SUMMER SCHOOL
Analysis and Number Theory

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Topics for the participants' lectures

Every participant will lecture on one of the following papers. Some papers are longer and will be discussed by two of the participants. Some of the papers will only be read partially, more detailed instructions are below. The papers are not listed in the order of difficulty nor in the order they will be presented, but somewhat ordered according to the subject. If you have problems obtaining any of these papers, the organizers can provide you with copies.

Every participant will lecture for 2 hours, split into two lectures of one hour each, one in the first half and one in the second half of the conference week.

Every participant submits a 4-6 page summary of his or her topic prior to the summer school. Preferrably in some form of tex or latex. A sample file can be retrieved here. Deadline for submission of the summary is August 15.

  1. A new iterative method in Waring's problem (R. C. Vaughan, Acta Math. 162 (1989) pp 1-72) Part I. We discuss only the case of Warings problem for $k=5$ one can trace back form section 5 what is needed to this end. Part I gives a brief introduction to the problem and presents Section 2 in detail (to the extend needed for $k=5$) and scetches (may be brief) Section 3 (to the extend needed for $k=5$). (Nick Crawford)

  2. A new iterative method in Waring's problem (R. C. Vaughan, Acta Math. 162 (1989) pp 1-72) Part II We discuss only the case of Warings problem for $k=5$ one can trace back form section 5 what is needed to this end. Part II gives a brief introduction to the problem and presents Section 4 and 5 (to the extend needed for $k=5$). (Georgiy Arutytunyants)

  3. On the order of z(1/2 + it) (E. Bombieri and H. Iwaniec, Annali della Scuola Normale Superiore de Pisa, classe de scienze, Ser IV Vol XIII, 3 (1986) pp. 449-472) This paper has a companion paper (see next topic), and thus can also be considered as Part I of a series. Indeed, we will leave the discussion of Lemmata 2.1,2.2,2.3,2.4, and 2.9 to the next topic. (Nets Katz)

  4. Some mean value theorems for exponential sums. (E. Bombieri and H. Iwaniec, Annali della Scuola Normale Superiore de Pisa, classe de scienze, Ser IV Vol XIII, 3 (1986) pp. 473-486) This paper has a companion paper (see previous topic), and thus can also be considered as Part II of a series. We mainly need the result of this article to the extend it is used in Part I. Also, to share the burden more evenly with Part I (though Part II does require some outside reading) this topic is also assumed to cover Lemmata 2.1,2.2,2.3,2.4, and 2.9 of the paper "On the order of z(1/2+it)" (Silvius Klein)

  5. Equidistribution of roots of quadratic congruence to prime moduli (Duke, Iwaniec, Friedlander, Ann. Math 141 (1995) no. 2 pp. 423-444. We are after the main theorem in this paper and its proof. Sections 2 and 3 require some outside reading. This outside material should also be presented or scetched in the talk to the extend reasonable. (Amanda Folsom)

  6. Fourier coefficients of modular forms of half integral weight (HJ. Iwaniec, Invent. Math. 87 (1987) pp 385-401) We are after Theorem 1 and its proof. This requires some outsiude reading on modular forms, which should be presented or sketched to the extend reasonable. The book by Sarnak "Some applicatiosn of modular forms" Cambridge Tracts in Mathematics 99, Cambridge University Press 1990 makes a good compnion reading to this lecture. (Nate Jones)

  7. Roth's theorem in the primes (Ben Green, avalilable through the arXiv, see Ben Green's webpage at www.dpmms.cam.ac.uk/~bjg23/) We present Theorem 5 in detail (depending on what time allows, some ingredients such as e.g. proposition 9 should be presented or sketched). There will probably be no time for Theorem 4. (John Bueti)

  8. Discrete analogues in harmonic analysis. Spherical averages. (Magyar, Stein, Wainger. Ann. math (2) 155 (2000) no.1 pp. 189-208 The lectures should present the main theorem of this paper and its proof. (Monica Visan)

  9. A new proof of Szemeredi's theorem for arithmetic progressions of length four. (T. Gowers. GAFA Vol. 8 (1998) pp. 529-221) The main theorem is Theorem 20 near the end of the paper. This theorem and its proof is the main objective of this lecture. We need Freiman's theorem, which is proved in the next topic. (Tonci Crmaric)

  10. Generalized arithmetic progressions and sumsets (I. Ruzsa, Acta Math. Hung. 65 (1994) pp. 379-388.) Part I We are interested in Freiman's theorem and Rusza's proof thereof. Instead of using Ruzsa's paper, one can find a summary of his proof in the book "Additive Number theory: Inverse theorems and the number of sums and products" by M. Nathanson, Springer New York 1996. Part I deals with sections 8.1,8.2,8.4 of chapter 8 in Nathanson's book, Section 8.3 is discussed in the next two topics. This lecture also presents the relevant theorems in chapter 7 (Plünnecke's inequality). If time allows (probably not) one could discuss the polynomial bound in "A polynomial bound in Freiman's theorem" (Mei-Chu Chang, Duke Math. J. 113 no 3 (2002) pp 399-419 The lecture presents and proves Theorem 1. This requires some reading in Nathanson's book (see references). (Lei Wu)

  11. Generalized arithmetic progressions and sumsets (I. Ruzsa, Acta Math. Hung. 65 (1994) pp. 379-388.) Part II We are interested in Freiman's theorem and Rusza's proof thereof. Instead of using Ruzsa's paper, one can find a summary of his proof in the book "Additive Number theory: Inverse theorems and the number of sums and products" by M. Nathanson, Springer New York 1996. Part II deals with section 8.3 "Bogolyubov's method. This requires reading and presenting some material from chapter 6 "geometry of numbers" (Ali Gurel)