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SUMMARY:Grigoriy Blekherman (Georgia Institute of Technology)
DTSTART:20210114T213000Z
DTEND:20210114T230000Z
DTSTAMP:20260423T021354Z
UID:FOTR/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FOTR/33/">Su
 ms of Squares: From Real to Commutative Algebra</a>\nby Grigoriy Blekherma
 n (Georgia Institute of Technology) as part of Fellowship of the Ring\n\n\
 nAbstract\nA real polynomial is called nonnegative if it takes only nonneg
 ative values. A sum of squares or real polynomials is clearly nonnegative.
  The relationship between nonnegative polynomials and sums of squares is o
 ne of the central questions in real algebraic geometry. A modern approach 
 is to look at nonnegative polynomials and sums of squares on a real variet
 y X\, where unexpected links to complex algebraic geometry and commutative
  algebra appear.\n\nIn the first half of the talk I will review the histor
 y of the problem\, do some examples\, and provide a brief overview of the 
 results. Our two guiding questions will be: the relationship between nonne
 gative polynomials and sums of squares\, and the number of squares needed 
 to write any sum of squares on X. I will explain the connection between th
 ese questions and properties of the free resolution of the ideal of X: the
  number of of steps that the resolution only has linear syzygies (property
  $N_{2\,p}$) and the number of steps that linear syzygies persist (the len
 gth of the linear strand).\n\nIn the second half\, I will concentrate on t
 he number of squares\, and introduce an invariant of X we call quadratic p
 ersistence. Quadratic persistence of X is equal to the least number of poi
 nts in X such that after projecting from (the span of) these points the id
 eal of the resulting variety has no quadrics. I will explain how quadratic
  persistence connects real algebraic geometry and commutative algebra. Joi
 nt work with Rainer Sinn\, Greg Smith and Mauricio Velasco.\n
LOCATION:https://researchseminars.org/talk/FOTR/33/
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