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SUMMARY:Jarek Buczyński (Warsaw)
DTSTART:20200520T150000Z
DTEND:20200520T160000Z
DTSTAMP:20260422T225700Z
UID:AppliedAlgebraicGeometry/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/AppliedAlgeb
 raicGeometry/1/">Apolarity\, border rank\, and multigraded Hilbert scheme<
 /a>\nby Jarek Buczyński (Warsaw) as part of Recent advances in border ran
 k and secant varieties of homogeneous varieties.\n\n\nAbstract\nThe rank o
 f a homogeneous polynomial F is the minimal number of summands r such that
  F can be expressed as sum of r powers of linear forms.\n The border rank 
 of F is a minimal r such that F is a limit of polynomials of rank at most 
 r. \nA classical tool to calculate or estimate the rank is called apolarit
 y lemma. In this talk we introduce an elementary analogue of the apolarity
  lemma\,\n which is a method to study the border rank.\nThis can be used t
 o describe the border rank of all cases uniformly\, including those very s
 pecial ones that resisted a systematic approach.\n We work in a general se
 tting\, where the base variety is not necessarily a Veronese variety\, but
  an arbitrary smooth toric projective variety\,\n and this includes the ca
 ses 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 studying tens
 ors and polynomials with large symmetries. In particular\, it can be appli
 ed to provide lower bounds for the border rank of some\n very interesting 
 tensors\, such as the matrix multiplication tensor. A critical ingredient 
 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://researchseminars.org/talk/AppliedAlgebraicGeometry/1/
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BEGIN:VEVENT
SUMMARY:JM Landsberg (Texas A&M)
DTSTART:20200527T150000Z
DTEND:20200527T160000Z
DTSTAMP:20260422T225700Z
UID:AppliedAlgebraicGeometry/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/AppliedAlgeb
 raicGeometry/2/">New border rank lower bounds for matrix multiplication</a
 >\nby JM Landsberg (Texas A&M) as part of Recent advances in border rank a
 nd secant varieties of homogeneous varieties.\n\n\nAbstract\nProgress on b
 oth upper and lower bounds for matrix\nmultiplication have been\nstalled i
 n the past few years. I will explain why it was stalled and how\nBuczynska
 -Buczynski's theory of border apolarity has opened doors to\nprogress on l
 ower\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://researchseminars.org/talk/AppliedAlgebraicGeometry/2/
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SUMMARY:Amy Huang (Texas A&M)
DTSTART:20200527T160000Z
DTEND:20200527T170000Z
DTSTAMP:20260422T225700Z
UID:AppliedAlgebraicGeometry/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/AppliedAlgeb
 raicGeometry/3/">Vanishing Hessian and wild polynomials</a>\nby Amy Huang 
 (Texas A&M) as part of Recent advances in border rank and secant varieties
  of homogeneous varieties.\n\n\nAbstract\nNotions of ranks and border rank
  abounds in the literature. Polynomials with vanishing hessian and their c
 lassification is also a classical problem. Motivated by an observation of 
 Ottaviani\, we will discuss why polynomials with vanishing Hessian and of 
 minimal border rank are wild\, i.e. their smoothable rank is strictly larg
 er than their border rank. If the polynomial is a cubic and of minimal bor
 der rank\, we will also talk about the equivalence of being wild and havin
 g vanishing Hessian. The main tool we are using is the recent work of Bucz
 ynska and Buczynski relating the border rank of polynomials and tensors to
  multigraded Hilbert scheme. From here\, we found two infinite series of w
 ild 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 Michaleck:\
 nhttps://arxiv.org/pdf/1912.13174.pdf\n
LOCATION:https://researchseminars.org/talk/AppliedAlgebraicGeometry/3/
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