On Thursday and Friday, August 18 and 19, 2016 we will hold a workshop in **Number Theory and Dynamics** at Chalmers University of Technology and University of Gothenburg in Sweden. The workshop is funded by GoCAS, the Gothenburg Center of Advanced Studies.

If you intend to come, please send email to one of the organizers. We currently cannot provide financial support for any participants except the speakers.

Landvetter Airport (airport code GOT) is close to Gothenburg and has international flights. You can come from the airport to the city:

- By taxi. Ask for fastpris to get a fixed price.
- By airport bus and public transportation
- The station for Flygbussen is next to the airport exit. You can purchase tickets on the bus. The bus brings you to the city.
- Public transportation in Gothenburg is decent. You can look up your further iternary on the Västtraffic website.

The conference will take place in the math building on the campus of Chalmers University of Technology, Tvärgata 3.

Talks take place in the lecture hall Pascal in the math building, room number H2022. A map is available - you can savely ignore the Swedish text on linked webpage.

The organizers are

- Dennis Eriksson dener@chalmers.se,
- Michael Björklund micbjo@chalmers.se, and
- Martin Westerholt-Raum raum@chalmers.se.

We are available for further inquiries at the above email addresses.

Confirmed speakers are:

- Gunther Cornelissen, Universiteit Utrecht
- Manfred Einsiedler, ETH Zurich
- Ksenia Fedosova, University of Bonn
- Alexander Gorodnik, University of Bristol
- Pär Kurlberg, KTH, Stockholm
- Anke Pohl, MPIM Bonn
- Andreas Strömbergsson, Uppsala University
- Trevor Wooley, University of Bristol

Local participants are:

- Julia Brandes
- Simone Calogero
- Laura Fainsilber
- Magnus Goeffeng
- Hanna Oppelmayer
- Ulf Persson
- Hjalmar Rosengren
- Per Sahlberger
- Mehrdad Taki
- Manh Hung Tran
- Elizabeth Wulcan

Participants from outside Gothenburg are:

- Kévin Destagnol, Institut de Mathématiques de Jussieu-Paris Rive Gauche
- Samual Edwards, University of Uppsala
- Lars Halle, KU Copenhagen
- Gustav Hammarhjelm, University of Uppsala
- Yair Hartman, Northwestern University
- Nika Laaksonen, KTH, Stockholm
- Junghun Lee, Nagoya University
- Pieter Moree, MPI for Mathematics, Bonn
- Fabien Pazuki, KU Copenhagen
- Olav Richter, UNT Texas

Thursday | Friday | |
---|---|---|

09.00 - 10.00 | Einsiedler | Gorodnik |

Break | Break | |

10.20 - 11.20 | Strömbergsson | Pohl |

Break | Break | |

11.30 - 12.30 | Cornelissen | Wooley |

Lunch | ||

14.00 - 15.00 | Kurlberg | |

Break | ||

15.20 - 16.20 | Fedosova | |

19.00 | Dinner |

I will show how to reconstruct the Ihara zeta function of a graph from its collection of edge-deleted subgraphs. In this way, one may recover several invariants of the action of the fundamental group of the graph on the boundary of its universal covering and the associated Patterson-Sullivan measures.

In joint work with E. Lindenstrauss we obtained the classification of positive entropy measures for new cases of diagonal actions.

To a hyperbolic surface and a finite-dimensional representation of its fundamental group, we associate a Selberg zeta function. The main goal of the talk is to show that under certain conditions, the Selberg zeta function admits a meromorphic extension to the whole complex plane. Our main tool is the use of transfer operators.

This is joint work with Anke Pohl.

We discuss discrepancy of distribution of the set of rational points lying on algebraic groups, and describe an approach for obtaining upper and lower bounds on discrepancy. It turns out that this problem leads to an interesting interplay between arithmetic geometry and the theory of automorphic representations.

This is a joint work A. Ghosh and A. Nevo.

The Laplacian acting on the standard two dimensional torus has spectral multiplicities related to the number of ways an integer can be written as a sum of two integer squares. Using these multiplicities we can endow each eigenspace with a Gaussian probability measure. This induces a notion of a random eigenfunction (aka “random wave”) on the torus, and we study the statistics of the lengths of nodal sets (i.e., the zero set) of the eigenfunctions in the “high energy limit”. In particular, we determine the variance for a generic sequence of energy levels, and also find that the variance can be different for certain “degenerate” subsequences; these degenerate subsequences are closely related to circles on which lattice points are very badly distributed. Time permitting we will discuss which probability measures on the unit circle that “comes from” lattice points on circles.

Given a Riemannian locally symmetric space, bounds for eigenfunctions of the Laplace operator or for joint eigenfunctions of the whole algebra of isometry-invariant differential operators are of great interest in several areas. For example, sup-norm estimates are intimately related to the multiplicity problem and to questions of quantum unique ergodicity. Methods from analysis allow to provide bounds (nowadays called `generic’) which are sharp for certain spaces.

If the Riemannian locally symmetric space is arithmetic and one restricts the consideration to the joint eigenfunctions of the algebra of differential operators and of the Hecke algebra then it is reasonable to expect that the generic bounds can be improved.

The archetypical result of such kind is due to Iwaniec and Sarnak in the situation of the modular surface and several other arithmetic Riemannian hyperbolic surfaces, dating back to 1957. In the following years, their way of approach was (and still is) used to deduce many similar results for various spaces of rank at most 1, and was only recently adapted to some higher rank spaces.

The first example of a subconvexity bound for a higher rank setup was provided by our joint work with Valentin Blomer, which we will discuss in this talk.

We study the error term R in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is a central limit theorem for R in the limit as n and x simultaneously tend to infinity, with x growing subexponentially with respect to n. We also prove a functional version of the result, giving convergence to Brownian motion. The proof goes via convergence of moments, and for the computations we develop a new version of a mean value formula over the space of lattices of C. A. Rogers. For the individual k:th moment of the variable R (normalized) we prove convergence to the corresponding Gaussian moment more generally for x allowed to grow as rapidly as exp(cn) with c>0 sufficiently small, and we determine the exact range of c-values for which this convergence holds.

Joint work with Anders Södergren (Copenhagen/Chalmers).

The subconvexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this subconvexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.