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SUMMARY:Combinatorics via Closed Orbits: Vertex Expansion and Graph Quantu
 m Ergodicity - Amitay Kamber (University of Cambridge) 
DTSTART:20210625T124500Z
DTEND:20210625T134500Z
UID:TALK160126@talks.cam.ac.uk
CONTACT:76015
DESCRIPTION:The symmetric space of $SL_2(R)$ is the hyperbolic plane\, and
  the fact that $SL_2(Z)$ is a lattice in $SL_2(R)$ implies that after taki
 ng a quotient we get a finite volume hyperbolic surface. \n\nWhen $SL_2(R)
 $ is replaced by the p-adic group $SL_2(Q_p)$ the symmetric space is a (q+
 1)-regular Bruhat-Tits tree. Ihara\, Margulis and Lubotzky-Phillips-Sarnak
  observed that when $SL_2(Z)$ is replaced by a lattice coming from a quate
 rnion algebra one gets a (q+1)-regular graph. Using deep number theoretic 
 results from the theory of automorphic forms\, related to the classical Ra
 manujan conjecture\, they showed that the resulting graphs are expanders w
 ith an optimal spectral gap\, i.e.\, “Ramanujan graphs”.\n\nThe number
 -theoretic Ramanujan graphs have a lot of combinatorial applications and w
 ere generalized to various combinatorial number theoretic constructions.\n
 \nHowever\, there are some notorious open questions about those constructi
 ons\, such as the vertex expansion of number theoretic Ramanujan graphs.\n
 \nIn the talk\, I will describe how one can construct extermal substructur
 es of some number-theoretic structures\, which provides counterexamples fo
 r many open problems. The idea is group-theoretic and simple - we use clos
 ed orbits of subgroups\, when those subgroups are available. The implement
 ation of the idea requires some number theory.\n\nI will try to appeal to 
 a wide audience\, and focus on the group theory involved.\n\nBased on join
 t work with Tali Kaufman.\n\n
LOCATION:Zoom https://maths-cam-ac-uk.zoom.us/j/95208706709
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