36th Autumn School in Algebraic Geometry

Power sum decompositions and apolarity, a geometric approach

September 1st-7th, 2013

Łukęcin, Poland



Inquires should be directed to Joachim Jelisiejew, (jj277546 at students mimuw edu pl).

Registration deadline was: June 15th, 2013.

Program of the meeting is posted here.

Notes from the lectures:

Topic of the school:

Worldwide, the study of tensor and polynomial decompositions from the geometric point of view is a very active area of science and has significant applications outside of pure mathematics. The underlying geometric problems have been often studied by mathematicians since 19th century, and it is a part of Italian mathematical tradition to investigate these objects. To illustrate, two fundamental varieties used as the main tool in this subject are named after two Italian mathematicians, Giuseppe Veronese and Corrado Segre. In Poland, the subject is relatively new, but a research group founded by Weronika Buczyńska and Jarosław Buczyński is growing and is very active in the subject. The school will be an opportunity for Polish students and researchers to learn about fundamental recent and classical achievements from world class experts, and to establish long lasting mathematical collaborations with European and extra-European scientists.

Given a homogeneous polynomial (a form) $F$ of degree $d$ over a field of characteristic $0$ (usually, complex numbers), one can write $F$ as a sum of powers of linear forms $F= l_1^d + ... + l_r^d$. The minimal $r$ allowing such power sum decomposition is called the rank of $F$ (also called Waring rank, or symmetric tensor rank). A decomposition $F= l_1^d + ...+ l_r^d$ with $r$ equal to the rank of $F$ is called a Waring decompostion of $F$. Generally, the Waring problem is to find bounds for the rank of $F$, but nowadays it has many variants. For instance, one can ask what is the rank of a general (random) form in a fixed number of variables and of a fixed degree. This is solved by the famous Alexander-Hirschowitz theorem. One can ask what is the rank of a particular $F$, such as monomial (solved recently by Enrico Carlini, Maria Virginia Catalisano, Anthony Geramita), or a determinant of a matrix, with variables as entries (unknown even for $3 \times 3$ matrix). One can also ask about possible forms of a Waring decomposition, is it unique, or if not, how to describe the space of all possible Waring decompositions (essentially this space is called the variety of sums of powers of $F$)?

These are seemingly purely algebraic problems, formulated in a relatively simple terms. However, the solutions, if known at all, tend to be far not simple, and involve vast areas of mathematics, including also:

During the school, we will concentrate on algebro-geometric methods and approaches to the Waring-type problems.

Abstracts of the lecture series

"Waring problems, Secant varieties and Sylvester algorithm", Alessandra Bernardi, 4 lectures

The polynomial version of the classical Waring problem can be formulated in three flavors:
(i) for any form,
(ii) for the generic form or
(iii) for a given form.
From a geometric point of view they correspond to (i) finding a bound for the maximum rank, (ii) the computation of the dimension of certain secant varieties of Veronese varieties and (iii) finding a generalization of the classical Sylvester algorithm. While the computation of the maximum rank of forms is still the most difficult problem, the two other ones can be successfully tackled via Apolarity and Inverse Systems. Such tools allow to translate these problems in terms of linear systems of hypersurfaces through fat points and Gorenstein $0$-dimensional schemes of minimal length.

"Identifiability and Weak defectivity", Giorgio Ottaviani, 4 lectures

Any tensor can be decomposed as a sum of decomposable tensors. In the symmetric case, this is the Waring decomposition of a polynomial as a sum of powers. This decomposition of a tensor is particularly useful in applications when it is unique, in this case we say the tensor is identifiable. We study results and criteria which guarantee that a tensor is identifiable. One of the most important is related to the geometric notion of weak defectiveness introduced by Chiantini and Ciliberto. If time allows, I will continue with applications of vector bundles (nonabelian apolarity).

Suggested preparatory reading: from J. Landsberg's book "Tensors: Geometry and Applications": chapter 4, chapter 5 from 5.1 to 5.4, chapter 6 from 6.1 to 6.4

"Apolarity and the variety of sums of powers of a form", Kristian Ranestad, 2 lectures

For an arithmetic Gorenstein variety of dimension $n$, one may associate an apolar form to the general codimension $n+1 $ linear section. The variety of sums of powers of the apolar form may be studied in this setup, and yields interesting and often surprising results for general forms in a number of special examples of arithmetic Gorenstein varieties.

Suggested preparatory reading: from Kristian Ranestad and Frank-Olaf Schreyer: "Varieties of Sums of Powers", J. reine angew. Math. 525 (2000), 147-181: sections 1,2,3.

Practical information

Location and accommodation

The school will take place in a Warsaw University pension in Łukęcin (look here for more information), on Western part of Polish Baltic sea shore. Participants are expected to arrive on Sunday, September 1st, evening. Lectures are Monday to Friday and Saturday is the departure day.

The accommodation (full board, double room) costs about 125 złoty (PLN) a day. 1 Euro $\simeq$ 4.2 złoty (approximately), but the exchange rate is not fixed.

Registration fee

A small registration fee of 60 złoty per person will apply. The fee is not covered by the financial support.

Financial support

Graduate students and young researchers with inadequate support from their home institutions are encouraged to apply for accommodation cost waiver. Please indicate your need for support in the registration form. You will need to provide the name of your scientific adviser and we may ask you to provide an additional letter of recommendation. The organizers will not pay for participants' travel, nor for the registration fee.

Map and travel to Łukęcin

Click map of Autumn School in Algebraic Geometry, Łukęcin, Poland for a larger map. Visa information: Polish Ministry of Foreign Affairs.

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  • Sponsors

    The school is supported by:

    NCN grant of Mariusz Koras  Institute of Mathematics
    of Univerisity of Warsaw