# Seminars

## TODAY

Analytic Proof of the Hairy Ball Theorem.

**Speaker:**

*Marcelo José*- IMPA

**Graduate Students’ Colloquium**

**Abstract: **We will show a demonstration of the Hairy Ball Theorem given by the mathematician John Milnor using only analytical tools.

## UPCOMING

Hamiltonian $S^1$-spaces with large equivariant pseudo-index

**Speaker:**

*Isabelle Charton*- Universität zu Köln

**Mathematical Physics**

**Abstract: **Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of embedded \(2\)-spheres in \(M\), we define the notion of equivariant pseudo-index of \(\mathcal{S}\): this is the minimum of the evaluation of the first Chern class \(c_1\) on the spheres of \(\mathcal{S}\). This can be seen as the analog in this category of the notion of pseudo-index for complex Fano varieties.

In this talk we discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of \(M\) are unimodal, we prove that it is at most \(n+1\) . Moreover, when it is exactly \(n+1\), \(M\) must be homotopically equivalent to \(\mathbb{C}P^n\).

Quantization of symplectic groupoid and multiplicative integrable models – Part I

**Speaker:**

*Francesco Bonechi*- Università Degli Studi di Firenze

**Mathematical Physics**

**Abstract: **I will present a class of non trivial examples where Weinstein's dream of quantizing Poisson manifolds through the quantization of the symplectic groupoid can be concretely realized. The construction uses singular polarizations, for instance those given by integrable models that are compatible with the groupoid structure. We call such models multiplicative. The main source of examples comes from Poisson–Nijenhuis structure defined on hermitian symmetric spaces.

Quantization of symplectic groupoid and multiplicative integrable models – Part II

**Speaker:**

*Francesco Bonechi*- Università Degli Studi di Firenze

**Mathematical Physics**

**Abstract: **I will present a class of non trivial examples where Weinstein's dream of quantizing Poisson manifolds through the quantization of the symplectic groupoid can be concretely realized. The construction uses singular polarizations, for instance those given by integrable models that are compatible with the groupoid structure. We call such models multiplicative. The main source of examples comes from Poisson–Nijenhuis structure defined on hermitian symmetric spaces.

Capillary Hydrodynamics: From thin films to oil recovery.

**Speaker:**

*Marcio Carvalho*- Departamento de Engenharia Mecânica, PUC-RJ

**Applied and Computational Mathematics**

**Abstract: **Multiphase flow behavior in microscale is governed by the balance between capillary and viscous, or viscoelastic, forces. Fundamental understanding of capillary hydrodynamics leads to important technology development in coating process of optical and specialty films, printed electronics, biological sensors, $CO_2$ sequestration and enhanced oil recovery. We discuss two examples of capillary hydrodynamic flows. The first is related to the manufacturing of functional films by coating processes. In many applications, such as solar panels and batteries, the coating liquid is a suspension of particles. We analyze coating flows of particle suspensions, investigating particle migration and orientation and how process parameters affect the final structure of the coated layer.The second example is associated with flows of complex liquids in porous media, with applications in oil recovery. We study the effect of complex dispersions (oil-water emulsions and soft microcapsules suspensions) in the pore scale. Visualization of the flow of complex fluids through a transparent network of micro-channels, which serves as a model of a porous media, reveals how the pore blocking by the dispersed phase improves pore-level displacement efficiency, leading to lower residual oil saturation.

On the geometry of euclidean hypersurfaces with constant Weingarten curvature

**Speaker:**

*Francisco Fontenele*- UFF

**Differential Geometry**

**Abstract: **I will talk about several results concerning the geometry of constant Weingarten curvature hypersurfaces immersed/embedded in euclidean space, and sketch the proof of some of these results.

Uma triangulação estendida para o algoritmo Marching Cubes 33

**Speaker:**

*Lis Ingrid Roque Lopes Custódio*- UERJ

**Computer Graphics**

**Abstract: **O algoritmo Marching Cubes é sem dúvida o mais popular dentre os algoritmos de extração de isosuperfície. Desde a sua criação, dois problemas persistiram, a saber, a qualidade da triangulação e a coerência topológica da malha resultante. Embora exista uma extensa literatura para resolvê-los, a coerência topológica é alcançada em detrimento da qualidade da triangulação e vice-versa. Nesta palestra apresentaremos uma versão estendida do algoritmo Marching Cubes 33 (uma variação do algoritmo Marching Cubes que garante a coerência topológica da malha gerada). No algoritmo proposto, os vértices da grade do dado volumétrico são rotulados com "+", "- " ou " = ", de acordo com a relação entre seu valor no campo escalar e o isovalor de interesse. A inclusão do rótulo "=" resulta em um processo de triangulação que naturalmente evita a criação de triângulos degenerados. Em seguida, apresentaremos a aplicação do método proposto na melhoria da qualidade da triangulação gerada, preservando ao máximo a topologia da malha.

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

**Speaker:**

*Ilkka Holopainen*- University of Helsinki

**Differential Geometry**

**Abstract: **I will talk on a recent joint paper with Jean-Baptiste Casteras and Jaime Ripoll.

Let $N$ be an $n$-dimensional Cartan--Hadamard manifold that satisfies the so-called strict convexity condition and has strictly negative upper bound for sectional curvatures, $K\le-\alpha^2<0$. Given a suitable subset $L\subset\partial_\infty N$ of the asymptotic boundary of $N$ and a continuous function $H\colon N\to [-H_0,H_0],\ H_0<(n-1)\alpha$, we prove the existence of an open subset $Q\subset N$ of locally finite perimeter whose boundary $M$ has generalized mean curvature $H$ towards $N\setminus Q$ and $\partial_\infty M=L$. By regularity theory, $M$ is a $C^2$-smooth $(n-1)$-dimensional submanifold up to a closed singular set of Hausdorff dimension at most $n-8$. In particular, $M$ is $C^2$-smooth if $n\le 7$. Moreover, if $H\in [-H_0,H_0]$ is constant and $n\le 7$, there are at least two disjoint hypersurfaces $M_1$, $M_2$ with constant mean curvature $H$ and $\partial_\infty M_i=L,\ i=1,2$.

Our results generalize those of Alencar and Rosenberg, Tonegawa, and others.