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.
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.
Christos Athanasiadis (University of Athens): Enumerative combinatorics from an algebraic-geometric point of view
Much of enumerative combinatorics studies natural statistics on sets of combinatorial objects, as well as various properties of the corresponding generating polynomials. One can often gain insight by interpreting these polynomials algebraically or geometrically. This introductory lecture will describe interesting situations of this type, by providing examples and explaining some theory. We will meet Eulerian and derangement polynomials, together with Coxeter group analogues and symmetric function generalizations, on the combinatorial side, and Stanley-Reisner rings, triangulations of spheres and balls and group actions on the cohomology of posets and toric varieties, on the algebraic-geometric side.
Bernard Mourrain (Inria Sophia Antipolis): Sparse representations from moments
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.
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.
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.
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.
A limited number of rooms has been set aside at the Intour Hotel until February 20th. Mention the keyword algebra when making a reservation.