Planning
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Thursday, December 8, 2022 |
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10:00
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17:00
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›10:30 (1h)
Andreas Wieser: Birkhoff genericity for points on curves in expanding horospheres and Diophantine applications
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.
› B405
›11:45 (2h)
› Faculty restaurant
›13:45 (1h)
Manuel Lüthi: Pieces of periodic horocycles in the level aspect and diophantine approximation for fractals
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.
› B405
›15:15 (1h)
Alexandre de Faveri: QUE for half-integral weight forms
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.
› B405
›16:30 (1h)
Ilya Khayutin: Local and Global Structure of Double Torus Invariants
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.
› B405
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B405
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