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# Linz-Wien workshop in March 2016

Location: in Znaim. http://www.penzionumikulase.cz/en/

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

Coordinators: Mehdi Makhul (mmakhul@risc.uni-linz.ac.at).

Participants: Jarek Buczynski, Christopher Chiu, Giancarlo Castellano , Zhangwen Guo, Herwig Hauser , Christoph Koutschan, Hana Kováčová, Zijia Li, Niels Lubbes, Mehdi Makhul, Stefan Perlega, Mateusz Piorkowski, Josef Schicho.

## Schedule

For details of the activities please see below.

Soon

## List of activities

Please add to this list any activity you are responsible for. 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>.

#### Applying Graph Theory in Topology

• Proposed by: Giancarlo Castellano
• Type: Talk/Lecture
• Duration: 60 minutes

A well-known theorem states that every planar graph has a subgraph which is a subdivision of K5 or K33. This talk contains a proof Jordan's curve theorem: every simply closed curve divides the plane into two parts. The proof uses the non-planarity of K33 - which can be proved by Eulers theorem on the number of faces, vertices, and edges of a planar graph - in an essential way.

#### Reconstructing Smooth Curves from Sample Points by CRUST

• Proposed by: Zhangwen Guo
• Type: Talk/Lecture
• Duration: 60 minutes

Given sample points of an unknown smooth curve, one wants to connect the points according to adjacency on the curve. This is possible assuming that the sample satisfies the following condition: for any point on the curve, the distance to the medial axes is less than 0.25 times the distance to the nearest sample point. This talk contains a simple and fast algorithm and the correctness proof (which is of course not so simple).

#### The Number of Permutations with a Fixed Partition

• Proposed by: Hana Kováčová:
• Type: Talk/Lecture
• Duration: 60 minutes

Using a deterministic procedure, one can associate to every permutation of {1..n} a partition and its Young tableau. This talk presents a formula for the number of permutations with a fixed partition. The formula uses the length of L-shaped paths in the tableau.

#### Counting Rational Curves in the Plane

• Proposed by: Josef Schicho
• Type: Talk/Lecture
• Duration: 60 minutes
• Minimum Requirements: basic definitions of algebraic geometry

Short description: After several attempts to familiarize myself with moduli spaces for stable maps as a tool to count rational curves, a colleague (Mehdi) draw my attention to the paper http://www.math.utah.edu/~yplee/teaching/gw/Koch.pdf, which made the method accessible to me. In this talk I try to make it accessible to you as well.

• Questions, comments, etc.: here you can ask the speaker a question or make a comment

#### Enumerative Algebraic Geometry of Conics

• Proposed by: Mehdi Makhul
• Type: Talk
• Duration:60 minutes
• Minimum Requirements: Basic of algebraic geometry

Short description : Given five conics in the plane, are there any conics that are tangent to all five? If so, how many are there?

#### Energy and Time in Classical and Quantum Physics

• Proposed by: Mateusz Piorkowski
• Type: Talk
• Duration:90 minutes
• Minimum Requirements: Basics of ordinary differential equations, high school level physics

Short description : This is a discussion of an interesting symmetry of physical laws. With the exception of the second law of thermodynamics, the physical laws are symmetric in time: in classical mechanics, one can postulate a universe where the second law of thermodynamics holds with reversed time. In quantum mechanics, time reversal also implies an interchange of position and momentum.

#### The Bennet condition for oriented lines

• Proposed by: Niels Lubbes
• Type: Talk
• Duration:60 minutes
• Minimum Requirements: basic projective geometry

Short description: The “Bennet condition” is a condition on 3 skew lines in Euclidean 3-space. This condition plays an important role in the classification of mechanical linkages whose self-motion along each joint can be modelled as a rotation along a line. A “Bennet congruence” is defined as a set of all lines, which satisfy the Bennet condition with two given skew lines. The problem which we would like to address in this talk is: How many lines do two Bennet congruences have in common? In order to answer this question we define the space of oriented lines as a quadric cone in projective 3-space, such that complex conjugation reverses the orientation. After reformulating the Bennet condition in terms of oriented lines, the solution of the stated problem becomes almost trivial. It is remarkable that complex solutions of algebraic equations translate in our setting to real lines in Euclidean 3-space.

#### Maps of Mori Dream Spaces

• Proposed by: Jarek Buczyński
• Type:Talk/Teaching
• Duration:90 minutes
• Minimum Requirments:algebraic geometry( in particular Weil/Cartier divisors and their sections),basic field theory (char 0 only): algebraic closure.

Short description: Any rational map between affine spaces or projective spaces can be described in terms of their (homogeneous) coordinates. Toric Varieties and Mori Dream Spaces are classes of algebraic varieties for which there exist a sensible analogue of homogeneous coordinate ring. I will present how to obtain a description of a map of Mori Dream Spaces (or Toric Varieties) in terms of such coordinate rings. More precisely (in the case of regular maps) I will show there exists a finite extension of the coordinate ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.

#### Arc spaces & the Grinberg-Kazhdan-Drinfeld theorem

• Proposed by:Christopher Chiu
• Type:Talk
• Duration:60 minutes
• Minimum Requirments:basic of algebraic geometry

Short description:I will give a short introduction to arc spaces and the Grinberg-Kazhdan-Drinfeld theorem on the formal neighborhood of a k-arc. If X is a positive-dimensional variety over k, then the corresponding space of arcs X_\infty is locally the spectrum of a quotient of a polynomial ring in countably many variables. Similarly, the formal neighborhood of a k-arc is the formal spectrum of a quotient of a power series ring in countably many variables. In my talk, I will investigate these two rings and their properties in contrast to polynomial/power series rings with only a finite number of indeterminates. 