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# Linz-Wien workshop in February 2013

Location: Laimbach am Ostrong, http://www.schreiners.at

This workshop will follow, as an experiment, the following rules: framework_freeform_workshop.txt.

Coordinators: Zijia Li (zijia.li@oeaw.ac.at) and Stefan Perlega.

Participants: Hamid Ahmadinezhad, Abraham Martin del Campo, Christopher Chiu, Matteo Gallet, Herwig Hauser, Christoph Koutschan, Zijia Li, Markus Patrick Mueller, Stefan Perlega, Georg Regensburger, Josef Schicho, Caroline Uhler, Nelly Villamizar

## Schedule

For details of the activities please see below.

 Time Activity Wednesday, 27.2.2013 Evening Arrival and dinner Thursday, 28.2.2013 09:00 - 10:15 Three Types of the Parallel 6R Linkages (Z. Li) 10:20 - 11:25 How to compute the dimension of a projective variety (M. Gallet) 11:30 - 12:35 14:00 - 15:00 15:00 - 16:30 Friday, 1.3.2013 08:45 - 10:15 10:30 - 11:45 12:00 - 13:00 13:00 - 14:15 14:30 - 15:45 16:00 - 17:00 Evening Departure

## List of activities

Please add to this list any activity you are responsible for. Also add yourself as a participant as you please, so we can estimate (you can add or remove yourself any time!). Finally, you can ask or comment in each entry (even anonymously, if you use the guest account).

If you need LaTeX do it as follows: <latex>\frac{a}{b}</latex>. Also BibTex available! Sample: we cite MR0286317 where <bibtex> @book {MR0286317,

  AUTHOR = {Knuth, Donald E.},
TITLE = {The art of computer programming. {V}ol. 1: {F}undamental
algorithms},
SERIES = {Second printing},

            Mills, Ont},
YEAR = {1969},
PAGES = {xxi+634},
MRCLASS = {68.00 (65.00)},
MRNUMBER = {0286317 (44 \#3530)},

MRREVIEWER = {M. Muller}, } </bibtex>

#### Three Types of the Parallel 6R Linkages

• Proposed by: Zijia Li
• Type: Talk and open problem
• Duration: 60 minutes
• Minimum Requirements: basic knowledge of quaternions.

Short description: We consider a special kind of overconstrained 6R closed linkage which we call parallel 6R linkages. These are linkages with the property that they have three pairs of parallel joint-axes. The three pairs of joint-axes include two pairs of parallel adjacent joint-axes and one pair of parallel opposite joint-axes. We give three constructions respect to three types of these linkages.

• Participants (tentative):

• Proposed by: Josef Schicho
• Type: Talk, with (limited) possibilities of interaction
• Duration: 60 minutes
• Minimum Requirements: multiplication of polynomials; creativity.

Short description: We proceed in the following steps.

• First, we construct a planar graph by pasting triangles and quadrangles according to a small set of rules.
• Second, we attach a revolution polynomial to each edge; two edges in each triangle or quadrangle are random, the others are determined by polynomial arithmetic.
• Third, we remove some edges (this step is optional).
• Fourth, we compute the axis of each revolution polynomial and connect edges with a common end point by links.

#### How to compute the dimension of a projective variety

• Proposed by: Matteo Gallet
• Type: Talk
• Duration: 60 minutes
• Minimum Requirements: basic knowledge of projective varieties and Hilbert function.

Short description: I would like to briefly consider the standard approach to the computation of the dimension of a projective algebraic variety, which relies on the computation of the Hilbert polynomial, and then introduce a technique developed by Marc Giusti (“Combinatorial Dimension Theory of Algebraic Varieties”, Journal of Symbolic Computation, 6, 1988), based on the concepts of lower (or upper) lexicographic dimension. In this way one can avoid some bottlenecks in the algorithm (which arise computing the Hilbert polynomial).

• Participants (tentative):

#### Random Walks in Multidimensional Lattices

• Proposed by: Christoph Koutschan
• Type: Talk
• Duration: 90 minutes
• Minimum Requirements: no special requirements

Short description: We give an introduction to the study of random walks on lattices. The focus is on the so-called face-centered cubic lattice and the main object of interest is the probability generating function. The latter allows to answer natural questions like “how likely is it that a random walker returns to the origin?”. Several computational approaches are presented and compared.

#### Finiteness of Symmetric Toric Ideals

• Proposed by: Abraham Martin del Campo
• Type: Talk with possibilities of interaction
• Duration: 60 minutes
• Minimum Requirements: Basic commutative algebra and Gr\“obner bases.

Short description: We consider a problem that arises in both chemistry and algebraic statistics, namely, to describe the algebraic relations between (a possible infinite number of) experimental measurements. These are symmetric ideals associated in a particular way to a fixed polynomial. In the special case when the fixed polynomial is a monomial, the ideals are toric ideals. This translates to study chains of toric ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. We present some recent finiteness results of the form: “there are a finite number of generators (for the chain of ideals) up to the action of the symmetric group.” We conclude by presenting some open problems.

#### Hypersurfaces and their singularities in causal inference

• Proposed by: Caroline Uhler
• Type: Interactive talk
• Duration: 60 minutes
• Minimum Requirements: What a statistician might know about algebraic geometry ;)

Short description: Many algorithms for inferring causality are based on partial correlation testing. Partial correlations define hypersurfaces in the parameter space of a directed Gaussian graphical model. The volumes obtained by bounding partial correlations play an important role for the performance of causal inference algorithms and the quantification of bias in causal inference. We develop an asymptotic theory for computing these volumes. Our analysis involves computing the singular loci of the partial correlation hypersurfaces and then applying the method of real log canonical thresholds. I'll explain the statistical problem and some preliminary results and you'll hopefully help me resolve some interesting singularities.

#### Uniqueness and existence of positive real solutions of parametrized polynomial systems

• Proposed by: Georg Regensburger
• Type: Talk with interactions
• Duration: 90 minutes
• Minimum Requirements: no special requirements

Short description: In applications, one is often mainly interested in positive real solutions of (parametrized) polynomial systems. We discuss conditions for uniqueness/existence of positive real solutions of systems coming from the intersection of monomial parametrizations and affine subspaces. If the corresponding subspaces are equal, existence and uniqueness is guaranteed by Birch's theorem. This is used, for example, in toric geometry, statistics, and chemical reaction networks. Our conditions, generalizing in particular Birch's theorem, are formulated in terms of sign vectors (oriented matroids) of these subspaces, and we illustrate them with examples. I'm also looking forward to discuss further relations to geometry and applications with you. The results are based on joint work with Stefan Müller in which we studied existence and uniqueness of positive equilibria of generalized mass action systems.