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SUMMARY:Quantitative geometry of arithmetic locally symmetric spaces
DTSTART:20260423T120000Z
DTEND:20260423T130000Z
DTSTAMP:20260411T175800Z
UID:indico-event-16415@indico.math.cnrs.fr
DESCRIPTION:Speakers: Lam Pham\n\nMotivic cohomology is a cohomology theor
 y that can be defined internally within Grothendieck's category of motives
 . Voevodsky developed this theory for smooth varieties\, demonstrating its
  profound connections to algebraic cycles and algebraic K-theory. However\
 , its behaviour in mixed-characteristic remains less well understood. Buil
 ding on recent advancements by Bachmann\, Elmanto\, Morrow\, and Bouis\, i
 n a joint work with Bouis\, we demonstrate a purity result over deeply-ram
 ified bases. I will also discuss an application of this r\n\nArithmetic lo
 cally symmetric spaces (in the broad sense) occur in several areas of math
 ematics. Archimedean\, non-archimedean\, and adelic constructions can be s
 tudied using arithmetic and algebraic groups. There is a remarkable interp
 lay of geometry\, algebra\, and arithmetic\, and these objects connect sev
 eral interesting areas (construction of expander graphs and high-dimension
 al expanders\, Diophantine geometry\, geometric analysis\, geometric group
  theory\, etc.). It is generally a difficult problem to understand this ge
 ometry quantitatively\, in infinite families. I will recall several import
 ant constructions in this unified framework and present some recent work o
 n conjectures of Margulis' (geometry) and Lehmer's (arithmetic) regarding 
 lower bounds on systoles of arithmetic locally symmetric spaces (length of
  the shortest closed geodesic). Part of this work is joint with F. Thilman
 y and M. Fraczyk.\n\nsult in p-adic Hodge theory.\n\nhttps://indico.math.c
 nrs.fr/event/16415/
URL:https://indico.math.cnrs.fr/event/16415/
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