SUMMER SCHOOL
Analysis and Number Theory
Home page
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.
-
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)
-
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)
-
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)
-
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)
-
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)
-
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)
-
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)
-
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)
-
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)
-
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)
-
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)