Andreas Wieser: Birkhoff genericity for points on curves in expanding horospheres and Diophantine applications
B405
Let $\{a(t):t \in \mathbb{R}\}$ be a diagonalizable subgroup of $SL(d,\mathbb{R})$ for which the expanded horosphere $U$ is abelian. By the Birkhoff ergodic theorem, for any point $x \in SL(d,\mathbb{R})/SL(d,\mathbb{Z})$ and almost every $u \in U$ the point $ux$ is Birkhoff generic for the flow $a(t)$. One may ask whether the same is true when the points in $U$ are sampled with respect to a measure singular to the Lebesgue measure. In this talk, we discuss work with Omri Solan proving that almost every point on an analytic curve within U is Birkhoff generic when the curve satisfies a non-degeneracy condition. This Birkhoff genericity result has various applications in Diophantine approximation, some of which we shall discuss during the talk. No preliminary knowledge of any of the above notions is assumed.
Manuel Lüthi: Pieces of periodic horocycles in the level aspect and diophantine approximation for fractals
B405
In recent work with Osama Khalil, we obtained the first instances of self similar fractals in Euclidean space satisfying a complete analogue to Khintchine's theorem in Diophantine approximation. For example, it was shown that the analogue holds for missing digit sets of sufficiently large Hausdorff codimension, including the example of numbers in the unit interval whose base 5 expansion does not contain the digit 2. The motivating question, posed by Kurt Mahler in 1984, was whether the middle third Cantor set satisfies an analogue of Khintchine's theorem and this remains still open.I will discuss the proof of our main theorem, namely the equidistribution of certain expansions of admissible measures on horocycles in the space of lattices and pinpoint the part of the argument which prevents extension to the middle third Cantor set. It turns out that the obstacle occurs via a volume dependence of the effective equidistribution of pieces of horocycle orbits in congruence quotients.
Alexandre de Faveri: QUE for half-integral weight forms
B405
Lindenstrauss pioneered the use of measure classification results to prove the quantum unique ergodicity conjecture (up to escape of mass) for Maass forms. I will discuss a generalization of such methods to metaplectic covers of SL_2(R), which implies QUE (up to escape of mass) for half-integral weight Maass forms. This is joint work with Alex Dunn.
Ilya Khayutin: Local and Global Structure of Double Torus Invariants
B405
In https://tinyurl.com/dxfs4kxn I have introduced double torus invariants as a tool to study the asymptotic entropy of packets of periodic torus orbits in Gamma\G for higher rank G. These invariants are a non-linear generalization of the polarization of the norm form on pure quaternions used by Linnik in rank 1. In this talk I will discuss the finer structure of the double torus invariants for forms of PGL3, including an analogue of Linnik’s Basic Lemma, new congruence relations and global structure results. At the moment, the new results provide modest improvements in the asymptotic entropy. These observations also partially generalize to other higher rank groups.
Nicolas de Saxcé: Arithmetic groups and diophantine approximation
B405
We revisit the theory of diophantine approximation using lattices spaces and reduction theory, due to Borel and Harish-Chandra. This will allow us to answer some questions of W. Schmidt on rational approximations to linear subspaces.
Claire Burrin: Windings of prime geodesics via the trace formula
B405
A standard construction in automorphic forms associates a multiplier system to any nonuniform Fuchsian group. I will explain how this construction extends to give a notion of winding number for closed geodesics on the linear space of the corresponding hyperbolic surface. I will then show a Dirichlet-type equidistribution theorem for various families of cusped hyperbolic surfaces: For every set A of integers with density d=d(A), the set of prime geodesics with winding number in A has density 1/d. This is joint work with Flemming von Essen.
Asbjørn Nordentoft: The distribution of closed geodesics in the homology of modular curves
B405
Given an element of the (wide) class group of a real quadratic field K, one can associate an (oriented) closed geodesic on the modular curve of square-free level N (when all primes dividing N splits in K). Duke proved that as the discriminant tends to infinity the (oriented) closed geodesics equidistribute. In this talk we will discuss the related problem regarding the distribution of the homology classes associated to the closed geodesics. Our main result is a convergence (in an appropriate sense that we will explain) to the Eisenstein element in the homology. This is the real quadratic analogue of the equidistribution of supersingular reduction of CM elliptic curves (as studied by Michel , Liu—Masri—Young, …). If time permits we will discuss a similar result for p-orbits of (oriented) closed geodesics (i.e. packets of closed geodesics associated to the units modulo p), as well as the connection to joint work with Peter Humphries on orbifolds associated to real quadratic fields.
Peter Humphries: Sparse equidistribution of hyperbolic orbifolds
B405
Duke, Imamoḡlu, and Tóth have recently constructed a new geometric invariant, a hyperbolic orbifold, associated to each narrow ideal class of a real quadratic field. Furthermore, they have shown that the projection of these hyperbolic orbifolds onto the modular surface equidistributes on average over a genus of the narrow class group as the fundamental discriminant of the real quadratic field tends to infinity. We discuss a refinement of this result, sparse equidistribution, and connect this to cycle integrals of automorphic forms and subconvexity for Rankin-Selberg L-functions. This is joint work with Asbjørn Nordentoft.
Maksym Radziwiłł: Non vanishing of twists of GL(4) L-functions
B405
I will discuss recent work with Lyiang Yang in which we establish the non vanishing of L(1/2, \pi \times \chi) for infinitely many primitive Dirichlet characters and where \pi is either a GL(4) cuspidal automorphic representation of cohomological type or arises from a functorial lift e.g Sym^3 g with g a GL(2) cuspidal automorphic representation. The exact family of \pi's for which our result works is quite a bit larger. I will describe the exact conditions under which the result holds.
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B405
