Webinar Tensors and Algebraic Geometry:
Recent advances in border rank and secant varieties of homogeneous varieties
Webinar organizzato da E. Angelini,
L. Chiantini,
M. Mella, G. Ottaviani.
Chiunque sia interessato a ricevere i link di accesso ai seminari scriva a elena.angelini@unisi.it
Si consiglia di connettersi 10 minuti circa prima dell'inizio del seminario.
Primo appuntamento:
mercoledì 20 maggio 2020, ore 17:00 (Central European time)
Jarek Buczynski (Warsaw) terrà il seguente seminario:
Titolo: Apolarity, border rank, and multigraded Hilbert scheme
Abstract: The rank of 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.
The border rank of F is a minimal r such that F is a limit of polynomials of rank at most r.
A classical tool to calculate or estimate the rank is called apolarity lemma.
In this talk we introduce an elementary analogue of the apolarity lemma, which is a method to study the border rank.
This can be used to describe the border rank of all cases uniformly,
including those very special ones that resisted a systematic approach.
We work in a general setting, where the base variety is not necessarily
a Veronese variety, but an arbitrary smooth toric projective variety,
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 in studying tensors and polynomials with large symmetries. In
particular, it can be applied to provide lower bounds for the border
rank of some very interesting tensors, such as the matrix multiplication tensor. A
critical ingredient of our work is an irreducible component of a
multigraded Hilbert scheme related to the toric variety in question.
The talk is based on a joint work with Weronika Buczynska,
http://arxiv.org/abs/1910.01944.
Materiale disponibile: slides .
Secondo appuntamento:
mercoledì 27 maggio 2020, ore 17:00 (Central European time)
JM Landsberg (Texas A&M) terrà il seguente seminario:
Titolo: New border rank lower bounds for matrix multiplication
Abstract: Progress on both upper and lower bounds for matrix
multiplication have been
stalled in the past few years. I will explain why it was stalled and how
Buczynska-Buczynski's theory of border apolarity has opened doors to
progress on lower
and perhaps even upper bounds. If time permits, I will also explain
new hurdles that will
need to be surmounted. This is joint work with A. Conner and A. Harper.
Materiale disponibile: slides .
Terzo appuntamento:
mercoledì 27 maggio 2020, ore 18:00 (Central European time)
Amy Huang (Texas A&M) terrà il seguente seminario:
Titolo: Vanishing Hessian and Wild Polynomials
Abstract: Notions of ranks and border rank abounds in the literature.
Polynomials with vanishing hessian and their classification 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 larger
than their border rank. If the polynomial is a cubic and of minimal border rank, we will
also talk about the equivalence of being wild and having vanishing Hessian. The main tool
we are using is the recent work of Buczynska and Buczynski relating the border rank of
polynomials and tensors to multigraded Hilbert scheme. From here, we found two infinite
series 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.
The talk is based on joint work with Emanuele Ventura and Mateusz Michaleck:
https://arxiv.org/pdf/1912.13174.pdf.
Materiale disponibile: slides .