Workshop Algebraic Geometry

Scientific program

Jan Legersky (JKU): Rigid Graphs - And yet they move

A graph is called generically rigid if there are only finitely many realizations in the plane inducing the same distances between adjacent points as a given generic realization, counted modulo rigid motions. In this talk, we focus on generically rigid graph for which a non-generic choice of edge lengths allows infinitely many injective realizations.

Cristopher Chiu (UVIE): A history of sound synthesis in the 1980s: Subtractive synthesis & Frequency modulation

In this talk we will give a brief introduction to various methods of sound synthesis techniques used in commercial electronic instruments of the late 1970s and early 1980s, mainly focusing on subtractive and FM synthesis. We will explain the basics of subtractive synthesis and then illustrate why it is not particularly adapt at synthesizing sounds with dynamic changes of timbre. After explaining the main idea behind John Chowning's seminal 1973 article using Fourier series and Bessel functions, we will then create a bell-like tone using a software implementation of FM  synthesis.

Josef Schicho (JKU): The Cross Ratio

We describe a construction of a quotient variety of n points in the projective line modulo projective transformation, based on the cross ratio of 4 points. The construction is isomorphic to the space of marked stable curves of genus zero by Knudsen, but more elementary. This is joint work with H. Hauser, in progress.

Marus Reibnegger (UVIE): Summation of divergent series.

Some methods for assigning finite values to divergent series in a systematic way are presented. In particulsr zeta function regularization and it's use in physics are discussed.

Jiayue Qi (JKU): 3D-realization for moving graphs.

We define a specific type of 3D-realization for moving graphs which we call L-model. We give a criterion which, when fulfilled, leads to the existence of a collision-free L-model of a given moving graph. Also we apply the result to some moving graph families.

Hana Melanova (UVIE): Geometric Invariants

Geometric invariants can be used in the resolution of singular plane algebraic curves. In this talk we will introduce the concept of geometric invariants and explain its geometrical meaning. After introducing higher algebraic curvatures we will show that they generate the whole field of geometric invariants. Finally, we will discuss which role they play in the resolution of singular plane  curves.

Audie Warren (RICAM): The Sum-product Phenomenon

We discuss the history of the sum-product phenomenon, and how incidence geometry is a main tool used to prove results in the area.

Chiara Novarini (UVIE): Statistics in Basketball

After presenting the basic statistics of basketball, we will proceed to analyze specifically how we can evaluate the difensive skills of a player. We will first present how to determine who is guarding whom from spatial data. From this we can infer how a difensive player impacts on efficiency and frequency of shooting of the offender he is guarding. Finally, we will
shortly describe the other statistics that one can deduce from these data, namely attention received and difensive entropy.

Elaine Wong (RICAM): Minimum Number of Monochromatic Schur Triples

We obtain a nice (exact) formula for the minimum number of monochromatic Schur triples over any 2-coloring of [n].

Sergey Yurkevich (UVIE): Some words on the queuing theory

Queues are very common in modern society and concern each of us: Waiting rooms at doctor's offices, airport management or even computer servers and online requests are some examples for them. A Queue that has gotten out of control often means welfare losses for agents involved in it. It turns out there is a wide mathematical theory on this topic, known as Queuing Theory. The main goal of the talk was to provide a short introduction into this theme to generate an interesting discussion on the mathematics and methods used in the study of queues. Even though the connection to Algebra is very vage, I am sure that the subject itself contains fascinating insights worth sharing.

Mehdi Makhul (RICAM): k-approximate groups

An approximate group is a subset of a group which behaves like a subgroup "up to a constant error". More precisely let g>1, a K_approximate subgroup in a group G is a finite symmetric subset of G contains the identity element and |AA|< K|A| (AA={ab: a,b in A}). A natural question is to find a classification for these sets, this question has been solved by Breuillar-Green Tao in 2011. A consequence of their result gives a short answer to the polynomial growth Gromov Theorem.

Levi Haunschmid (UVIE): The Casas-Alvero Conjecture

The Casas-Alvero Conjecture states that any complex polynomial in one variable having a common root with all of its derivatives must be a power of a linear polynomial. In 2006 this conjecture was proven if the degree of the polynomial is a prime power. In the talk this proof will be discussed and some more recent developments will be outlined.

Georg Grasegger (RICAM): From Matchstick Graphs To Movable Graphs

In this talk I present how problems from matchstick graphs turned out to yield examples of interesting flexible motions of genereically rigid graphs. I show how flexible matchstick embeddings can be found algorithmically for these graphs.

David Stinner (UVIE): Integral Dimension of a noetherian ring

In my talk, the integral dimension of a noetherian ring, denoted by i(R), was introduced. While this definition relies heavily on the integral closure of ideals and seems to be completely different to the ordinary Krull-dimension, one can still find for many classes of noetherian rings a connection between these two notions. Some basic facts about the behaviour of i(R) under localization and a conjecture about finiteness where discussed.

Niels Lubbes (RICAM): Surfaces that are the union of circles

We first recall the classification of surfaces that are covered by two families of lines. After, we state the classification of surfaces that are covered by two families of circles.