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SUMMARY:Jarek Buczyński (Warsaw)
DTSTART;VALUE=DATE-TIME:20200520T150000Z
DTEND;VALUE=DATE-TIME:20200520T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T071255Z
UID:AppliedAlgebraicGeometry/1
DESCRIPTION:Title: 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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SUMMARY:JM Landsberg (Texas A&M)
DTSTART;VALUE=DATE-TIME:20200527T150000Z
DTEND;VALUE=DATE-TIME:20200527T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T071255Z
UID:AppliedAlgebraicGeometry/2
DESCRIPTION:Title: New border rank lower bounds for matrix multiplication\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;VALUE=DATE-TIME:20200527T160000Z
DTEND;VALUE=DATE-TIME:20200527T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T071255Z
UID:AppliedAlgebraicGeometry/3
DESCRIPTION:Title: Vanishing Hessian and wild polynomials\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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