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DTSTART:20151025T020000
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UID:calendar.50882.field_date.0@mathematics.huji.ac.il
DTSTAMP:20191116T232726Z
CREATED:20180115T080013Z
DESCRIPTION:Date: \n\n9:45am to 11:00am\n\n\n\n\nSee also: Groups & Dynam
ics\, Events & Seminars\, SeminarsLocation: \n\nManchester building\, Hebr
ew University of Jerusalem\, (Room 209)\n\n\nTitle: Arithmetic of Double
Torus Quotients and the Distribution of Periodic Torus Orbits\nAbstract:\n
In this talk I will describe some new arithmetic invariants for pairs of t
orus orbits on inner forms of PGLn and SLn. These invariants allow us to s
ignificantly strengthen results towards the equidistribution of packets of
periodic torus orbits on higher rank S-arithmetic quotients. An important
aspect of our method is that it applies to packets of periodic orbits of
maximal tori which are only partially split. \nPackets of periodic torus o
rbits are natural collections of torus orbits coming from a single rationa
l adelic torus and are closely related to class groups of number fields. T
he distribution of these orbits is akin to the distribution of integral po
ints on homogeneous algebraic varieties with a torus stabilizer.\nThe dist
ribution of packets of periodic torus orbit has been studied using dynamic
al methods in the pioneering work of Linnik in the rank 1 case (equidistri
bution on the 2-sphere) and by Einsiedler\, Lindenstrauss\, Michel and Ven
katesh (ELMV) in higher rank. We note that in rank 1\, stronger equidistri
bution results for packets of periodic orbits have been established by Duk
e and Iwaniec using the theory of automorphic functions.\nThe dynamical ap
proach typically consists of two main ingredients: an arithmetic one\, whi
ch implies that the toral packets have high asymptotic metric entropy\, an
d a measure rigidity argument\, which deduces from the entropy result a st
atement regarding the limit distribution of the orbits. While thanks to Ei
nsiedler\, Katok and Lindenstrauss we have very powerful measure rigidity
tools for higher rank toral actions\, we know much less regarding the arit
hmetic of these packets in higher rank. A notable exception is the work of
ELMV which synergies the dynamical approach with harmonic analysis to pro
ve an equidistribution theorem similar to Linnik’s in the split rank 2 cas
e. In the other cases the current known results\, due to the same authors\
, are significantly weaker. \nOur methods generalize Linnik's original ar
ithmetic approach in a different direction. We derive new invariants\, aki
n to the discriminant inner product used by Linnik. These invariants come
from studying double quotients of a reductive group by a torus using geome
tric invariant theory.\nWe then derive a sharper lower bound for the entro
py from these invariants using the action of the Galois group of the torus
’ splitting field and the algebraic relations between the invariants. This
lower bound gives new qualitative restrictions on the possible limit meas
ures and applies also to partially split maximal tori.\n\n Export\n \n\n
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DTSTART;TZID=Asia/Jerusalem:20151105T094500
DTEND;TZID=Asia/Jerusalem:20151105T110000
LAST-MODIFIED:20191103T130312Z
SUMMARY:Groups & Dynamics : Ilya Khayutin (HUJI)
URL;TYPE=URI:https://mathematics.huji.ac.il/event/groups-dynamics-ilya-khay
utin-huji-1
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