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SUMMARY:Jarek Buczyński (Warsaw)
DTSTART:20200520T150000Z
DTEND:20200520T160000Z
DTSTAMP:20260404T110642Z
UID:AppliedAlgebraicGeometry/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Appli
 edAlgebraicGeometry/1/">Apolarity\, border rank\, and multigraded Hilbert 
 scheme</a>\nby Jarek Buczyński (Warsaw) as part of Recent advances in bor
 der rank and secant varieties of homogeneous varieties.\n\n\nAbstract\nThe
  rank of a homogeneous polynomial F is the minimal number of summands r su
 ch that F can be expressed as sum of r powers of linear forms.\n The borde
 r rank of F is a minimal r such that F is a limit of polynomials of rank a
 t most r. \nA classical tool to calculate or estimate the rank is called a
 polarity lemma. In this talk we introduce an elementary analogue of the ap
 olarity lemma\,\n which is a method to study the border rank.\nThis can be
  used to describe the border rank of all cases uniformly\, including those
  very special ones that resisted a systematic approach.\n We work in a gen
 eral setting\, where the base variety is not necessarily a Veronese variet
 y\, but an arbitrary smooth toric projective variety\,\n and this includes
  the cases of border rank of tensors. We also define a border rank version
  of the variety of sums of powers and analyse how it is useful\n in studyi
 ng tensors and polynomials with large symmetries. In particular\, it can b
 e applied to provide lower bounds for the border rank of some\n very inter
 esting tensors\, such as the matrix multiplication tensor. A critical ingr
 edient of our work is an irreducible component of a\n multigraded Hilbert 
 scheme related to the toric variety in question.\n\nThe talk is based on a
  joint work with Weronika Buczyńska\, http://arxiv.org/abs/1910.01944\n
LOCATION:https://stable.researchseminars.org/talk/AppliedAlgebraicGeometry
 /1/
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BEGIN:VEVENT
SUMMARY:JM Landsberg (Texas A&M)
DTSTART:20200527T150000Z
DTEND:20200527T160000Z
DTSTAMP:20260404T110642Z
UID:AppliedAlgebraicGeometry/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Appli
 edAlgebraicGeometry/2/">New border rank lower bounds for matrix multiplica
 tion</a>\nby JM Landsberg (Texas A&M) as part of Recent advances in border
  rank and secant varieties of homogeneous varieties.\n\n\nAbstract\nProgre
 ss on both upper and lower bounds for matrix\nmultiplication have been\nst
 alled in the past few years. I will explain why it was stalled and how\nBu
 czynska-Buczynski's theory of border apolarity has opened doors to\nprogre
 ss on lower\nand perhaps even upper bounds. If time permits\, I will also 
 explain\nnew hurdles that will\nneed to be surmounted. This is joint work 
 with A. Conner and A. Harper.\n
LOCATION:https://stable.researchseminars.org/talk/AppliedAlgebraicGeometry
 /2/
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BEGIN:VEVENT
SUMMARY:Amy Huang (Texas A&M)
DTSTART:20200527T160000Z
DTEND:20200527T170000Z
DTSTAMP:20260404T110642Z
UID:AppliedAlgebraicGeometry/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Appli
 edAlgebraicGeometry/3/">Vanishing Hessian and wild polynomials</a>\nby Amy
  Huang (Texas A&M) as part of Recent advances in border rank and secant va
 rieties of homogeneous varieties.\n\n\nAbstract\nNotions of ranks and bord
 er rank abounds in the literature. Polynomials with vanishing hessian and 
 their classification is also a classical problem. Motivated by an observat
 ion of Ottaviani\, we will discuss why polynomials with vanishing Hessian 
 and of minimal border rank are wild\, i.e. their smoothable rank is strict
 ly larger than their border rank. If the polynomial is a cubic and of mini
 mal border rank\, we will also talk about the equivalence of being wild an
 d having vanishing Hessian. The main tool we are using is the recent work 
 of Buczynska and Buczynski relating the border rank of polynomials and ten
 sors to multigraded Hilbert scheme. From here\, we found two infinite seri
 es of wild polynomials and we will try to describe their border varieties 
 of sums of powers\, which is an analogue of the variety of sums of powers.
 \n\nThe talk is based on joint work with Emanuele Ventura and Mateusz Mich
 aleck:\nhttps://arxiv.org/pdf/1912.13174.pdf\n
LOCATION:https://stable.researchseminars.org/talk/AppliedAlgebraicGeometry
 /3/
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