Dusa, Gábor, ed. Félúton: Contemporary Theatre and Dance in Custer, Marianne. Telephone Interview. 23 April 2007. Hungary. Budapest: IETM, 1994. Goda, Gábor. Personal interview. 17 January 2007. 21 March 2008. and Educational Exchange Fábri, Péter, ed. A Shabby Paradise: Contemporary Hungarian Theatre 2004. Huszti, Péter. Personal interview. 10 January 2007. Busapest: Hungarian Centre of the International Theatre in Hungary Institute, 2004. Kinga Keszthelyi. Personal interview. 18 January 2007. 25 February 2008. Féral, Josette. Pluralism in Art or Interculturalism? http://www.powerofculture.nl/uk/archive/amsterdam/ Laban, Katalan. Personal interview. 7 January 2007. ukverslag_feral.html Linka, Peter. Personal interview. 10 January 2007. 16 October Hakan Seyalioglu Hadas, Miklós and Vörös, Miklós, eds. Colonisation or 2007. Partnership? Eastern Europe and Western Social Sciences. Budapest: Replika Lenyel, György. Personal interview. 15 January 2007. Hungarian Social Science ...... Quarterly, 1996. Marton, László. Personal interview. 18 January 2007. 5 November 2007. College of William & Mary Hungarian Academy of Sciences Huszár, Ildikó, ed. Fulbright Student Conference Papers: Williamsburg, VA Alfréd Rényi Institute of Mathematics

Academic Years 2002/2003 and 2003/2004. Nagy, András. Personal interview. 11 January 2007. 1 February www.wm.edu 1053 Budapest, Reáltanoda utca 13-15 Budapest: Hungarian-American Commission for Educational 2008. [email protected] www.renyi.hu Exchange, 2004. Advisers: Antal Balog, András Biró Nicola, Jim. Personal interview. 19 October 2006. Müller, Péter P. and Lakos, Anna, eds. Collision: Essays on ...... Contemporary Hungarian Smith, Molly. Personal interview. 2 February 2007. Drama. Hungarian Theatre Museum and Institute, 2004.

Szász, János. Personal Interview. 28 March 2008. The purpose of this project was to spend a year studying number theory at the Alfréd Rényi Tompa, Andrea and Nánay, István, eds. Hungarian Theatre. Institute of Mathematics while promoting Hungarian-American educational exchange. The Budapest: Hungarian Theatre Museum and Institute, 2000. Tompa, Andrea. Personal interview. 16 January 2007. 20 project consisted of coursework, independent research, and helping compile course notes for a October 2007. 3 March 2008. number theory class taught in the Budapest Semesters in Mathematics exchange program. Weiss, Peter. The Persecution and Assasination of Jean-Paul The report begins with background on the subject of , which was my Marat As Performed by the www.en.wikipedia.org/wiki/Peter_Weiss focus, centering on recent developments and Hungarian contributions. The report also contains Inmates of the Asylum of Charenton Under the Direction of the www.iti-worldwide.org specifics of my academic tenure in Hungary as well as an overview of the course whose notes I am Marquis De Sade. www.nytw.org/artist_development.asp completing. London: Dramatic Publishing, 1965. www.presidency.ucsb.edu/ws/index.php?pid=30576 www.squattheatre.com Arnoult, Philip. Personal interview. 23 September 2006 and 8 www.thunderbayus.org October 2006 www.toaca.ro/Thetre_project1.php www.us.fulbrightonline.org/about.html Coigney, Martha. Personal interview. 9 December 2006.

130 131 AY 2007-2008 Hakan Seyalioglu: Number Theory and Educational Exchange

and some of the concepts introduced by discipline. Such functions are somewhat 1. Introduction 2. Development of Analytic the ancient would unique in all of mathematics. While they Hungary has a knack for beautiful Number Theory later be revitalized to provide the basis are easy to define, their properties are still for some of the most advanced modern mysterious, after over a century of study. mathematics. The results of Pál Erdős Before discussing the specifics of my concepts in the field. Their structure and some very basic cannot be overstated, but one should project and research, I would like to The disciplines provide a natural properties are completely unknown and also not forget the other legendary give the reader some background by balance for each other, though the they represent one of the most mystifying mathematicians whose portraits line explaining some of the fundamental reason is hard to discern at first. Number classes of functions in mathematics. the main stairway in the humble orange aspects of analytic number theory. Some theory deals with the discrete. Counting The next major achievement in building on Reáltanoda Street. It was topics included are the development of primes and analyzing their properties the discipline was the proof of the a thrill to be among the great minds at sieve theory, the Green-Tao theorem on seems somehow incompatible with the celebrated theorem. Proved the Rényi Institute and to have access primes in arithmetic progressions, and continuous foundation of mathematical by Hadamard and de la Vallée Poussin to their advice and guidance. My time the results of Goldston-Pintz-Yıldırım analysis. However, the natural regularity in 1896 [1], it gives an asymptotic for as a Fulbright Scholar has proven a on small gaps between primes. Some that the primes offer serves as a perfect the number of primes under a given great opportunity for me to develop as terms from higher mathematics are used avenue by which to apply the principles magnitude. The proof uses methods a mathematician and has influenced not without explanation if understanding the of analysis. While a function which pioneered by Euler and Dirichlet and only my abilities, but also the focus of my specific concept is not necessary. counts primes under a given magnitude relies on showing a lack of zeros of the research. My stay at the Rényi Institute consisted mainly of studying a subfield may be an incontinuous step function, Riemann zeta function on the extremal of number theory known as analytic 2.1 Origins of the Discipline when the average is taken, it follows a line of the critical strip. While the number theory. This report begins with ‘somewhat’1 continuous pattern. Since specific technical details of the proof are background on this subfield with focus Mathematical analysis provided a the prime counting function allowed such not important, one should note that the on major contributions by Hungarian foundation for many disciplines that regularity, Dirichlet capitalized on the methods used in the proof are highly mathematicians. emerged in the last two centuries. First rich theory mathematical analysis had to analytical. Apart from my studies in number theory, developed as a method to rigorously offer in order to prove what would serve define calculus, elementary proofs in as the foundation for analytic number I was also affiliated with the Budapest 2.2 Developments Semesters in Mathematics program, analysis have a distinctive flavor and theory. Dirichlet’s theorem [1], states that helping Professor Csaba Szabó complete language, concerned mainly with limits for any x and y which share no common in the 20th Century of functions and . However, factors, there are infinitely many primes the course notes for his introductory Even with these landmark achievements, number theory course, one of the most the techniques of mathematical analysis congruent to x modulo y. spread into other disciplines quickly. Dirichlet’s proof not only states the some of the greatest problems in number popular courses in the program. The theory remained unanswered at the course is one of sentimental value for me, Its power would become apparent even infinitude of such primes, but also that when applied to one of the most ancient the sum of the reciprocals of such primes dawn of the new century. Fermat’s Last as it is where I was first exposed to number Theorem taunted mathematicians around theory, over two years ago. The course disciplines, number theory. tends to infinity in a similar fashion to Number theory is, in general, the study a well-known proof of the existence the world and the conjecture gave me an opportunity I would not have (the existence of infinitely many pairs had at my home institution and I hope that of whole integers and their properties. It of infinitely many primes. The proof is considered one of the oldest and most uses the concepts of the Riemann zeta of primes with a difference of two) still by finishing the course notes, I will make loomed menacingly, as did the seemingly the course more accessible to American beautiful disciplines in mathematics, function and related L-functions heavily, with seminal results which date back tools which are still at the core of the immovable Goldbach conjecture (that undergraduates so that they can gain every even integer greater than two is access to the same Hungarian treatment of millennia still being studied. The proof of the infinitude of primes is considered the sum of two primes). The final two the subject that I found so enthralling. 1 The notion of ‘somewhat’ is characterized in analytic number one of the earliest results of the discipline problems would provide a basis for some theory as an error term, and is a rigorously developed concept.

132 133 AY 2007-2008 Hakan Seyalioglu: Number Theory and Educational Exchange of the most exciting developments in One of the greatest results of the many works further contributed greatly arbitrarily large progressions is quite analytic number theory, and while they discipline came from the work of the to our understanding of the structure of striking. Another nice result dealing are still without proof, related statements Hungarian mathematician, Alfréd Rényi prime numbers. Some of the most notable with arithmetic progressions of primes have given great insight into the structure concerning the large sieve. The large results in analytic number theory include comes from Antal Balog, a researcher at of prime numbers. In particular, the most sieve was first developed by the Russian the Green-Tao theorem on arithmetic the Rényi and co-adviser to my project. succesful attacks, mounted through sieve mathematician Yuri Linnik. Alfréd Rényi progressions and the Goldston-Pintz- His result (praised as ‘beautiful’ in the methods, provided a springboard for further developed the concept of the large Yıldırım results on small gaps between paper proving the Green-Tao theorem) breakthroughs in many related problems. sieve in order to prove one of the most prime numbers. shows that there are arbitrarily large sets One of the most important techniques celebrated results in number theory, that The question of primes in arithmetic of primes such that the average of any developed in analytic number theory in the there exists a constant K such that every progressions has been a constant source two primes in the set is prime. The most last half-century is sieve theory. The basic even prime number can be written as of inspiration for new and exciting significant result to my project, however, idea behind it is simple and elegant. The the sum of at most K primes [1]. Rényi’s problems. Arithmetic progressions is the result of Dan Goldston, János Pintz theory owes its foundation to the ancient result was the first that proved a weak alone provide a stunning number of and Cem Yıldırım concerning small gaps Greek mathematician Eratosthenes. One version of the Goldbach conjecture and questions, many related to Szemerédi’s between primes [7]. defines a simple set, usually all numbers is one of the first steps towards a proof proof [4] of the Erdős–Turán conjecture under a certain magnitude. One then of one of greatest unsolved problems in [5] and Erdős’ conjecture on arithmetic attempts to sift out of this set, all numbers mathematics. progressions. Each of these three 2.4 Small Gaps Between Primes other than the ones under analysis, usually The Chinese mathematician Chen Hungarian mathematicians tried to The recent results concerning small primes of a certain form. As an example, Jungrun also proved many problems discern necessary conditions to place gaps between primes are not only consider the aforementioned sieve of related to the Goldbach and twin prime on a set of integers to guarantee that the exciting findings, but they followed Eratosthenes. The goal of the sieve is to conjectures with the same tools. Using set contains arbitrarily long arithmetic quite a turbulent path to discovery. find the number of primes under a certain sieve methods, he proved that there are progressions. Szemerédi’s theorem gives With the original findings published by magnitude. In order to find this, one infinitely many primes p, such that p+2 is necessary results when dealing with Dan Goldston and Cem Yıldırım which considers the set of all numbers up to the either a prime or a semi-prime (a product the density of a set. Erdős’ conjecture showed that one can find arbitrarily designated threshhold and calculates how of two primes), a statement closely related takes the statement a step further, asking big pairs of successive primes whose many of them are divisible by two. Next, to the twin prime conjecture. Similarly, he whether the divergence of the sum of the difference is arbitrarily small when one subtracts this number from the size showed that every sufficiently large even reciprocals of a set of integers implies that compared to their logarithm, the of the total set, in effect, sifting out the number is either the sum of two primes or the set contains arithmetic progressions mathematical community was very even numbers and leaving the size of the a prime and a semi-prime, closely related of arbitrary length. excited. Not only had they improved the set of odd numbers under the designated to the Goldbach conjecture [2, 3]. best known ratio of the difference to the magnitude. The process is continued for It is a well known result that the sum of logarithm, but they showed that it could each number below the square root of the reciprocals of primes diverges. The 2.3 Recent Discoveries be taken as small as one desires, the best the desired level. The sieve does not quite Green-Tao theorem [6] demonstrates possible result. work, however, in getting a good estimate The last fifteen years have proven to Erdős’ conjecture in the case where the But there was a major setback. After for the number of primes under a given be some of the most productive in the set is taken to be the primes. It is a great the proof had been released and was in magnitude, due to technical limitations history of number theory. Andrew Wiles’ theorem with very deep consequences. the process of being scrutinized by the and can only sieve out numbers with small proof, using concepts from elliptic curve Considering that the largest known global mathematics community, a mistake 2 prime factors. theory, shook the world. In addition, arithmetic progression of primes to was discovered. However, through the this day is only of about twenty terms, expertise of Professor Pintz of the Rényi the statement about the existence of 2 Eratosthenes’ sieve fails quickly when trying to sieve out all the main term as we continue sifting, rendering the method only useful Institute, a way to correct the mistake was such primes as one can only find exact counts for how many numbers in developing asymptotic estimates for numbers with only large prime are divisible by a number d when the number d is fixed, otherwise, an factors. estimate must be used. The error term in this estimate encapsulates 134 135 AY 2007-2008 Hakan Seyalioglu: Number Theory and Educational Exchange found and the celebrated result stood. The of prominent Hungarian mathematicians themselves at it with such vigor if no from Kannan Soundararajan’s recent improvement was such a big step forward, and it provides a home base for an practical application to their work could work in moments of the Riemann zeta that many related problems’ results were extraordinary collection of contemporary ever be found? However, with the advent function to topics relevant to my research improved trivially by these new findings. researchers. of electronic communications and the such as Helmut Maier’s famous matrix It is on improving results related to some The Institute, housed in the heart of the need for security, the Queen found method (which was used in a follow-up to of these problems through these new city, has provided an excellent center from another avenue through which she would Goldston-Pintz-Yıldırım’s first paper to methods that my research has focused. which to conduct my project. My personal compete for attention. improve it by a linear factor). lodgings have been turbulent and there In modern times, number theory has In addition to these weekly meetings, in were a few occasions during which if I did great practicality. It serves as a basis for the first semester, I took two classes, one 3. Research, Classes not have access to the technical offerings innumerable cryptographic algorithms offered by the Rényi Institute’s Gergely and Exchange of the Institute, my work would have and one can be certain that practically Harcos on Elementary Methods in Prime suffered a major setback. They provided any virtual transaction one makes owes a Number Theory and a second, offered by In this section, the focus will be my excellent physical accommodations and great debt to the works of Euler, Fermat the Budapest Semesters in Mathematics project and my activities as a visiting access to an expansive library, where and Gauss. The notion of quadratic program, taught by Professor Balog, scholar. I will talk about my time at the I could be confident that any source residuosity has served as one of the first Topics in Number Theory. Both classes Rényi Institute and as a student as well as material I needed would be readily utensils for the theory of zero-knowledge served as a good way to acquaint myself about the notes I am completing. available. I recommend the Rényi protocols and discrete logarithms with these new topics. Professor Harcos’ My stay in Hungary has allowed me to Institute as a host for future Fulbright proved to be the backbone of the first presentation of Postnikov and Romanov’s develop greatly as a mathematician with Grantees without reservations and hope non-classified public key encryption elementary proof of the prime number attentive advisers and a wonderful work that the great interchange between algorithm, both central concepts in theorem was a highlight. I am including environment. I would be hard pressed the Commission and the Institute is modern cryptography. With my interest a similar proof in the notes for Professor to exaggerate, or even do justice to the maintained. in cryptography, a year to study number Szabó’s class. hospitality the Rényi Institute provided theory in Hungary was a perfect In the second semester, I shifted my me and the patience my advisers have opportunity. I would be allowed to study focus from classes and study to research. demonstrated as I learn this new branch 3.2 Learning a New Field alongside some of the most prominent I took a second class offered by Professor of mathematics. While most of my modern mathematicians in a style which Harcos entitled Modular Forms and personal research is not complete, I would While number theory has always been a great passion, it had not been a primary I might never have been exposed to in the L-Functions, which, while a little more like to take this opportunity to give some United States in a very relevant discipline. removed from the focus of my study, background on my host institution and focus of mine until this year. My personal research had been mostly restricted to proved to be a very rewarding class that the Budapest Semesters in Mathematics gave insight into some of the biggest program, my second affiliation. applications in cryptography. While I am 3.3 Independent Study not sure if analytic number theory will be recent discoveries in mathematic. It was my permanent focus, I decided to spend and Classes during this time that I also decided to take a part in the Budapest Semester in 3.1 The Alfréd Rényi Institute a year conducting research in number Serving as my advisers, Professors Antal theory because it is such an exciting field. Mathematics program to encourage their Founded by the Hungarian Academy Balog and András Biró have done more goal of educational exchange. Referred to as the Queen of Mathematics, to introduce me to higher mathematics of Sciences in 1949, the Alfréd Rényi number theory used to be a subject that Institute of Mathematics has housed than anyone in my academic career. Since seemed so far removed from applicability the beginning of the academic year, I one of the most prominent research that this ‘uselessness’ was a testament staffs in mathematics since its inception. have had weekly meetings with Professor its beauty. Why else would generation Biró where we have discussed recent Temporary and permanent members after generation of mathematician hurl have included an extremely high portion advancements in number theory, ranging

136 137 AY 2007-2008 Hakan Seyalioglu: Number Theory and Educational Exchange

3.4 Mathematics and Fulbright Commission in a mathematical The way the material is delivered is 4.1 Results and Conclusions setting, Professor Balog suggested I speak very different from most traditional Educational Exchange to Professor Csaba Szabó, one of the undergraduate courses in number theory. This section is dedicated to the current introductory number theory instructors, The course begins with distinguishing the I participated in the Budapest Semesters status of the project and the direction it about helping him finish compiling the concepts of irreducibility and primality, in Mathematics program, first, two years will be continued in during the end of the course notes for his class. which is often neglected in such courses. ago, as a student. It was my study abroad grant period and the upcoming summer. This introduction is considerably more experience as an undergraduate and one My research has been productive, algebraic than most, as these concepts, of the primary motivators in my decision however, as our results are not finalized, 3.5 Number Theory, while interchangeable when discussing to return to Budapest. It delivered a I have not included them in this report. Hungarian Style prime numbers over the integers, are two semester heavily focused on advanced They are related to small gaps between very important and separate concepts in groups of primes and use a modification mathematics, in a beautiful new culture, The above title is the subtitle for the algebra. In addition, the course focuses of the Goldston-Pintz-Yıldırım sieve with instruction by some of the most course notes for Professor Szabó’s course. heavily on problem solving, after all, half devoted teachers I had encountered. It in a related setting. Both advisers have of the course hours are dedicated problem expressed optimism about results and as was as a student that I met Professor I was enrolled in Professor Szabó’s class sessions. soon as the project is finished, I will make Dezs Miklós, who served as my first as an undergraduate and was so excited by ő Problem sessions are a concept the work available. There are very subtle contact when deciding to apply for a his teaching style that I also enrolled in his regrettably absent in most American points in modifying the argument, which Fulbright grant for study in Hungary. Galois Theory course, a topic far removed institutions. In his class, every Thursday, may take a substantial amount of time to He was excited to hear I was interested from what I saw as my academic focus. His Professor Szabó will distribute a long complete. in returning and put me in touch with teaching style is, as an understatement, sheet of problems, walk to the podium The course notes are still in their both Professors Balog and Biró. I owe a unique. There are not many professors of and exclaim several numbers. These infancy, as I began the project in the great debt to his generosity in helping me mathematics who introduce the topic of will be the first problems that the class second semester of my grant period. The arrange the specifics of my project as well quadratic residues with shouts and yells, attempts as Professor Szabó makes his first three chapters are virtually complete as being a very reliable source of advice. who think throwing chalk at the board round, observing the students’ attempts with about half left, corresponding to Budapest Semesters in Mathematics is an acceptable substitute for pointing at solutions and giving them advice. the current state of the lectures. I have owes its creation to the mathematical and who knowingly put erroneous This gives students a chance for personal inserted several appendixes to the notes, legends, László Lovász, László Babai, information on the board to keep students interaction with the teacher as well as one on number theory in public key Vera Sós, and Pál Erd s. In operation for attentive. A few years ago, a student had ő an opportunity for the professor to get a cryptography, corresponding to my more than twenty years, it provides an begun compiling notes for the course, better understanding of the class’ grasp of interest in the subject, as well as one on opportunity for English speaking students but as she was taking her first course in the course material. The absence of these an elementary proof of the prime number to take courses offered primarily by number theory at the time and was a full sessions from American classrooms is a theorem. The proof of the prime number professors from Eötvös Loránd University time student, the notes are incomplete shame. Perhaps the reason for the absence theorem that is being used is due to Hedi and the Rényi Institute. It boasts chances and in need of technical polish. Professor is the pressure such scrutiny places on Daboussi and is important in that it is for the students to learn, among other Szabó and I have begun working students; having a professor peering over the earliest that does not use Selberg’s topics, number theory and combinatorics together to finish them. I attend weekly one’s shoulder every week is certainly not Lemma. It is a very nice proof that was in Hungary, one of the countries most lectures and every week, deliver him my the most relaxing environment in which given to me by Professor Balog, which I dedicated to their study. These classes update of the typed notes along with my to learn. However, I found the experience have been translating from French, as I frequently feature the highest enrollment hand written notes. We try to capture very compatible with my own learning have not found a readily available English and both offer multiple sections. As I a conversational tone with occasional style and think it would be a good way to translation. I think that using this proof looked for an opportunity to contribute to anecdotes from the class thrown in for supplement courses for willing students. is a good choice because it is unique in the educational exchange mission of the variety.

138 139 relation to Selberg’s Lemma and is one 4.2 Works Cited that may not be available otherwise. I Egy amerikai az Amerikai have also included heuristics for the first [1] H. Davenport, Multiplicative Number Theory, Third elementary proof of the prime number Edition. Graduate Texts in Mathematics, No. 74, Springer- Kuckóban theorem. Verlag, (2000). I hope to continue my academic [2] J. R. Chen, On the representation of a large even integer as relationship with both my advisers the sum of a prime and the product of at most two primes, Kexue and Professor Szabó long after I leave Tongbao 17 (1966), 385-386. Budapest. If our ideas bear fruit, it is [3] J. R. Chen, On the representation of a larger even integer as likely that my advisers and I will continue the sum of a prime and the product of at most two primes, Sci. to apply them to similar problems. I see Sinica 16 (1973), 157-176. Matthew D. Smith my experience as a Fulbright Scholar [4] E. Szemerédi, On sets of integers containing no k elements as having affected my educational in arithmetic progression”, Acta Arithmetica 27 (1975), 199–245 experience in a very substantial way and [5] P. Erdős, P. Turán, On some sequences of integers, Journal of do not know of a way to demonstrate my the London Mathematical Society 11 (1936), 261–264...... gratitude. I hope to continue interacting [6] B Green, T Tao, Arxiv preprint math.NT/0404188 (2004) - with the Commission in the future and arxiv.org University of Pittsburgh University of Pécs hope Hungary remains a popular host [7] D Goldston, Y. Motohashi, J. Pintz, C. Y. Yildirim, Small Pittsburgh, PA 15260 7624 Pécs Ifújsag utca 6 country for students of mathematics. gaps between primes exist, Proc. Japan Acad., 82A (2006), 61-65. www.pitt.edu www.pte.hu [email protected] advisor: Dr. Kurdi Mária supervisor: Ms. Nagy Zsuzsanna

......

There are four American Corners in Hungary. My placement is at the American Corner in Pécs where I take on numerous roles. This paper describes my roles and responsibilities at the American Corner in Pécs. I pay particular attention to advising-related activities and the neccessitiy for me to seek opportunities to advise students at the Amereican Corner. I also decribe the academic and cultural programs that I helped the American Corner in Pécs present during the 2007-2008 academic year. Finally, I enlisted the help of my American Corner and Fulbright colleagues from the other American Corner cities to give their input and feedback on what an American grantee can do for an American Corner. The objective of this paper is to share my experiences so that future Fulbright grantees to Hungary can know how they can better interact with American Corners in Hungary.

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