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VERSION:2.0
PRODID:icalendar-ruby
CALSCALE:GREGORIAN
X-WR-CALNAME:MIT Probability Seminar
X-WR-TIMEZONE:Eastern Time (US & Canada)
BEGIN:VEVENT
DTSTAMP:20260616T124258Z
UID:tag:localist.com\,2008:EventInstance_40810956158788
DTSTART:20221031T201500Z
DTEND:20221031T211500Z
DESCRIPTION:Speaker: Robert Hough (Stony Brook University) \n\nTitle: Cover
 ing systems of congruences.\n\nAbstract: \n\n\noindent A distinct covering
  system of congruences is a list of congruences\n\\[\na_i \\bmod m_i\, \\q
 quad i = 1\, 2\, ...\, k\n\\]\nwhose union is the integers.  Erd\\H{o}s as
 ked if the least modulus $m_1$ of a distinct covering system of congruence
 s can be arbitrarily large (the minimum modulus problem for covering syste
 ms\, \\$ 1000 ) and if there exist distinct covering systems of congruence
 s all of whose moduli are odd (the odd problem for covering systems\, \\$ 
 25).  I'll discuss my proof of a negative answer to the minimum modulus pr
 oblem\, and a quantitative refinement with Pace Nielsen that proves that a
 ny distinct covering system of congruences has a modulus divisible by eith
 er 2 or 3.  The proofs use the probabilistic method and in particular use 
 a sequence of pseudorandom probability measures adapted to the covering pr
 ocess.  Time permitting\, I may briefly discuss a reformulation of our met
 hod due to Balister\, Bollob\\'{a}s\, Morris\, Sahasrabudhe and Tiba which
  solves a conjecture of Shinzel (any distinct covering system of congruenc
 es has one modulus that divides another) and gives a negative answer to th
 e square-free version of the odd problem.
GEO:42.358262;-71.090045
LOCATION:Building 2\, 2-147
SUMMARY:MIT Probability Seminar
URL;VALUE=URI:https://calendar.mit.edu/event/mit_probability_seminar_202209
 26
CATEGORIES:Conferences/Seminars/Lectures
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