Graduate Student Meeting on Applied Algebra and Combinatorics

NEW! There was a second meeting in Leipzig!

The meeting is addressed to PhD students, recent graduates and post-docs working in Applied Algebra and/or Combinatorics. We strongly encourage all interested students to participate actively and give a talk.

NEW! The final schedule, abstracts and list of participants are now online.

See our poster.

Important information

The meeting took place from 22.03. to 24.03.2018 at the University of Osnabrück, Institute of Mathematics (building 69), Albrechtstraße 28a, room 125. We started on Thursday morning and finished on Saturday around 15:30.

Have a look at the schedule. Read the abstracts. See who else is participating.

Keynote speakers

Christos Athanasiadis (University of Athens): Enumerative combinatorics from an algebraic-geometric point of view

Click here to see Christos Athanasiadis' abstract

Bernard Mourrain (Inria Sophia Antipolis): Sparse representations from moments

Click here to see Bernard Mourrain's abstract

Recovering a hidden structure from measurements, observations, evaluation, statistics etc. is a problem that is encountered in many domains such functionnal approximation, signal processing, geometric modeling, medical imaging, etc. It has a long history going back to the work of G. de Prony on the decomposition of a function as a sum of exponential functions or the work of J.J. Sylvester on the decomposition of a binary form as a sum of powers of linear forms, or even more recently Berlekamp-Massey approach for decoding algebraic codes.

One of the motivation of the presentation is to show that these different problems fall in the same framework, which applies in several dimensions. Another motivation is to show that these decomposition problems can be analyzed and solved efficiently by algebraic methods.
We will see that they reduce to the problem of decomposition of series as polynomial-exponential series.
We will show that polynomial-exponential series are naturally in correspondance with Artinian Gorenstein algebras. This leads to a characterization of Hankel operators of finite rank, relating the rank of the operator with inverse systems of multiple points.

By exploiting standard eigenvector methods for solving polynomial equations, we can compute the frequencies and weights of a minimal polynomial-exponential series, using truncated pieces of the Hankel operator. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carathéodory-Fejèr decomposition of multivariate Hankel matrices. We will also describe an algorithm to compute a border basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process.

The approach will be illustrated in different multivariate decomposition problems: convolution operator of finite rank, reconstruction of measures as weighted sums of Dirac measures, representation of multivariate polynomial-exponential functions, sparse interpolation of polylog functions, tensor decomposition.
Some connections with polynomial optimisation, semi-definite programming, compressed sensing and super-resolution shall also be discussed, if there is a moment left.

Slides

Claudia Andrei (University of Bucharest): The coordinate ring of a convex polyomino. Slides

Christos Athanasiadis (University of Athens): Enumerative combinatorics from an algebraic-geometric point of view. Slides

Christopher Borger (Otto-von-Guericke-Universität Magdeburg): Defectivity of families of full-dimensional point configurations. Slides

Óscar Iglesias-Valiño (University of Cantabria): The classification of empty 4-simplices. Slides

Alexandru Iosif (Otto-von-Guericke-Universität Magdeburg): Mass-action networks with the isolation property. Slides

Bernard Mourrain (Inria Sophia Antipolis): Sparse representations from moments. Slides

Jonathan Toledo (Cinvestav IPN): Strong persistence property for monomial ideals. Slides

We would like to collect all the slides from the workshop. If you wish to have yours uploaded here please send a pdf to one of the organizers.

Description

The aim of the meeting is to bring together PhD students working in some area associated with Applied Algebra and/or Combinatorics and to encourage an active interaction and networking among them.

Topics of interest cover a wide spectrum of areas of research such as

Combinatorial and Computational Commutative Algebra
Topological Combinatorics
Algebraic and Discrete Geometry
Tensor Methods
Algebraic Statistics

and the possible interplay between these subjects as well as applications in other areas of science.

The first two days will be opened by an introductory 60 minutes talk (plus 10 minutes of questions) by a keynote speaker. The rest of the schedule is dedicated to 35-minute talks by PhD students and recent graduates. We are also planning a poster session.

Organizers & Scientific Committee

Jan-Marten Brunink
Alexandros Grosdos
Lorenzo Venturello
Markus Wageringel

The scientific committee consists of Dinh Van Le (head) and the organizers.

Registration

The Workshop is now over. Thank you all for participating.

Please register here. There is no workshop fee. The deadline for talk submission and funding has passed. You can still register for participation and poster presentation.

There is limited funding for PhD students who cannot be supported by their home institution to cover accommodation and travel costs, and preference will be given to the students who are willing to give a talk.

Recommended Accommodation

A limited number of rooms has been set aside at the Intour Hotel until February 20th. Mention the keyword algebra when making a reservation.