- Tim Austin (Brown) - Equidistribution of joinings under off-diagonal polynomial flows. (abstract)
Furstenberg proved the original Multiple Recurrence Theorem for probability-preserving systems in order to give an ergodic-theoretic proof of Szemeredi's Theorem in additive combinatorics. In the thirty years since that work, the study of the multiple ergodic averages that underlie proofs of multiple recurrence has become a sophisticated theory in its own right. In addition to the positivity that implies multiple recurrence, their convergence and a description of their limits are of particular interest.
I will discuss some recent work on the continuous-time multiple averages associated to a tuple of polynomial flows in an acting nilpotent Lie group. This work relies on the formulation of convergence and recurrence phenomena in terms of the equidistribution of certain self-joinings on a Cartesian power of the original system under an off-diagonal polynomial flow. I will emphasize two key ingredients in the proof of this equidistribution that seem to indicate a parallel with the study of unipotent flows on homogeneous spaces, although the technical details of the proofs remain quite different. The first ingredient is a kind of measure-classification that constrains the possible structure of any subsequential limit joinings, and builds on several older works studying `characteristic factors'. The second ingredient is an auxiliary result promising that given a family of such off-diagonal flows parametrized by polynomials in some extra real parameters, a `generic' flow in this family (suitably interpreted) gives equidistribution to a limit joining that is independent of the parameter and is invariant under the group of off-diagonal transformations generated by all the flows in the whole family: this amounts to a kind of Pugh-Shub phenomenon among joinings.
- Don Blasius (UCLA) - Recent progress concerning existence of automorphic forms of prescribed type. (abstract)
We will review recent work and open problems. Especially we will discuss recent work of W. Conley, extending to unitary groups work of J. Weinstein for Hilbert modular forms, on construction of automorphic forms of prescribed local type (up to suitable unramified twists) at each place. Several conjectures and open problems for cusp forms of regular algebraic type at infinity will be given.
- S.G. Dani (TIFR) - Complex continued fractions and
values of binary quadratic forms
on Gaussian integers. (abstract)
For a non-degenerate quadratic form in 3 or more complex
variables it is known that the set of values on the tuples of
Gaussian integers form a dense set of complex numbers, unless
the form is a scalar multiple of a rational form. For binary
forms however the analogue of this does not hold, and there is
no simple characterization of the forms for which the value-set
is dense. A similar phenomenon holds for indefinite real quadratic
forms as well, with regard to the values on integer pairs. For
$(x-ay)(x-by)$ some answers to the question are known in terms
of the (classical) continued fraction expansions of $a$ and $b$.
In this talk we shall discuss an analogous development of continued
fractions for complex numbers in terms of the Gaussian integers,
and applications to values of complex binary forms on the set of
pairs of Gaussian integers.
- Manfred Einsiedler (ETH) - Measure rigidity for torus actions. (abstract)
In this talk we will discuss the measure rigidity theorem for maximal diagonalizable subgroups of homogeneous spaces.
Unlike the case of unipotent dynamics, we do not understand the full classification of invariant measures instead we have to
assume positive entropy. Another main difference with the case of unipotent flows is that the answer depends crucially on the
lattice in the given group (or in other words on the Q-structure of the given algebraic group). We will discuss these phenomena
and the general theorem. This is joint work with Elon Lindenstrauss.
- Alex Gamburd (UCSC/CUNY) - Generalization of Selberg's 3/16 Theorem and
Affine Sieve. (abstract)
TBD
- Alex Gorodnik (Bristol) - Large sieve for
arithmetic groups.
(abstract)
We develop a version of the large sieve for sets in arithmetic groups indexed by height functions, and discuss several applications. In particular, we explain how to compute the asymptotics of the number of elements with generic Galois groups. This is a joint work with Amos Nevo.
- Patrick Ingram (Colorado) - Post-critically finite maps. (abstract)
Post-critically finite maps (rational functions whose critical points all have finite forward orbit under iteration) have been of interest for some time in complex holomorphic dynamics, but have only recently been studied from the perspective of number theory. We will survey some recent results on the arithmetic of these rational functions, focussing on their distribution in the moduli space.
- Dmitry Jakobson (McGill) - L^p norms and limits of eigenfunctions on arithmetic flat tori. (abstract)
We shall give a survey of some old and more recent results
on limits of eigenfunctions on arithmetic flat tori, as well as related
results about L^p norms. If time permits, we shall also discuss analogous
results for solutions of the Schrodinger flow (the latter is joint work
with T. Aissiou and F. Macia).
- Florent Jouve (Orsay) - Probabilistic Galois theory on arithmetic groups. (abstract)
Let G be a connected reductive group over a number field k.
Up to considering a faithful k-representation of G, the splitting
field k_g of the characteristic polynomial of an integral point g of
G(Z_k) is a well-defined object. From the point of view of
probabilistic Galois theory it is natural to wonder what the typical
Galois group Gal(k_g/k) should be if g is picked at random. In this
talk I will report on joint work with E. Kowalski and D. Zywina where
we show that the probability for a random walk on G(Z_k) (defined
using a set of generators of the group of integral points of G) to
lead to a g in G(Z_k) whose Galois group Gal(k_g/k) is "as big as
possible" grows exponentially fast in terms of the number of steps.
- Dubi Kelmer (Boston) - On the spectrum and length spectrum of hyperbolic manifolds. (abstract)
There is a very close relation between the Laplace spectrum and the lengths of closed geodesics on negatively curved manifolds. For hyperbolic surfaces the Laplace spectrum and the length spectrum determine each other. However, in higher dimensions the connection is more mysterious. In this talk I will describe new results showing that the Laplace spectrum of a compact hyperbolic manifold is determined by its length spectrum, and discuss how much of the length spectrum is actually needed in order to recover the Laplace spectrum.
- Alex Kontorovich (Yale) - On Zaremba's Conjecture. (abstract)
It is folklore that modular multiplication is "random". Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. We will prove that a density one set satisfies Zaremba's conjecture. This is joint work with Jean Bourgain.
- Pär Kurlberg (KTH) - Nodal length statistics for arithmetic random waves. (abstract)
Using spectral multiplicities of the Laplacian acting on the
standard two-torus, we endow each eigenspace with a Gaussian probability
measure. This induces a notion of a random eigenfunction on the torus,
and we study the statistics of nodal lengths 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.
- Philippe Michel (EPFL) - Hybrid bounds for Rankin-Selberg L-functions. (abstract)
Inspired by some work of R. Holowinsky and R. Munshi, we explain how the period approach
to the subconvexity problem initiated by A. Venkatesh and expanded in our joint work, allied to sufficiently good approximations to the Ramanujan-Petersson
conjecture and to sufficiently good subconvex bounds for auxiliary L-functions, yields to subconvex bounds for $GL_2$ Rankin-Selberg L-function $L(\pi_1\times \pi_2,1/2)$ when both $\pi_1$ and $\pi_2$ are allowed to vary (but under the condition that they
remain "away" from each other).
- Amir Mohammadi (Chicago) - Orbit closure in the absence of polynomial divergence. (abstract)
We will describe how existence of certain "super harmonic" functions on homogeneous spaces and moduli space, can play the role of polynomial divergence of unipotent orbits. This is a joint work with A. Eskin and M. Mirzakhani.
- Shahar Mozes (HUJI) - Invariant measures and divisibility. (abstract)
In a joint work with Manfred Einsiedler we discuss a relationship between the dynamical properties of a maximal diagonalizable group $A$ on certain arithmetic quotients and arithmetic properties of the lattice. In particular, given a finite set of odd primes with at least two elements we consider the semigroup of all integer quaternions that have norm equal to a product of powers of primes from the set. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given quaternion from the semigroup has subexponential growth.
- Paul Nelson (EPFL) - Some twisted variants of QUE. (abstract)
We will discuss some recent work on "twisted" variants of quantum unique ergodicity involving restriction problems for Hilbert modular forms, modular forms on compact arithmetic surfaces and the numerical computation of the latter, for which the primary motivation is to compute Heegner points arising from modular parametrizations in the absence of a q-expansion and in contexts where known methods of p-adic uniformization do not apply.
- Barak Weiss (BGU) - Parking Garages with optimal dynamics. (abstract)
In a celebrated work, Veech showed that so-called "lattice polygons" with an exceptionally large group of affine symmetries exhibit a striking dynamical dichotomy, called the Veech dichotomy. It is not known whether other (non-lattice) polygons might have the same properties. We construct generalized polygons (`parking garages') in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov-Katok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties. This is joint work with Meital Cohen. This is a survey lecture, all relevant terms will be defined.
- Tamar Ziegler (Technion) - Linear equation in primes and dynamics on nilmanifolds. (abstract)
A classical theorem of Dirichlet establishes the existence of infinitely many primes in arithmetic progressions, so long as there are no local obstructions. In 2006 Green and Tao set up a programme for proving a vast generalization of this theorem. They conjectured a relation between the existence of linear patterns in primes and dynamics on nilmanifolds. In recent joint work with Green and Tao we completed the final step of this programme.