is context-free and not context-sensitive c) L is recursively enumerable but not recursive d) We will be able to say something about L only after we resolve the NP complementation issue Q14. Nobody knows if P=NP at present. Consider a language L as defined below L=(0+1)* if satisfiability is in P L=(0*1)0* if satisfiability is not in P L=(1*0)1* if 3-sat is in P L=(0*1*)* if 3-sat is not in P L=(0*1*0*1*)* if 0/1 knapsack problem is in P L=(1*0*1*0*)* if 0/1 knapsack problem is not in P L=(0*(00)*(1*11*)*) * if max-clique problem is in P L=(0*(00)*(1*11*)*) * if node-cover problem is not in P L=(0*1*)****(010)* if edge-cover problem is not in P L=(0* + 1* + (00)* + (11)*)*(0100101010)* if the chromatic problem is not in P What can we say about the string 0000111100001111=x a) x is always in L b) whether x is in L or not will be known after we resolve P=NP c) the definition of L is contradictory d) x can never be in L Q15. An arbitrary turing machine M will be given to you and we define a language L as follows L=(0+00)* if M accepts at least one string L=(0+00+000)* if M accepts at least two strings L=(0+00+000+0000)* if M accepts at least three strings --------- --------- L=(0+00+000+---+0^n) *if M accepts at least n-1 strings Choose the correct statement. a) We cannot say anything about L as the question of whether a turing machine accepts a string is undecidable b) L is context-sensitive but not regular c) L is context-free but not regular d) L is not a finite set Q16. We are given two context-free languages L1 and L2 and L defined as below a) L=(0+1)* if L1=L2 b) L=((0+00+000)*(1+11+111)*)* if L1 is contained in L2 c) L=((0(10)*)*(1(01)*)* if L2 is contained in L1 d) L=(00+11+0+1)*(0+00+000)* if L1 and L2 are incomparable a) As all the conditions relating to L1 and L2 are undecidable we cannot say anything about L b) L is recursively enumerable c) L is recursive but not context-sensitive d) L is context-sensitive but not context-free e) L is context-free but not regular Q17. It is undecidable if an arbitrary cfl is inherently ambiguous. We are given a cfg G and the language L is defined as below L= (0+1)*01(0+1)* U 1*0* if L(G) is inherently ambiguous L=(0+1)*10(0+1)* U 0*1* if L(G) is not inherently ambiguous Choose the incorrect statement a) L is regular b) L iscontext-free c) L is context-sensitive d) The above choices can be resolved only if we know if L(G) is inherently ambiguous or not Q18. We are given an arbitrary turing machine M and define the language L as below L=(0*+1*)* if M halts on blank tape L=(0+1*)* if M ever prints a 1 L=(0*+1)* if M ever enters a designated state q L=((0+1+00+11+000+111)+)* if M accepts an infinite set L=0*(10*)* if M accepts a finite set L=1*(01*)* if M accepts exactly 45 strings Choose the correct statement with reference to the string x=00000111111000000111111 a) x is in L b) x is not in L c) we can never decide if x is in L as all the problems of the turing machine are undecidable d) whether x is in L depends on the particular turing machine M Q19. We are given a language L defined as follows L=(0+1)* if the Hamiltonian circuit problem is in P L=(0*1*+0)* if the Traveling salesman problem is not in P L=(0*1*1)*0* if the bin packing problem is in P a) the definition of L is contradictory b) What L is will be known after we resolve the P=NP question c) L if a finite set d) The string 01010101001010110010101 is in L Q20. The intersection of two cfls can simulate a turing machine computation. We are given two cfls L1 and L2 and define the language L as below a) L=(00)* if the intersection of L1 and L2 is empty b) L=((0(00)*)(0(00)*))* if L1 is regular c) L=(00+0000+000000)* if L2 is not regular d) L=(00)*+(0000)* if the complement of L1 is a cfl a) L is a finite set b) L is a regular set c) L is undecidable d) L is recursive but not context-free Result Page:- 1-20 | 21-40 | 41-60 | 61-80 | 81-100 | 101-120 |

Ashtavaidyans

Ashtavaidyans are believed to be the traditional Ayurvedic physicians of Kerala and are from Namboothiri community. They are masters of the eight branches of medicine from which the word Ashtavaidyan is originated. They wrote several books incorporating their observations and clinical experiences. "Chikitsa Manjari", "Yogamithram", "Abhidhana Manjari", "Alathur Manipravalam", "Sindoora Manjari" and "Kairaly Commentary on Ashtanga Hridayam" are some of them. They come under the family of Vaagbhatachaaryan, one of the members of Brihat Trayee. Brihat Trayees are three authentic Aachaaryans, namely Susruthan, Charakan and Vaagbhatan.

According to Mr. N V K Varier's "Ayurveda Charithram", the word Ashtavaidyans does not refer to eight designated families of physicians, but rather to 18 Ashtaangavaidyans each one designated to 18 Sabhaamadhams (Vedam Schools) serving the many (32) Graamams of Kerala. These families were learned experts proficient in all the eight branches (Ashtaangams) of Ayurveda system (Poorna Vaidyans or complete physicians). The word Ashtaangavaidyans were later apparently reduced to Ashtavaidyans. It so happens that, in the absence of male members, several of these families had to be finally merged into eight of these families. The families are listed below with the existing families in the left column. Except Aalathiyoor and Kaarathol who are Nambis, all others are Moosses.

1.	Aalathiyoor Nambi	1. Aalathiyoor Nambi 2. Kaarathol Nambi 3. Choondal Mooss
2.	Elayidath Thaikkatt Mooss	4. Elayidath Thaikkatt Mooss 5. Kuriyedath Mooss (Njarakkal Mooss) 6. Kurumbempilly Mooss 7. Paduthol Mooss
3.	Pazhanellippurath Thaikkatt Mooss	8. Pazhanellippurath Thaikkatt Mooss 9. Peringavu Mooss 10. Parappur Mooss
4.	Kuttancherry Mooss	11. Kuttancherry Mooss 12. Vatuthala Mooss 13. Akalaanath Mooss
5.	Vayaskara Mooss	14. Vayaskara Mooss
6.	Chirattamon Mooss	15. Chirattamon Mooss (Olassa Mooss)
7.	Velluttu Mooss	16. Velluttu Mooss 17. Ubhayur Mooss
8.	Pulamanthol Mooss	18. Pulamanthol Mooss

Of these, Kaarathol Nambi either became extinct without any male children, or became Vaidyamadham. Moreover, there are no practising physicians in the families of Kuttancherry, Vayaskara and Velluttu Mooss, at present.

Another version is that it was Lord Parasuraman who brought Brahmanans (Namboothiris) to Kerala, assigned eight of the families as physicians, and these families came to be known as Ashtavaidyans. There is a third view which states that eight prominent disciples of Vaagbhata and their families continued the Ashtaangahridayam method of treatment, thus prompting the dual meaning of the word. Some believe that Vaagbhata came to Kerala and composed Ashtaangahridayam sitting on a rock near Thiruvizha temple, though historians contest this. Anyway, while the rest of the country follows Charaka and Sushrutha, Kerala follows Vaagbhata's Ashtaangahridayam, and this strict method of treatment is world-renowned.

As mentioned earlier, all the families are addressed as Moosses rather than Namboothiris, except Aalathiyoor and Kaarathol who are called Nambis. Ashtavaidyans are given a slightly depressed status perhaps because they have to examine dead bodies, perform surgical operations and use and follow Budhist Granthams (Treatises). However, considerable respect and place are given to them by the Namboothiri community.

Owing to the slightly lower status for Moosses, they are not permitted inside Yaaga saalaas, a place where Yaagams are performed. It is, however essential to have a physician nearby. This was assigned to Vaidyamadham. It is likely that Kaarathol Nambis were upgraded to Namboothiris and brought to Mezhathol for this purpose. Vaidyamadham was said to be the physician for all the 99 Yaagams performed by Mezhathol Agnihothri. The family follows Aalathiyoor Nambi's treatment methods, which points to the possibility of his ancestry to Kaarathol Nambi, who was himself trained under Aalathiyoor Nambi.

There are not many historical studies nor records documenting this rich heritage. Their knowledge, ideas, experiences and ideals will be of great value not only to the present generation, but also to the future ones to come. Kerala is in a way fortunate to have had a number of people taught and trained by the Ashtavaidyans. The heritage has even transgressed to other communities and religious faiths.

Saalaavaidyan

Aayurvedam had developed along two scientific streams - "Sasthrakriya" (surgery) and "Chikilsa" (treatment). The surgeons are called "Dhanwanthareeyans" and the physicians, "Bhaaradwaajeeyans". The luminaries of that period were Susruthan and Charakan. Since treatises, "Susrutha samhitha" and "Charaka samhitha" are observed to have been revised, it may have to be surmised that what we see today are not the original Granthams.

Since those days, it was the great Vaagbhataachaaryan who tried to rejuvenate and modernise Aayurvedam. It must be mentioned that Vaagbhataachaaryan was a liberal, when one looks at his life and works. His works reveal his intention to integrate the two branches. He composed his treatises in a meticulous and orderly manner, choosing apt words, and with even a poetic ring to them. Unfortunately, he had to suffer a lowering of his social status (a minor defilement) throughout his life for his Budhist beliefs. Consequently, most people did not accept his Granthham. During that period, when he reached Kerala, he was given a very warm and hearty welcome. That made him happy and it is said that he spent rest of his life here in Kerala. He is rightly considered as the "Aachaaryan" of the Aayurvedic tradition in Kerala. His work "Ashtaanga hrudayam" describes the eight branches and , also led to the popular "Ashtavaidyam" ( Click Here for "Aayurvedam"). The full form of this term should have been "Ashtangaayurveda vaidyam" (the eight branches of the medical/ health sciences). One has to view the Kerala situation which existed at that time.

It must be surmised that during that period, just as there was a dearth of an effective health science, there was also a dearth of Vedic Karmams and Yajnams. As if to fill that void, was born the great Mezhathol Agnihothri ( Click Here ). It must have been due to his singular efforts that the Yajnam culture was rejuvenated in this area. Vaagbhatan must have been born before Agnihothri, because, when Agnihothri started Yaagams, the hereditary practice of Ashtaangaayurvedam was already prevalent here. There are clear indications of the essentiality of a physician in Yaagasaala, (the place where Yaagam is performed) to take care of the medical problems of the performers ("Rithwiks") even during the Vedic period. It is believed that in those days, the "Aswineedevans" were given the physicians' lower grade ("Paathithyam") and prevented from entering the Yaagasaala, but then, sage Chyavanan, through his blessings, is presumed to have removed this problem. The same must also have happened in Kerala during the revival of Vedic culture.

The Ashtaanga Aayurveda doctors of Kerala, who follow the Vaagbhata school used to practice both streams - surgery and treatment, but the lowering of the grade was assigned only for the surgeons. Thus it became necessary to find an individual or a family of physicians to be assigned to the Yaagasaala needs. That is how and when the Vaidyamadham family of Mezhathur was honoured with this task, selected perhaps from one of the several families of the Vaagbhata tradition.

This assignment may not have been to an individual, since there is a 300-plus published and unpublished palm leaf Granthham collection pointing to the hereditary tradition of the "Vaidyamadham swaroopam". Such a huge ancestral collection would not possibly have been there if it were assigned to an individual. Historically, there were 18 families with Ashtavaidyam tradition, but many became extinct ( Click Here for Ashtavaidyans). Kaarathol Nambi was one such, and lived somewhere close to Aalathur Nambi and related to them. Considering the many commonalities in the treatment techniques of Vaidyamadham and Kaarathol Nambi, some believe that the latter was inducted as the Yaagasaala doctor through the efforts of Agnihothri and was resettled at Mezhathur. These conclusions are, at best, only logical conjectures, and beyond solid proof.

There is another possibility. Vaagbhataachaaryan apparently had two great disciples, one with pen name "Indu" and the other "Jarjatan", as mentioned in the Vaagbhata invocation song ("Dhyaana slokam"). Vaidyamadham may have their ancestry in the family of the greater one of the two, Indu. This surmise is derived from the fact that the two of the three palm-leaf copies of the "Vyaakhyaanams" (explanations) of "Ashtaangahrudayam" and "Sangraham" (summary) written by Indu were in the Vaidyamadham collection. One of these two copies was taken by an eminent and renowned member of the family known as Kunchu Apphan, some 130 years ago (say, around AD 1870) to Edappally where he was staying as physician to the Kochi Royal family. He used it for reference when he had to teach "Vaidyam" to students. It was apparently lost after this time. The other copy is still in the family collection. So much is the background of Vaidyamadham family.

Today, Vaidyamadham swaroopam is the only family in Kerala with the Bhaaradwaajeeya tradition. They are not permitted to do surgery ("Salyasaalaakya" or "Sasthrakriya") that causes "paathithyam". This was perhaps the reason for their Vedic rights and assignment as "Saalaavaidyan". The normal practice in Yajna culture is for the Yajamaanan (master, the person actually doing the Yajnam) to consult and get the permission from the Rithwiks and the Vaidyan before deciding on the Yaagam. Once decided, more than one person requests the Raja of Kollangode for the "Soma" ( Click Here ) and the leather. He is called the "Gandharvan" who protects the Soma. The age-old rule is for the Saalaavaidyan to be always present in the Saala from the beginning to the end of the Yaagam, for looking after the health and medical needs of the Yajamaanan and Rithwiks, as they are not permitted to leave the premises to the end. Till today the formality continues, though he may not be present always and every day.

The Vaidyan's position in the Yaagasaala is in the area called "Ulkkaram". He is the only person who is provided with an "Aavanappalaka" (a special low wooden seat) to sit on. If, for any reason (say "Pula" or defilement), he cannot be present, he usually sends a replacement. In his absence, the standard fee ("Prathipphalam") of 16 Panam is kept on his seat. It is of special interest to note that while the "Paradevatha" of all other Ashtavaidya families is Dhanwanthari, Vaidyamadham's Kuladevatha" is Dakshinaamoorthy (form of Lord Siva in meditation).

Vaidyamadham is supposed to participate as Saalaavaidyan in any Yaagam performed in Kerala, and this has so far been adhered to. In Sukapuram and Perumanam Graamams there used to be a practice of giving a share to the Karmis. It is done in any one of the eight days (only three in Perumanam temple) from Chithra to Uthraatam star of Medom (Malayalam month - mid-April to mid-June) after a bath in the temple tank followed by worship. The share is called "Pazhuthi". At Sukapuram the traditional way of wearing the cloth ("Thattudukkal") is necessary before worshipping in the temple.

Srinivasa Ramanujan

It is one of the most romantic stories in the history of mathematics: in 1913, the English mathematician G. H. Hardy received a strange letter from an unknown clerk in Madras, India. The ten-page letter contained about 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory (Here is a .dvi file with a sample of these results). Every prominent mathematician gets letters from cranks, and at first glance Hardy no doubt put this letter in that class. But something about the formulas made him take a second look, and show it to his collaborator J. E. Littlewood. After a few hours, they concluded that the results "must be true because, if they were not true, no one would have had the imagination to invent them".

Thus was Srinivasa Ramanujan (1887-1920) introduced to the mathematical world. Born in South India, Ramanujan was a promising student, winning academic prizes in high school. But at age 16 his life took a decisive turn after he obtained a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics. The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. It was in no sense a mathematical classic; rather, it was written as an aid to coaching English mathematics students facing the notoriously difficult Tripos examination, which involved a great deal of wholesale memorization. But in Ramanujan it inspired a burst of feverish mathematical activity, as he worked through the book's results and beyond. Unfortunately, his total immersion in mathematics was disastrous for Ramanujan's academic career: ignoring all his other subjects, he repeatedly failed his college exams.

As a college dropout from a poor family, Ramanujan's position was precarious. He lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work. Finally he met with modest success when the Indian mathematician Ramachandra Rao provided him with first a modest subsidy, and later a clerkship at the Madras Port Trust. During this period Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. Still no one was quite sure if Ramanujan was a real genius or a crank. With the encouragement of friends, he wrote to mathematicians in Cambridge seeking validation of his work. Twice he wrote with no response; on the third try, he found Hardy.

Hardy wrote enthusiastically back to Ramanujan, and Hardy's stamp of approval improved Ramanujan's status almost immediately. Ramanujan was named a research scholar at the University of Madras, receiving double his clerk's salary and required only to submit quarterly reports on his work. But Hardy was determined that Ramanujan be brought to England. Ramanujan's mother resisted at first--high-caste Indians shunned travel to foreign lands--but finally gave in, ostensibly after a vision. In March 1914, Ramanujan boarded a steamer for England.

Ramanujan's arrival at Cambridge was the beginning of a very successful five-year collaboration with Hardy. In some ways the two made an odd pair: Hardy was a great exponent of rigor in analysis, while Ramanujan's results were (as Hardy put it) "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account". Hardy did his best to fill in the gaps in Ramanujan's education without discouraging him. He was amazed by Ramanujan's uncanny formal intuition in manipulating infinite series, continued fractions, and the like: "I have never met his equal, and can compare him only with Euler or Jacobi."

One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) of partitions of a number n. A partition of a positive integer n is just an expression for n as a sum of positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. The problem of finding p(n) was studied by Euler, who found a formula for the generating function of p(n) (that is, for the infinite series whose nth term is p(n)xⁿ). While this allows one to calculate p(n) recursively, it doesn't lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)).

Ramanujan's years in England were mathematically productive, and he gained the recognition he hoped for. Cambridge granted him a Bachelor of Science degree "by research" in 1916, and he was elected a Fellow of the Royal Society (the first Indian to be so honored) in 1918. But the alien climate and culture took a toll on his health. Ramanujan had always lived in a tropical climate and had his mother (later his wife) to cook for him: now he faced the English winter, and he had to do all his own cooking to adhere to his caste's strict dietary rules. Wartime shortages only made things worse. In 1917 he was hospitalized, his doctors fearing for his life. By late 1918 his health had improved; he returned to India in 1919. But his health failed again, and he died the next year.

Besides his published work, Ramanujan left behind several notebooks, which have been the object of much study. The English mathematician G. N. Watson wrote a long series of papers about them. More recently the American mathematician Bruce C. Berndt has written a multi-volume study of the notebooks. In 1997 The Ramanujan Journal was launched to publish work "in areas of mathematics influenced by Ramanujan".

Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras. When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. In December 1889 he contracted smallpox.

When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19^th century. The book, published in 1856, was of course well out of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery.

Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book.

Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General's Office in Madras. It was then suggested that he approach Ramachandra Rao who was a Collector at Nellore. Ramachandra Rao was a founder member of the Indian Mathematical Society who had helped start the mathematics library. He writes in [J.%20Indian%20Math.%20Soc.%2012%20(1920),%2087-90.',30)" onmouseover="window.status='Click to see reference';return true">30]:-

A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature-shining eyes- walked in with a frayed notebook under his arm. He was miserably poor. ... He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. ... I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches.

Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. In his letter of application he wrote [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]:-

I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject.

Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras. Middlemast, a graduate of St John's College, Cambridge, wrote [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]:-

I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work.

On the strength of the recommendation Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with a training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the Madras Engineering College C L T Griffith was also interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.

Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E W Hobson and H F Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. In Ramanujan's letter to Hardy he introduced himself and his work [Ramanujan%20:%20Am%20inspiration%202%20Vols.%20(Madras,%201968).',10)" onmouseover="window.status='Click to see reference';return true">10]:-

I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.

Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. On 8 February he replied to Ramanujan [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3], the letter beginning:-

I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes:
(1) there are a number of results that are already known, or easily deducible from known theorems;
(2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance;
(3) there are results which appear to be new and important...

Ramanujan was delighted with Hardy's reply and when he wrote again he said [Collected%20Papers%20(Cambridge,%201927).',8)" onmouseover="window.status='Click to see reference';return true">8]:-

I have found a friend in you who views my labours sympathetically. ... I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government.

Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy's at Trinity College and who met with Ramanujan while lecturing in India.

Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville's home before moving into rooms in Trinity College on 30^th April. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education. He wrote [Dictionary%20of%20Scientific%20Biography%20(New%20York%201970-1990).',1)" onmouseover="window.status='Click to see reference';return true">1]:-

What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity.

Littlewood was asked to help teach Ramanujan rigorous mathematical methods. However he said ([Minerva%2029%20(1991),%20393-419.',31)" onmouseover="window.status='Click to see reference';return true">31]):-

... that it was extremely difficult because every time some matter, which it was thought that Ramanujan needed to know, was mentioned, Ramanujan's response was an avalanche of original ideas which made it almost impossible for Littlewood to persist in his original intention.

The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. In February 1918 Hardy wrote (see [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]):-

Batty Shaw found out, what other doctors did not know, that he had undergone an operation about four years ago. His worst theory was that this had really been for the removal of a malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory - the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. ... Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.

The honours which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his effors at producing mathematics. By the end of November 1918 Ramanujan's health had greatly improved. Hardy wrote in a letter [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]:-

I think we may now hope that he has turned to corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight. ... There has never been any sign of any diminuation in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. ....

He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success - indeed all that is wanted is to get him to realise that he really is a success.

Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died there the following year.

The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. On the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.

Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death.

In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.

The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75^th anniversary of his birth.

Hertha Marks Ayrton

April 28, 1854 - August 23, 1923

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Phoebe Sarah Marks was born in Portsea, England in 1854. She changed her first name to Hertha when she was a teenager. After passing the Cambridge University Examination for Women with honors in English and mathematics, she attended Girton College at Cambridge University, the first residential college for women in England. Charlotte Scott also attended Girton at this time, and she and Marks helped form a mathematics club to "find problems for the club to solve and 'discuss any mathematical question that may arise'" [1]. Marks passed the Mathematical Tripos in 1880, although with a disappointing Third Class performance. Because Cambridge did not confer degrees to women at this time, just certificates, she successfully completed an external examination and received a B.Sc. degree from the University of London.

From 1881 to 1883, Marks worked as a private mathematics tutor, as well as tutoring other subjects. In 1884 she invented a draftsman's device that could be used for dividing up a line into equal parts as well as for enlarging and reducing figures. She was also active in devising and solving mathematical problems, many of which were published in the Mathematical Questions and Their Solutions from the "Educational Times". Tattersall and McMurran write that "Her many solutions indicate without a doubt that she possessed remarkable geometric insight and was quite a clever student of mathematics."

Marks began her scientific studies by attending evening classes in physics at Finsbury Technical College given by Professor William Ayrton, whom she married in 1885. She assisted her husband with his experiments in physics and electricity, becoming an acknowledged expert on the subject of the electric arc. She published several papers from her own research in electric arcs in the Proceedings of the Royal Society of London and The Electrician, and published the book The Electric Arc in 1902. According to Tattersall and McMurran,

The text included descriptions and many illustrations of her experiments, succinct chapter reviews, a comprehensive index, an extensive bibliography, and a chapter devoted to tracing the history of the electric arc. Her historical account provided detailed explanations of previous experiments and results involving the arc and concluded with the most recent research of the author and her colleagues...The book was widely accepted as tour de force on the electrical arc and received favorable reviews on the continent where a German journal enthusiastically praised if for its clear exposition and relevant conclusions.

Hertha Ayrton had been elected the first female member of the Institution of Electrical Engineers in 1899. In 1902 she became the first woman nominated a Fellow of the Royal Society of London. Because she was married, however, legal counsel advised that the charter of the Royal Society did not allow the Society to elect her to this distinction (this advice was reversed in 1923, but the first woman was still not admitted to the Royal Society until twenty years later.) However, in 1904 Ayrton did become the first woman to read her own paper before the Royal Society. This paper was on "The origin and growth of ripple-mark" [Abstract] and was later published in the Proceedings of the Royal Society. In 1906 Ayrton received the Royal Society's Hughes Medal for her experimental investigations on the electric arc, and also on sand ripples. She was the fifth recipient of this prize, award annually since 1902 in recognition of an original discovery in the physical sciences, particularly electricity and magnetism or their applications, and as of 2005, the only woman so honored.

After her husband's death in 1908, Ayrton continued her research. One set of experiments validated Lord Rayleigh's mathematical theory of vortices. She also invented a fan that could create spiral vortices to repel gas attacks. These became known as Ayrton fans, but were never widely used.

Ayrton was an active member of the Woman's Social and Political Union and participated in many suffrage rallies between 1906 and 1913. She was a founding member of the International Federation of University Women and the National Union of Scientific Workers. She served as vice-president of the British Federation of University Women and vice-president of the National Union of Women's Suffrages Societies. Two years after her death in 1923, her lifelong friend Ottilie Hancock endowed the Hertha Ayrton Research Fellowship at Girton College.

Nobel Prize in Mathematics

A trick question! There is no Nobel prize in mathematics. Why not? That question has created numerous stories, myths, and anecdotes. The most popular is that Nobel's wife had an affair with a mathematician, usually said to be Mittag-Leffler, and in revenge Nobel refused to endow one of his prizes in mathematics. Too bad for this story that Nobel was a life-long bachelor! The other common story is that Mittag-Leffler, the leading Swedish mathematician of Nobel's time, antagonized Nobel and so Nobel gave no prize in mathematics to prevent Mittag-Leffler from becoming a winner. This story is also suspect, however, because Nobel and Mittag-Leffler had almost no contact with each other. Most likely Nobel simply never gave any thought to including mathematics among his list of prize areas.

References:

1. Garding, Lars and Lars Hormander. "Why is there no Nobel prize in mathematics?" The Mathematical Intelligencer, 7(3)(1985), 73-74.
2. Ross, Peter. "Why isn't there a Nobel prize in mathematics?" Math Horizons, November 1995, p9. [Reprint from the Math Forum]
3. Why is there no Nobel Prize in Mathematics?, http://www.almaz.com/Nobel/why_no_math.html, The Nobel Prize Internet Archive

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Fields Medal

The Fields Medal is considered to be the equivalent of the Nobel prize for mathematics. John Charles Fields (1863-1932), a Canadian mathematician, endowed funds in his will for an award for mathematical achievement and promise that would emphasize the international character of the mathematical endeavor. The first Fields Medal was awarded at the International Congress of Mathematics meeting in Oslo in 1936. Since 1950 the medal has been awarded every four years at the International Mathematical Congress to between 2 and 4 mathematicians. Although there is no specific age restriction in Fields' will, he did wish that the awards recognize both existing work and the promise of future achievement, so the medals have been restricted to mathematicians under the age of 40. No woman mathematician has ever won a Fields Medal.

Reference:

1. Fields Medals and Rolf Nevalinna Prize, http://www.emis.math.ca/EMIS/mirror/IMU/medals/ [contains complete list of all winners and pictures of the front and back of the medal]
2. Historical Introduction by Alex Lopez-Ortiz, part of his FAQ site on mathematics.

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Ruth Lyttle Satter Prize in Mathematics

[Description from the Notices of the American Mathematical Society]
The Ruth Lyttle Satter Prize in Mathematics was established in 1990 using funds donated to the American Mathematical Society by Joan S. Birman of Columbia University in memory of her sister, Ruth Lyttle Satter. Professor Satter earned a bachelor's degree in mathematics and then joined the research staff at AT&T Bell Laboratories during World War II. After raising a family, she received a Ph.D. in botany at the age of forty-three from the University of Connecticut at Storrs, where she later became a faculty member. Her research on the biological clocks in plants earned her recognition in the U.S. and abroad. Professor Birman requested that the prize be established to homor her sister's commitment to research and to encouraging women in science. The prize is awarded every two years to recognize an outstanding contribution to mathematics research by a woman in the previous five years. The winners have been:

• 1991 Dussa McDuff
• 1993 Lai-Sang Young
• 1995 Sun-Yung Alice Chang
• 1997 Ingrid Daubechies
• 1999 Bernadette Perrin-Riou
• 2001 Karen E. Smith and Sijue Wu
• 2003 Abigail Thompson
• 2005 Sveltana Jitomirskaya
• 2007 Claire Voisin

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Louise Hay Award for Contributions to Mathematics Education

[Description from the Notices of the American Mathematical Society]
The Executive Committee of the Association for Women in Mathematics established the annual Louise Hay Award for Contributions to Mathematics Education. The purpose of this award is to recognize outstanding achievements in any area of mathematics education, to be interpreted in the broadest possible sense. While Louise Hay was widely recognized for her contributions to mathematical logic and for her strong leadership as head of the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago, her devotion to students and her lifelong commitment to nurturing the talent of young women and men secure her reputation as a consummate educator. The annual presentation of this award is intended to highlight the importance of mathematical education and to evoke the memory of all that Hay exemplified as a teacher, scholar, administrator, and human being.

The winners have been:

• 1991 Shirley Frye, National Council for Teachers of Mathematics
• 1992 Olga Beaver, Williams College
• 1993 Naomi Fisher, University of Illnois at Chicago
• 1994 Kaye A. de Ruiz, U.S. Air Force
• 1995 Etta Falconer, Spelman College [Biography]
• 1996 Glenda T. Lappan, Michigan State University, and Judith Roitman, University of Kansas [Biography]
• 1997 Marilyn Burns, Marilyn Burns Education Associates
• 1998 Deborah Hughes Hallett, Harvard University and the University of Arizona
• 1999 Martha K. Smith, University of Texas at Austin
• 2000 Joan Ferrini-Mundy, Michigan State University
• 2001 Patricia D. Shure, University of Michigan
• 2002 Annie Sheldon, Tennessee Technological University
• 2003 Katherine Puckett Layton, Beverly Hills High School and UCLA Graduate School of Education
• 2004 Bozenna Pasik-Duncan, University of Kansas
• 2005 Susanna S. Epp, DePaul University
• 2006 Patricia Kenshaft, Monclair State University
• 2007 Virginia McShane Warfield, University of Washington

For more information about the award and the recipients, visit Louise Hay Award at the Association for Women in Mathematics web site.

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Leroy P. Steele Prize for Seminal Contributions to Research

The Steele Prizes were established in 1970. In 1993, the AMS formalized three categories for the prizes. The prize for "seminal contributions to research" is awarded for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research.

Women mathematicians who have won the prize are:

• 2007 Karen Uhlenbeck, "Removable singularities in Yang-Mills fields," Comm. Math. Phys. 83 (1982), 11-29; and "Connections with L^p bounds on curvature," Comm. Math. Phys. 83 (1982), 31-42.

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Chauvenet Prize

The Chauvenet Prize is awarded annually by the Mathematical Association of America to the author of an outstanding expository article on a mathematical topic by a member of the association. First awarded in 1925, the Prize is named for William Chauvenet, a professor of mathematics at the United States Naval Academy. It was established through a gift in 1925 from J.L. Coolidge, then MAA President. Winners of the Chauvent Prize are among the most distinguished of mathematical expositors.

Women mathematicians who have won the prize are:

• 1996 Joan Birman, "New Points of View in Knot Theory," AMS Bulletin, 28(1993).
• 2001 Carolyn S. Gordon (with David L. Webb), "You can't hear the shape of a drum", American Scientist 84 (1996), 46-55.
• 2002 Ellen Gethner (with Stan Wagon and Brian Wick), "A Stroll through the Gaussian Primes", American Mathematical Monthly, vol 105, no. 4 (1998), 327-337.

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MacArthur Fellowships

MacArthur fellowships, popularly known as the "genius awards," cannot be applied for; rather, candidates are drawn from a pool of initial nominations by an anonymous group of 100 people. The John D. and Catherine T. MacArthur Foundation aims to recognize people whose achievements in the arts, humanities, sciences, social sciences, and public affairs show the promise of even greater accomplishments in the future. There are no strings attached. Recipients can spend the money, usually anywhere from $150,000 to $375,000 over a period of five years, anyway they want. The fellowships were established in 1981.

Women mathematicians who have received MacArthur Fellowships are:

• 1983 Julia Robinson
• 1983 Karen Uhlenbeck
• 1990 Nancy Kopell
• 1992 Ingrid Daubechies

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Alice T. Schafer Prize

The Schafer Prize is awarded to an undergraduate woman in recognition of excellence in mathematics and is sponsored by the Association for Women in Mathematics The Schafer Prize was established in 1990 by the executive committee of the AWM and is named for former AWM president and one of its founding members, Alice T. Schafer, who has contributed a great deal to women in mathematics throughout her career. The criteria for selection includes, but is not limited to, the quality of the nominees' performance in mathematics courses and special programs, exhibition of real interest in mathematics, ability to do independent work, and if applicable, performance in mathematical competitions.

The winners of the Schafer Prize have been:

• 1991 Linda Green (University of Chicago) and Elizabeth Wilmer (Harvard University)
• 1992 Jeanne Nielsen (Duke University)
• 1993 Zvezdelina E. Stankova (Bryn Mawr College)
• 1994 Catherine O'Neil (University of California) and Dana Pascovici (Dartmouth College)
• 1995 Jing Rebecca Li (University of Michigan)
• 1996 Ruth Britto-Pacumio (Massachusetts Institute of Technology)
• 1997 Ioana Dumitriu (New York University's Courant Institute of Mathematical Sciences)
• 1998 Sharon Ann Lozano (University of Texas at Austin) and Jessica A. Shepherd (University of Utah)
• 1999 Caroline J. Klivans (Cornell University)
• 2000 Mariana E. Campbell (University of California, San Diego)
• 2001 Jaclyn (Kohles) Anderson (University of Nebraska at Lincoln)
• 2002 Kay Kickpatrick (Montana State University) and Melanie Wood (Duke University)
• 2003 Kate Gruher (University of Chicago)
• 2004 Kimberley Spears (University of California)
• 2005 Melody Chan (Yale University)
• 2006 Alexandra Ovetsky (Princeton University)
• 2007 Ana Caraiani (Princeton University)

For more information about the Alice T. Schafer Prize for Excellence in Mathematics by an Undergraduate Woman, see Alice T. Schafer Prize at the Association for Women in Mathematics web site.

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MAA Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics

The Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics is the most prestigious award made by the Mathematical Association of America. This award, first given in 1990, is the successor to the Award for Distinguished Service to Mathematics, awarded since 1962.

Women mathematicians who have won this award or the previous Distinguished Service Award are:

• 1962 Mina Rees
• 1991 Shirley Hill
• 1995 Anneli Lax
• 1997 Deborah Tepper Haimo
• 1998 Alice T. Schafer
• 2004 T. Christine Stevens

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Sylvester Medal of the Royal Society of London

The Sylvester Medal has been awarded by the Royal Society of London every three years since 1901 for the encouragement of mathematical research without regard to nationality. It is given in honor of Professor J. J. Sylvester.

Women mathematicians who have won the Sylvester Medal are:

• 1964 Mary L. Cartwright

Complete list of winners of the Sylvester Medal

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De Morgan Medal of the London Mathematical Society

The De Morgan Medal, the London Mathematical Society's premier award, is awarded every third year in memory of Professor A. De Morgan, the Society's first President. The only criteria for the award is the candidate's contributions to mathematics. The medal was first awarded in 1884.

Women mathematicians who have won the De Morgan Medal are:

• 1968 Mary L. Cartwright

Complete list of winners of the De Morgan Medal

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Adams Prize

The Adams Prize, given annually by the University of Cambridge to a British mathematician under the age of 40, commemorates the discovery by John Couch Adams of the planet Neptune through calculation of the discrepancies in the orbit of Uranus. It was enowed by members of St John's College, Cambridge, and approved by the Senate of the University in 1848. Each year applications are invited from mathematicians who have worked in a specific area of mathematics.

Women mathematicians who have won the Adams Prize are:

• 2002 Susan Howson, University of Nottingham (Number Theory)

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CRM-Fields-PIMS Prize

The CRM-Fields-PIMS prize is intended to be the premier mathematics prize in Canada. The prize recognizes exceptional achievement in the mathematical sciences. The winner's research should have been conducted primarily in Canada or in affiliation with a Canadian university. The main selection criterion is outstanding contribution to the advancement of research. The prize was established by the Centre de recherches mathematiques and the Fields Institute as the CRM-Fields prize in 1994. In 2005, Pacific Institute for the Mathematical Sciences (PIMS) became an equal partner.

Women mathematicians who have won the CRM-Fields-PIMS prize are:

• 2006 Nicole Tomczak-Jaegermann, University of Alberta

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AWM Emmy Noether Lecturers

The Association for Women in Mathematics established the Emmy Noether Lectures to honor women who have made fundamental and sustained contributions to the mathematical sciences. These one-hour expository lectures are presented at the Joint Mathematics Meetings each January. The Emmy Noether Lecturers have been:

• 1980 F. Jessie MacWilliams, "Survey of Coding Theory"
• 1981 Olga Taussky-Todd, "Many Aspects of Pythagorean Triangles"
• 1982 Julia Robinson, "Functional Equations in Arithmetic"
• 1983 Cathleen S. Morawetz, "How Do Perturbations of the Wave Equation Work"
• 1984 Mary Ellen Rudin, "Paracompactness"
• 1985 Jane Cronin Scanlon, "A Model of Cardiac Fiber: Problems in Singularly Perturbed Systems"
• 1986 Yvonne Choquet-Bruhat, "On Partial Differentials Equation of Gauge Theories and General Relativity"
• 1987 Joan S. Birman, "Studying Links via Braids"
• 1988 Karen K. Uhlenbeck, "Moment Maps in Stable Bundles: Where Analysis Algebra and Topology Meet"
• 1989 Mary F. Wheeler, "Large Scale Modelling of Problems Arising in Flow in Pourous Media"
• 1990 Bhama Srinivasan, "The Invasion of Geometry into Finite Group Theory"
• 1991 Alexandra Bellow, "Almost Everywhere Convergence: The Case for the Ergodic Viewpoint"
• 1992 Nancy Kopell, "Oscillators and Networks of Them: Which Differences Make a Difference"
• 1993 Linda Keen, "Hyperbolic Geometry and Spaces of Riemann Surfaces"
• 1994 Lesley Sibner, "Analysis in Gauge Theory"
• 1995 Judith D. Sally, "Measuring Noetherian Rings"
• 1996 Olga Oleinik, "On Some Homogenization Problems For Differential Operators"
• 1997 Linda Rothschild, "How do Real Manifolds live in a Complex Space"
• 1998 Dusa McDuff, "Symplectic Structures-A New Approach to Geometry"
• 1999 Krystyna M. Kuperberg, "Aperiodic Dynamical Systems"
• 2000 Margaret Wright, "The Mathematics of Optimization"
• 2001 Sun-Yung Alice Chang, "Nonlinear Equations in Conformal Geometry"
• 2002 Lenore Blum, "Computing over the Reals: Where Turing meets Newton"
• 2003 Jean E. Taylor, "Five Little Crystals and How They Grew"
• 2004 Svetlana R. Katok, "Symbolic dynamics for geodesic flows"
• 2005 Lai-Sang Young, "From limit cycles to strange attractors"
• 2006 Ingrid Daubechies, "Mathematical results and challenges in learning theory"
• 2007 Karen Vogtmann, "Automorphisms of free groups, outer space, and beyond"

AWM web site about the Emmy Noether Lectures.

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Emmy Noether Lecturers, International Congress of Mathematicians

The Emmy Noether Lectures at the International Congress of Mathematicians, held every four years, is jointly organized by European Women in Mathematics, the Committee on Women of the Canadian Mathematical Society, and the Association for Women in Mathematics.

• 1994 Olga Ladyzhenskaya, "On some evolutionary fully nonlinear equations of geometrical nature"
• 1998 Cathleen Morawetz, "Variations on conservation laws for the wave equation"
• 2002 Hesheng Hu, "Two-dimensional harmonic maps"

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AWM/MAA Falconer Lecturers

The Association for Women in Mathematics and the Mathematical Association of America annually present the Etta Z. Falconer Lectures to honor women who have made distinguished contributions to the mathematical sciences or mathematics education. These one-hour expository lectures are presented at Mathfest each summer. While the lectures began with Mathfest 1996, the title "Etta Z. Falconer Lecture" was established in 2004 in memory of Falconer's profound vision and accomplishments in enhancing the movement of minorities and women into scientific careers. The Falconer Lecturers have been:

• 1996 Karen E. Smith, MIT, "Calculus mod p"
• 1997 Suzanne M. Lenhart, University of Tennessee, "Applications of Optimal Control to Various Population Models"
• 1998 Margaret H. Wright, Bell Labs, "The Interior-Point Revolution in Constrained Optimization"
• 1999 Chuu-Lian Terng, Northeastern University, "Geometry and Visualization of Surfaces"
• 2000 Audrey Terras, University of California at San Diego, "Finite Quantum Chaos"
• 2001 Pat Shure, University of Michigan, "The Scholarship of Learning and Teaching: A Look Back and a Look Ahead"
• 2002 Annie Selden, Tennessee Technological University, "Two Research Traditions Separated by a Common Subject: Mathematics and Mathematics Education"
• 2003 Katherine P. Layton, Beverly Hills High School, "What I Learned in Forty Years in Beverly Hills 90212"
• 2004 Bozenna Pasik-Duncan, University of Kansas "Mathematics Education of Tomorrow"
• 2005 Fern Hunt, National Institute of Standards and Technology, "Techniques for Visualizing Frequency Patterns in DNA"
• 2006 Trachette Jackson, University of Michigan, "Cancer Modeling: From the Classical to the Contemporary"
• 2007 Katherine St. John, City University of New York, "Polygenetic Trees"

AWM web site about the Falconer Lectures.

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AWM-SIAM Sonia Kovalevsky Lecturers

The Assocation for Women in Mathematics in cooperation with the Society for Industrial and Applied Mathematics (SIAM) sponsers the AWM-SIAM Sonia Kovalevksy Lecture Series. The lecture is given annually at the SIAM Annual Meeting by a woman who has made distinguished contributions in applied or computational mathematics. The lectureship may be awarded to any woman in the scientific or engineering community. The Kovalevsky Lecturers have been:

• 2003 Linda R. Petzold, University of California, Santa Barbara, "Towards the Multiscale Simulation of Biochemical Networks"
• 2004 Joyce R. McLaughlin, Rensselaer Polytechnic Institute, "Interior Elastodynamics Inverse Problems: Creating Shear Wave Speed Images of Tissue"
• 2005 Ingrid Daubechies, Princeton University, "Superfast and (Super)sparse Algorithms"
• 2006 Irene Fonseca, Carnegie-Mellon University, "New Challenges in the Calculus of Variations"
• 2007 Lai-Sang Young, Courant Institute of Mathematical Sciences

AWM web site about the Sonia Kovalevsky Lecturers.

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Krieger-Nelson Prize Lectureship for Distinguished Research by Women in Mathematics

The Canadian Mathematical Society inaugurated the The Krieger-Nelson Prize to recognize outstanding research by a female mathematician. The first prize was awarded in 1995. The winners have been:

• 1995 Nancy Reid, University of Toronto
• 1996 Olga Kharlampovich, McGill University
• 1997 Cathleen Morawetz, New York University
• 1998 Catherine Sulem, University of Toronto
• 1999 Nicole Tomczak-Jaefermann, University of Alberta
• 2000 C. Kanta Gupta, University of Manitoba
• 2001 Lisa Jeffrey, University of Toronto
• 2002 Priscilla Greenwood, University of British Columbia and Arizona State University
• 2003 Leah Keshet, University of British Columbia
• 2004 Not awarded
• 2005 Barbara Keyfitz, University of Houston
• 2006 Penny Haxell, University of Waterloo
• 2007 Pauline van den Driessche, University of Victoria
• 2008 Izabella Laba, University of British Columbia

As part of its celebrations of the World Mathematical Year in 2000, the Canadian Mathematical Society sponsored the creation of a poster on women in mathematics. The poster features the six outstanding women mathematicians who were awarded the Krieger-Nelson prize from 1995 to 2000.

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American Mathematical Society Colloquium Lecturers

The American Mathematical Society Colloquium Lectures have been presented since 1896. Women mathematicians who have presented lectures are:

• 1927 Anna Pell-Wheeler
• 1980 Julia Robinson
• 1985 Karen Uhlenbeck

Complete list of the AMS Colloquium Lecturers.

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Josiah Willard Gibbs Lecturers

To commemorate the name of Professor Gibbs, the American Mathematical Society established an honarary lectureship in 1923 to be known as the Josiah Willard Gibbs Lectureship. The lectures are of a semipopular nature and are given by invitation. They are usually devoted to mathematics or its applications. It is hoped that these lectures will enable the public and the academic community to become aware of the contribution that mathematics is making to present-day thinking and to modern civilization.

Women mathematicians who have presented the Josiah Willard Gibbs Lectures have been:

• 1981 Cathleen S. Morawetz, "The mathematical approach to the sound barrier"
• 1999 Nancy J. Kopell, "We got rhythm: Dynamical systems of the nervous system" (published in the AMS Notices, January 2000)
• 2005 Ingrid Daubechies, "The Interplay Between Analysis and Algorithms"

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Earle Raymond Hedrick Lecturers

The Earle Raymond Hedrick Lectures were established by the Mathematical Association of America in 1952 to present to the Association a lecturer of known skill as an expositor of mathematics "who will present a series of at most three lectures accessible to a large fraction of those who teach college mathematics."

Women mathematicians who have presented the Earle Raymond Hedrick Lectures have been:

• 1979 Mary E. Rudin
• 1995 Doris Schattschneider
• 1998 Jean E. Taylor
• 2001 Ingrid Daubechies

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J. Sutherland Frame Lectures

The J. Sutherland Frame Lectures were established by Pi Mu Epsilon to honor James Sutherland Frame who was instrumental in founding the Pi Mul Epsilon Journal and in creating the Pi Mu Epsilon Summer Student Paper Conferences in conjunction with the American Mathematical Society and the Mathematical Association of America. The lectures are presented at the summer meeting of the Mathematical Association of America.

Women mathematicians who have presented the J. Sutherland Frame Lectures have been:

• 1988 Doris Schattschneider, "You Too Can Tile the Conway Way"
• 1989 Jame Cronin Scanlon, "Entrainment of Frequency
• 1995 Marjorie Senechal, "Tilings as Differntial Games"
• 2004 Joan P. Hutchinson, "When Five Colors Suffice"

Complete List of J. Sutherland Frame Lecturers.

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Presidents of the Association for Women in Mathematics

The Association for Women in Mathematics was established in 1971 to encourage women to enter careers in mathematics and related areas, and to promote equal opportunity and equal treatment of women in the mathematical community. The Presidents of the AWM have been:

• 1971-1973 Mary Gray
• 1973-1975 Alice T. Schafer
• 1975-1979 Lenore Blum
• 1979-1981 Judy Roitman
• 1981-1983 Bhama Srinivasan
• 1983-1985 Linda Rothschild
• 1985-1987 Linda Keen
• 1987-1989 Rhonda Hughes
• 1989-1991 Jill Mesirov
• 1991-1993 Carol Wood
• 1993-1995 Cora Sadosky
• 1995-1997 Chuu-Lian Terng
• 1997-1999 Sylvia Wiegand
• 1999-2001 Jean Taylor
• 2001-2003 Suzanne Lenhart
• 2003-2005 Carolyn Gordon
• 2005-2007 Barbara Keyfitz
• 2007-2009 Cathy Kessel

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Presidents of the Mathematical Association of America

In December 1915, ten women and 96 men met at The Ohio State University to established the organization that became the Mathematical Association of America. Women who have served as President of the MAA have been

• 1979-1981 Dorothy L. Bernstein, Goucher College
• 1989-1990 Lida K. Barrett, Mississippi State University
• 1991-1992 Deborah Tepper Haimo, University of Missouri at Saint Louis
• 2001-2002 Ann Watkins, California State University, Northridge

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Presidents of the American Mathematical Society

The American Mathematical Society was founded in 1889. Since then, women who have served as President of the AMS have been

• 1983-1984 Julia Robinson, University of California at Berkeley
• 1995-1996 Cathleen Morawetz, Courant Institute of Mathematical Sciences