Institutionen för matematik, KTH | Matematiska institutionen, Uppsala

DNA-seminariet (Dynamiska system, talteori, analys)

Nästa seminarium:
December 18, Monday, 2006, kl 13.15-14.15, KTH-Matematik, rum 3721.
Speaker: Tsachik Gelander, Hebrew University, Israel
Title: For n>1, a generic n--tuple of elements in a connected compact non-abelian Lie group G generates a free group. Margulis and Soifer conjectured that every such tuple can be slightly deformed to one which generates a group which is not virtually free. I will explain a proof of this conjecture, and actually show that for n>2 and for an arbitrary dense subgroup D, with some restriction on the minimal size of a generating set, the set of deformations of F_n whose image is D is dense in the variety of all deformations. The proof relies on the product replacement method. Using the same ideas I will also give a proof of a conjecture of Goldman on the ergodicity of the action of  Out(F_n) on Hom(F_n,G)/G when n>3. For n=2, I will explain how to produce for any pair (a,b) an arbitrarily close pair (a',b') which generates an infinite group which has Serre property FA and in particular is not virtually free.

Previously:

December 11, Monday, 2006, kl 13.15-14.15, KTH-Matematik, rum 3721.
Speaker: Stephan Baier, IU Bremen, Germany
Title: The Sato-Tate and Lang-Trotter conjectures about elliptic curves on average
Abstract: Let $E$ be an elliptic curve over $\ratq$. For any prime number $p$ of good reduction, let $\lambda_E(p)$ be the trace of the Frobenius morphism of $E/\fie_p$. Then the number of points on the reduced curve modulo $p$ equals $p+1- \lambda_E(p)$. By Hasse's theorem, there exists a unique angle $0\le \theta\le \pi$ such that $$\lambda_E(p)=\sqrt{p}\left(e^{i\theta_E(p)}+e^{-i\theta_E(p)}\right)= 2\sqrt{p}\cos \theta_E(p).$$ For the case when $E$ does not have complex multiplication, Sato and Tate formulated a conjecture on the distribution of $\theta_E(p)$ as $p$ varies. In a recent preprint, R. E. Taylor succeeded in proving the Sato-Tate conjecture for elliptic curves $E$ that satisfy a certain mild condition.\newline In this talk, we will discuss the Sato-Tate distribution in {\it small} sectors on average over a family of elliptic curves. Moreover, we will talk about the Lang-Trotter conjecture on average. The last-mentioned conjecture predicts an asymptotics for the number of primes $p$ with $\lambda_E(p)=r$, where $r$ is fixed. \newline Some of these results are joined work with Liangyi Zhao.
December 11, Monday, 2006, kl 14.45-15.45, KTH rum D33.
Speaker: Francis Brown, Institut Mittag-Leffler
Title: Arithmetic of zeta values and the irrationality of $\zeta(2)$ and $\zeta(3)$.
Abstract: A folklore conjecture states that the values of the Riemann zeta function at odd positive integers are algebraically independent over $\Q$. The main results in this direction are that $\zeta(3)$ is irrational (Ap\'ery), and that there are infinitely many irrational numbers amongst $\zeta(2n+1)$ (Rivoal). It is not known, however, whether $\zeta(5)$ is irrational or not. In this talk, I will begin by surveying the known and conjectured diophantine properties of multiple zeta values. I will then explain Beukers' elementary proof that $\zeta(3)$ is irrational, and describe the group structures corresponding to $\zeta(2)$ and $\zeta(3)$, which give the best bounds for their irrationality measures known to date. In the last part of the talk I will explain how the moduli spaces of curves of genus 0 give a unifying geometric interpretation of these proofs.
Sept 11, Monday, 2006, kl 13.15-14.15, KTH-Matematik, rum 3721.
Speaker: Jens Marklof, University of Bristol, U.K.
Title: Spectral Theta Series
Abstract: The theta series $\vartheta(z)=\sum \exp(2\pi\i n^2 z)$ is a classical example of a modular form. In this talk we argue that the trace $\vartheta_P(z)=\Tr\, \exp(2\pi\i P^2 z)$, where $P$ is a first-order self-adjoint pseudo-differential operator with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of $\vartheta_P(z)$ near the real axis, and the proof of logarithm laws and limit theorems for its value distribution.

Sept 21, Thursday, 2006, kl 13.15-14.15, KTH-Matematik, rum 3721.
Speaker: John Friedlander, University of Toronto, Canada
Title: Hyperbolic Prime Number Theorem
Abstract: It is known since Fermat and Euler that prime numbers of the form $4n+1$ are precisely the ones (in addition to 2) which can be written as the sum of two squares. Because of the simplest case of the prime ideal theorem this means that we can count asymptotically the number of primes $p=x^2+y^2$ within a large disc $x^2+y^2 \le X$ in the Euclidean plane. In joint work with Henryk Iwaniec we study some natural generalizations of this question with particular emphasis on analogues concerning points in a large disc in the hyperbplic plane.


Sept 4 , Monday, 2006, kl 13.15-14.15, Uppsala, rum 64119.
Speaker: Fredrik Stromberg, Inst. of Theoretical Physics, TU-Clausthal, Germany
Title Transfer Operators for Hecke Triangle Groups
Abstract: It is well-known that the classical geodesic flow on the modular surface can be coded with the help of simple (Gauss) continued fractions and that the reduction theory of binary quadratic forms can be used to establish a connection between the transfer operator of the geodesic flow and the Selberg zeta function of the surface. In this setting it is also possible to relate eigenfunctions of the transfer operator directly to Maass waveforms via functional equations and cohomology (Lewis-Zagier theory of period functions). I will talk about work by Tobias Mühlenbruch, Dieter Mayer and myself aimed at generalizing these kinds of results from the Modular group to Hecke triangle groups.

    Detta är ett nytt gemensamt seminarium mellan KTH och Uppsala som är tänkt att presentera nya resultat och metoder inom gränsområdena mellan dynamiska system, talteori och analys. Avsikten är att detta ska ses ur ett brett perspektiv, så att även besläktade aspekter av ergodteori, kvantkaos, statistisk mekanik, sannolikhetsteori och matematisk fysik tas med. Vi planerar för närvarande att seminariet ska hållas varannan vecka. Målsättningen är att föredragen ska rikta sig till en bred publik, bland annat genom att tydligt peka på öppna problem. Doktorander är särskilt välkomna att delta i seminariet.

The DNA seminar (Dynamical systems, Number theory, Analysis)

    This new joint seminar between KTH and Uppsala is intended to highlight some of the recent developments occurring at the intersection of dynamics, number theory, and analysis. We plan for the seminar to take a wide view, so included here will also be related aspects of ergodic theory, quantum chaos, statistical mechanics, probability, and mathematical physics. We currently anticipate the seminar will meet twice monthly. The talks will be designed to appeal to a wide audience, featuring for instance clear indications of open problems. Graduate students are especially invited to participate.

9 februari 2006, kl 13.15-14.15, KTH, Seminarierum 3721.

Moon Duchin (University of California, Davis): Random moves in the space of metrics

Abstract. For a fixed topological surface, the Teichmuller space is a parameter space for its possible metrics. These are considered up to conformal equivalence and with a "marking" by a choice of curves. I will discuss random walks in Teichmuller space where the moves are changes of marking. Adapting a geometric strategy from work of Karlsson-Margulis, I will show that sample paths for this random walk are tracked by geodesics in the Teichmuller space, which have a nice description as metric deformations.

2 mars 2006, kl 13.15-14.15, Uppsala, rum 3:513.

Jeffrey Steif (Chalmers): Statistical mechanical systems on complete graphs, infinite exchangeability and finite extensions

Abstract. We discuss the notion of infinite exchangeability in the context of statistical mechanical systems. It turns out that many ferromagnetic systems (like Ising, Potts and Heisenberg models) can be extended to so-called infinite exchangeable processes. For the antiferromagnetic Ising model, we obtain sharp quantitative results on how much the system can be extended. One approach for doing this is via a solution of a new type of moment problem. This is joint work with Tom Liggett and Balint Toth.

6 mars 2006, kl 13.15-15.00, KTH, sal D35 (Två timmar!).

Michael Benedicks (KTH): Non uniformly hyperbolic attractors - invertible and non-invertible.

Abstract. I will describe the theory of Hénon attractors and Hénon-like attractors with chaotic behaviour and the corresponding ergodic theory: existence of SRB measures and the description of their basins, decay of correlation, stability under random perturbations. The maps in Hénon- and Hénon-like case are diffeomorphims. Recently, Pesin and Yurchenko have noticed similar phenomena for non-inversible maps appearing from coupled map lattice models of certain PDES. I will discuss an attempt to generalize the theory of Hénon- and Hénon-like maps to certain non-invertible cases (work in progress).

20 april 2006, kl 13.15-14.15, Uppsala, rum 3:513.

Andrew Booker (University of Michigan): Convergent Hejhal-type algorithms

Abstract. There have been many numerical investigations of the spectrum of the Laplace operator on non-compact, finite volume hyperbolic surfaces. The algorithm of D. Hejhal has proven to be robust and has yielded good results in many cases. However, there is no rigorous proof of either the convergence of the algorithm or that the results it gives are correct. In the talk I will give an overview of Hejhal's algorithms for the case of the modular group, and discuss some recent joint work with A. Strömbergsson and A. Venkatesh in which we compute and prove correct the first few eigenvalues to high precision. I will then show how to adapt the method to give algorithms similar to those of Hejhal, but for which one can prove convergence. If time permits I will discuss some related questions concerning large eigenvalue computations.

24 april 2006, kl 13:15-14:15, KTH, sal 3721.

Peter Storm (Stanford University): Topological lower volume bounds for hyperbolic 3-manifolds

Abstract. I will outline some of the geometric and topological tools used to obtain lower volume bounds for hyperbolic 3-manifolds with boundary. I'll focus on a few specific examples which should elucidate the origin of these lower bounds. The techniques are mostly based on ideas of Besson-Courtois-Gallot. However, the recent work of Perelman can be applied to yield strictly stronger results.

2 maj 2006, kl 15.15-16.15, KTH, sal 3733.

Jean Lafont (Ohio State University): Simplicial volume of locally symmetric spaces of non-compact type.

Abstract. I will define simplicial volume of a topological manifold, and motivate the interest in the positivity of this invariant. I will then outline a proof that the simplicial volume of locally symmetric spaces of non-compact type is in fact positive. This was joint work with Ben Schmidt.

11 maj 2006, kl 13.15-14.15, Uppsala, rum 3:513.

Mark Pollicott (Warwick University): Hausdorff dimension for randomly perturbed self-affine attractors

Tidigare seminarier: hösten 2005 (fall 2005)

12 september, Anders Öberg (Uppsala): Uniqueness of $g$-measures

12 september, Andreas Strömbergsson (Uppsala): Maass waveforms and Galois representations

3 oktober 2005, Dmitry Jakobson (McGill University): On nodal sets, critical points, and quasi-symmetry properties of eigenfunctions

17 oktober 2005, Andrew Booker (University of Michigan): On decidability of Artin's conjecture

12 december 2005, Yiannis Petridis (City University of New York, Lehman College): Distribution of closed geodesics with constraints

våren 2005 (spring 2005)

28 februari, Jörg Schmeling (Lund): Estimating the dimension of a model arising from horospherically tame Kleinian groups

14 mars, Stefan Lemurell (Chalmers): Deformations of Maass forms

14 mars, David Farmer (AIM): Derivatives of polynomials and L-functions

14 april, DNA Workshop: Number Theory and Quantum Chaos

18 april, Andreas Juhl (Humboldt-University, Berlin): Automorphic distributions, Selberg zeta functions and conformal geometry

21 april, Carles Simó (Universitat de Barcelona): Chaos and quasi-periodicity in diffeomorphisms of the solid torus

21 april, Robert MacKay (Warwick Mathematics Institute): Coupled map lattices with non-unique phase

25 april, Eric Bedford (Indiana University): Characterization of the horseshoes in the Henon family

2 maj, Peter Forrester (University of Melbourne): Sampling from eigenvalue distributions for matrix ensembles

9 maj, Peter Walters (Warwick Mathematics Institute): Cohomology for Shift Transformations

hösten 2004 (fall 2004)

20 september, Charles Favre (Université Denis Diderot): Equidistribution of points of small heights

27 september, Martin Olbrich (Technische Universität Clausthal and Georg-August-Universität Göttingen): Relations between length spectrum and topology of certain infinite volume hyperbolic manifolds

11 oktober, Dimitri Gioev (KTH): Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles

1 november, Torbjörn Lundh (Chalmers): DNA has not all the answers

15 november, Jens Marklof (University of Bristol): Recent developments in quantum chaos and number theory

29 november, Thomas Guhr (Matematisk fysik, LTH, Lunds universitet): Random Matrices in Physics and Supersymmetric Methods

29 november, Kurt Johansson (KTH): Determinantal processes and number variance saturation

20 december, Holger Then: Maass waveforms and an application to cosmology

våren 2004 (spring 2004)

5 april, Andreas Strömbergsson: On the values of a random linear form modulo one

19 april, Anders Karlsson: Metrics, isometries, and ergodic theory

3 maj, Mark Pollicott (University of Manchester): Zeta functions and hyperbolic dynamical systems

17 maj, Peter W. Jones (Yale University): Schwarzian derivatives for SLE mappings and approximation by Julia Sets

17 maj 2004, Marco Martens (University of Groningen): Renormalization of Lorenz systems