ErgodicPNT

We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We get some first results on the number of prime divisors of ``most'' elements of an elliptic divisibility sequence, and we develop in some detail ``probabilistic'' sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some step of a random walk on SL(n,Z). In addition to the sieve principle, the applications depend on bounds for a large sieve constant. To prove such bounds involves a variety of deep results, including Property (T) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields. It seems likely that this sieve can have further applications.

Random matrix theory has been a hot topic in number theory, particularly since the Rudnick and Sarnak landmark work on the spacing of consecutive zeros of L-functions. This highly accessible talk has a far more elementary flavour, focusing on eigenvalues of random integer matrices instead of the Gaussian Unitary Ensemble. For a fixed n, consider a random n×n integer matrix with entries bounded by the parameter k. I'll give a simple proof that such a matrix almost certainly has no rational eigenvalues (as k increases). Then we'll delve into more detail on the exact eigenvalue distribution of the 2×2 case. Along the way we'll rediscover a forgotten determinant identity and tackle some quadruple sums. This is joint work with Greg Martin.

A new norm related to the Gowers U^3 norm Add to your list(s) Download to your calendar using vCal Pablo Candela Pokorna Monday 16 February 2009, 16:00-17:00 MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB. If you have a question about this talk, please contact Anton Evseev. The uniformity norms (or U^d norms, for d>1 a positive integer) were introduced about ten years ago by Gowers in his effective proof of Szemerédi’s theorem, and have played an important role in arithmetic combinatorics ever since. The U^2 norm is naturally related to Fourier analysis, and a very active trend in current research aims to develop an analogue of Fourier analysis for each U^d norm with d>2. The body of results of this research for d=3 is known as quadratic Fourier analysis. After an introduction to this area we will consider a new norm related to the U^3 norm, and discuss some of its applications in quadratic Fourier analysis, including a strengthening of a central theorem of Green and Tao (the inverse theorem for the U^3 norm), and how this stronger version of the theorem can be used to give a new proof of a recent decomposition-theorem of Gowers and Wolf. This talk is part of the Junior Algebra/Combinatorics/Number Theory seminar series.

Hall, Brian C. Lie groups, Lie algebras, and representations. An elementary introduction. Graduate Texts in Mathematics, 222. Springer-Verlag, New York, 2003. This is only a recommended text, but it is highly recommended. By emphasizing matrix groups, the book covers most of the important examples occuring in nature while avoiding a lot of the technical difficulties necessary in a more general treatment. It gives an excellent presentation of most of what we'll talk about. I think it will be a great book to read to supplement the lectures. Looking around on the web, I found many copies that were very reasonably priced.

Humphreys, James E. Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. A classic. Would have been my choice for a textbook, but unfortunately only covers Lie algebras.

Fulton, William; Harris, Joe. Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. A beautiful book to read. Very useful for self-study. Bump, Daniel. Lie groups. Graduate Texts in Mathematics, 225. Springer-Verlag, New York, 2004. Perhaps too hard for beginners, but it contains an excellent collection of topics in the final part.

We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d small-biased generators with error ε2O(d) is a pseudorandom generator against degree d polynomials with error ε. This gives a generator with seed length 2O(d) log(n/ε). Our construction follows the recent breakthrough result of Bogadnov and Viola. Their work shows that the sum of d small-biased generators is a pseudo-random generator against degree d polynomials, assuming the Inverse Gowers Conjecture. However, this conjecture is only proven for d=2,3. The main advantage of our work is that it does not rely on any unproven conjectures.

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.

Résumé - AbstractLet K=ℚ(-D) be an imaginary quadratic field, and denote by h its class number. It is shown that there is an absolute constant c>0 such that for sufficiently large D at least c·h∏ p∣D (1-p -1 ) of the h distinct L-functions L K (s,χ) do not vanish at the central point s=1/2.

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I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime k-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes p for which the gap q-p between p and the next prime q is smaller than ε times the average gap, for arbitrary small ε>0.