# Blog de la Biblioteca de Matemàtiques i Informàtica ## SIMBa: On a theorem of Waring for plane algebraic curves El proper dimecres, 22 de maig, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Andrés Rojas González
Universitat: Universitat de Barcelona

Data: Dimecres, 22 de maig de 2019
Lloc: Aula B1 Facultat de Matemàtiques de la Universitat de Barcelona.
Idioma: Anglès

Títol: On a theorem of Waring for plane algebraic curves
Resum: Waring’s theorem is a classic result about the intersection of plane algebraic curves. Roughly speaking, it asserts that the barycenter of the intersection points is determined by the asymptotes of the curves.
After a short historical introduction, in this talk we present the ideas that allow to obtain an extension of Waring’s theorem. Finally, we see some consequences; the most important of them will be a theorem due to Chasles.

Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.

## SIMBa: Algebraic versus semi-algebraic conditions for phylogenetic varieties El proper dimecres, 8 de maig, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Marina Garrote López
Universitat: Universitat Politècnica de Catalunya

Data: Dimecres, 8 de maig de 2019
Lloc: Aula B1 Facultat de Matemàtiques de la Universitat de Barcelona.
Idioma: Anglès

Títol: Algebraic versus semi-algebraic conditions for phylogenetic varieties
Resum: It is well known that there exists a close relationship between Phylogenetics and Algebraic Geometry. It is common to modelevolution adopting a parametric statistical model which allowsto define a joint probability distribution at the leaves of the trees. When these models are algebraic, one is able to deduce polynomial relationships between these probabilities, and the study of these polynomials and the geometry of the algebraic varieties that arisefrom them can be used to reconstruct phylogenetic trees. However not every point in this algebraic varieties has biological sense. In this talk we would like to discuss the importance of studying the subset of these varieties with biological sense and explore the extent to which restricting to these subsets can provide insight into existent methods of phylogenetic reconstruction.

Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.

## SIMBa: Studying the Fatou set for a generalization of Milnor’s family ofcubic maps El proper dimecres, 10 d’abril, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Dan Alexandru Paraschiv
Universitat: Universitat de Barcelona

Data: Dimecres, 10 d’abril de 2019
Lloc: Aula B1 Facultat de Matemàtiques de la Universitat de Barcelona.
Idioma: Anglès

Títol: Studying the Fatou set for a generalization of Milnor’s family ofcubic maps.
Resum: The first attempts of studying iteration of holomorphic functionswere succesfully realized by Julia and Fatou in the 1920s. However,due to technological limitations, having a good intuition of thearising fractal-like patterns was very difficult, and the problemstarted being studied again in the 1980s by Mandelbrot, startingwith the quadratic family $P_{c}=z^{2} + c$, where $c \in \mathbb{C}$.

Nowadays, among rational maps on the Riemann sphere, a heigh-tened interest in perturbation maps exist. We’ll summarilly introduce 3 families of maps (2 with perturbations, one of polynomials of degree 3) which are helpful in understanding ourwork. We will conclude with some results obtained by us for aspecific case of perturbation of bicritical hyperbolic polynomials ( $D_\lambda = z^{n+1}- az^{n} + \frac{\lambda}{z^d}$ for fixed $a$ belonging to a specific hyperbolic component of the parameter plane of $M_a = z^{n+1} - az^{n},\lambda$ small enough and $n,d$ natural numbers such that $\frac{1}{n}+\frac{1}{2}<1)$.

Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.

## SIMBa: Sampling, Interpolation and Fekete Points El proper dimecres, 5 de desembre, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Carlos Cruz
Universitat: Universitat de Barcelona

Data: Dimecres, 5 de desembre de 2018
Lloc: Aula B1 Facultat de Matemàtiques de la Universitat de Barcelona.
Idioma: Castellà

Títol: Sampling, Interpolation and Fekete Points
Resum: In this talk, we will explain a characterization of sampling or interpolating arrays in the space of polynomials $\mathcal{P}_k(\mathbb{C})$. This allows to understand, in a simple model, this problem in a more general case. Moreover, the aim of this talk is to introduce some interesting objects like the Fekete points. These points are so interesting and are an example of sampling and interpolating arrays.

Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.

## SIMBa: The escape trichotomy for singularly perturbed polynomials El proper dimecres, 21 de novembre, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Dan Alexandru Paraschiv
Universitat: Universitat de Barcelona

Data: Dimecres, 21 de novembre de 2018
Lloc: Aula B1 Facultat de Matemàtiques de la Universitat de Barcelona.
Idioma: English

Títol: The escape trichotomy for singularly perturbed polynomials
Resum: Holomorphic dynamics studies the iteration of holomorphic functions over different spaces.
The dichotomy for quadratic polynomials, which generates the Mandelbrot set, is one of the fundamental results.
Devaney, Look and Uminsky have managed to extend this result to the family of singularly perturbed polynomials, that is: $F_{\lambda}(z)=z^{n}+\frac{\lambda}{z^d}$, where $\lambda \in \mathbb{C}, z \in \hat{\mathbb{C}},n, d \in \mathbb{N^*}$ and $d \geq 2$.
The scape trichotomy shows that, according to the position of the critical values of the map, there exist 3 possible kinds of Julia sets: Cantor sets, Cantor sets of quasicircles and Sierpinski curves. For a clear understanding of the result, there are first going to be presented several basic notions and results from holomorphic dynamics.

Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.

## SIMBa: Osculating spaces and Lefschetz properties El proper dimecres, 14 de novembre, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Martí Salat Moltó
Universitat: Universitat de Barcelona

Data: Dimecres, 14 de novembre de 2018
Lloc: Aula B1 Facultat de Matemàtiques de la Universitat de Barcelona.
Idioma: Castellà

Títol: Osculating spaces and Lefschetz properties
Resum: Let $X \in \mathbb{P}^N$ be an algebraic variety. The sth osculating space of $X$ at a point $p \in X$ is the vector space $T_{p}^{(s)}X$ spanned by all the sth partial derivatives of a local parametrization of $X$ at $p$. Hence it can be seen as a natural generalization of the well-known notion of tangent space. It is a wide and longstanding problem to study and classify varieties whose osculating space is defective (i.e. it does not reach the maximal dimension).
In the first part of this talk we will recall the notion of osculating space and we will see some examples. In the second part we will explore the connection made by Mezzetti, Ottaviani and Miró-Roig. This connection explains why some varieties have a defective osculating space by means of the –a priori unrelated notion– of the weak Lefschetz property. This result motivates the definition of Togliatti systems. Finally, if the time allows us to do so, we will give an insight to the classification problem of Togliatti systems using combinatorial methods of Toric varieties.

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## SIMBa: An Introduction to post-quantum cryptography El proper dimecres, 24 d’octubre, se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Sergi Rovira Cisterna
Universitat: Universitat de Barcelona

Data: Dimecres, 24 d’octubre de 2018
Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.