Talk

Given a Lie algebroid structure on a vector bundle $A$ and a leaf $L$ of this Lie algebroid, a natural question is whether all nearby Lie algebroid structures on A have a leaf which is diffeomorphic to $L$. This question was answered by M. Crainic and R. Fernandes. In this talk, I will show that this question is an instance of a general question about differential graded Lie algebras: Given a differential graded Lie algebra $\mathfrak{g}$, a differential graded Lie subalgebra $\mathfrak{h}$, and a Maurer-Cartan element $Q$ of the subalgebra, are all Maurer-Cartan elements of $\mathfrak{g}$ near $Q$ gauge equivalent to an element of $\mathfrak{h}$? In the case that $\mathfrak{h}$ has finite codimension in $\mathfrak{g}$, I will give a sufficient criterium for a positive answer to the question. As a consequence, I will mention some stability results for zero-dimensional leaves of various geometric structures that can be obtained from this.

Contact: Aru Ray

Hybrid. Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

Euler sums are real numbers defined by iterated integrals on the projective line minus $0,\infty,1,-1$.  In this talk, we introduce a family of linear relations among Euler sums which exhausts all motivic linear relations. This gives an explicit description of the level two motivic Galois group. We also show that the level two motivic Galois group coincides with the cyclotomic Grothendieck-Teichmüller group introduce by Benjamin Enriquez. Some part of this talk is based on a joint work with Nobuo Sato.

Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

A theorem of Serre states that almost all plane conics over Q have no rational point. We prove an analogue of this for a family of conics parametrised by certain elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve. Another way to think about this result is: we show that 0% of points on elliptic curves have a denominator which is a sum of two squares.

Hybrid. Contact: Christian Kaiser (kaise @ mpim-bonn.mpg.de)

In this talk, I will explain how quantum toroidal algebras lead to a novel family of commuting difference operators whose joint eigenfunctions are the wreath Macdonald polynomials. The latter polynomials arise from the study of symplectic quotient singularities involving the wreath products of symmetric groups with an arbitrary finite cyclic group. When the cyclic group is trivial, one recovers the usual (type A) Macdonald difference operators and polynomials, whose theory has been extensively developed. In contrast, very little is known about wreath Macdonald polynomials in general. Our 'wreath Macdonald operators' provide a new, more direct characterization of wreath Macdonald polynomials for arbitrary finite cyclic groups. This is joint work with Mark Shimozono and Joshua Wen.

Hybrid. Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

Landau was the first to obtain an asymptotic formula for the number of integers up to a given number that are sum of two coprime squares, where he used analytic method. Later, this procedure was further developed by Delange and Selberg allowing them to obtain asymptotic for partial sums of arithmetic functions whose Dirichlet series can be written in terms of complex powers of the Riemann zeta-function. In 1967 Levin and Fainleib using differential equation established the logarithmic density of the same set by an elementary argument under more general conditions. We generalise the Levin and Fainleib approach and allow more general hypotheses as well. When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for the mean value of f(n)/n when summed over n <x provided that we have the mean value over primes, namely, f(p)(logp)/p summed over p < Q.

This is a joint work with Olivier Ramare and Rita Sharma.

A classical set of equidistribution problems concerns torus orbits of growing discriminants inside projective units of quaternion algebras. Examples include Heegner points or closed geodesics on the modular surface, integral points on ellipsoids, and supersingular reduction of CM elliptic curves.  A new generation of equidistribution problems, motivated by the mixing conjecture of Michel and Venkatesh, considers torus orbits inside pairs of quaternion algebras which in special cases has a rather concrete arithmetic description. I will show how automorphic forms and L-functions shed light on such questions.

 

I will discuss some joint work (in progress) with Ichino. In a previous paper, we showed that the Jacquet-Langlands correspondence for cohomological Hilbert modular forms preserves rational Hodge structures, as predicted by the Tate conjecture. In this talk, I will discuss a related result in the case of weight one forms. Since weight one forms are not cohomological, the Tate conjecture does not apply and thus it is not at all obvious what the content of such a result should be. I will motivate and explain the statement, which is suggested by another recent development, namely the conjectural connection between motivic cohomology and the cohomology of locally symmetric spaces.

We consider the problem of whether the diagonal restrictions of a Hilbert theta series over a totally real number field span the whole space of elliptic modular forms. We transfer this problem to definite quaternionic Hilbert modular forms via theta correspondences and get the surjectivity of the restriction map by applying equidistribution properties of unipotent orbits in SL(2) as well as Jacquet-Langlands correspondences.