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SUMMARY:Uriya First (University of Haifa)
DTSTART:20251028T203000Z
DTEND:20251028T213000Z
DTSTAMP:20260423T130124Z
UID:MITNT/123
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/123/">
 Higher Essential Dimension: First Steps</a>\nby Uriya First (University of
  Haifa) as part of MIT number theory seminar\n\nLecture held in Room 2-449
  in the Simons Building (building 2).\n\nAbstract\nLet $G$ be a linear alg
 ebraic group over a field $k$.\nLoosely speaking\, the essential dimension
  of $G$ measures\nthe number of independent parameters that are required t
 o define\na $G$-torsor over a $k$-field. It measures the complexity of $G$
 -torsors and equivalent objects.\nOne formal way to define it is to say\nt
 hat the essential dimension of $G$ is $\\leq m$ if every $G$-torsor over a
  finite-type $k$-scheme is\, away from some codimension-$1$ closed subsche
 me\, the specialization of a $G$-torsor over a finite-type $k$-scheme of d
 imension $m$.\n\nRecently\, for every integer $d\\geq 0$\, we\ndefined the
  $d$-essential dimension of $G$\,\ndenoted $\\mathrm{ed}^{(d)}(G)$\, by re
 placing ``codimension-$1$'' with ``codimension-$(d+1)$''.\nAfter recalling
  ordinary essential dimension and its usages\, I will discuss work in prog
 ress about the new sequence of invariants $\\{\\mathrm{ed}^{(d)}(G)\\}_{d\
 \geq 0}$ and its asymptotic behavior as $d\\to \\infty$.\nFor example\, $\
 \mathrm{ed}^{(d)}(\\mathbf{G}_m)=d$\, $\\mathrm{ed}^{(d)}(\\mathbf{\\mu}_n
 )=d+1$ and\n$\\mathrm{ed}^{(d)}(\\mathbf{G}_m\\times\\mathbf{G}_m)=2d$ in 
 characteristic $0$. Moreover\, there is a dichotomy between unipotent and 
 non-unipotent\ngroups: If $G$ is unipotent\, the sequence $\\{\\mathrm{ed}
 ^{(d)}(G)\\}_{d\\geq 0}$ is bounded\,\nwhereas  if $G$ is not unipotent\, 
 then $\\mathrm{ed}^{(d)}(G)\\geq d-C_G$ for some constant $C_G$.\nThere ar
 e also some interesting anomalies.\n
LOCATION:https://researchseminars.org/talk/MITNT/123/
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