In this talk we present the results obtained in collaboration with F. Dillen, Y. Fu and M.I. Munteanu in the study of constant angle property for surfaces in product spaces. First results consist in the classification of constant angle surfaces in \(M^2\times R_1\). Next, we study the surfaces endowed with a canonical principal direction in \(M^2\times R_1\) and \(H^2\times R\), and finally the surfaces making constant angle with a Killing vector field in the Euclidean space \(E^3\) are classified.

We investigate the trajectories of charged particles moving in a space modeled by the 3-space \(M^2(c) \times R\) under the action of the Killing magnetic fields. One explicitly determines all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space (c=0). See [1]. We give the local description of the magnetic trajectories associated to Killing vector fields in \(S^2 \times R\), providing their complete classification (c=1). Moreover, some interpretations in terms of geometric properties are given. See [2]. Then, the geometry of normal magnetic curves in a Sasakian (respectively cosymplectic) manifold of arbitrary dimension is explained. Some results about the reduction of the codimension of a normal magnetic curve in a Sasakian space form are given. See [3].

This talk is based on the following joint papers:

[1] S.L. Druta-Romaniuc, M.I. Munteanu, Magnetic curves corresponding to Killing magnetic fields in \(E^3\), J. Math. Phys. 52 (11) (2011), art. no. 113506.

[2] M.I. Munteanu, A.I. Nistor, The classification of Killing magnetic curves in \(S^2 \times R\), J. Geom. Phys. 62 (2) (2012), 170 - 182.

[3] S.L. Druta-Romaniuc, J. Inoguchi, M.I. Munteanu, A.I. Nistor, Magnetic curves in Sasakian and cosymplectic manifolds, submitted.

Nesta palestra vamos apresentar como se deu o nascimento da teoria (linear e não-linear) dos operadores absolutamente somantes e citar alguns de seus principais resultados. Entre esses resultados, apresentaremos versões completamente não-lineares do Teorema da Dominação de Pietsch e do Teorema de Inclusão. Tais resultados são frutos de pesquisas recentes nessa área.

The parabolic Anderson model in a dynamic random environment is a differential equation, which describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate \(2d\kappa\), \(\kappa > 0\), split into two and die at rates determined by the environment. We denote by \(u(x,t)\) the mean number of particles at site \(x\) at time \(t\) conditioned on the evolution of the environment. My main object of interest is the quenched Lyapunov exponent \(\lambda(\kappa)= \lim_{t \to \infty} 1/t \log u(0,t)\), which describes the exponential growth rate of the solution at the origin as time tends to infinity. In this talk I will discuss some results concerning its qualitative behaviour as a function of \(\kappa\).

The Edwards-Sokal coupling is a well known method that links the uniqueness of Gibbs measure in the Potts model with the lack of percolation under the associated random cluster measure. Comparing stochastically this random cluster measure with the Bernoulli bond percolation measure, it is posible to estimate the critical temperature of the Potts model. This coupling has also been used in other models such as the Ashkin-Teller model, the disordered ferromagnetic Potts model, the Edwards-Anderson spin-glass model and the Widom-Rowlinson model. These issues are developed in [1] and [2].

In this talk we show how to use this method to prove the positive correlation property for the \(q\)-state clock model and to find a temperature regime where there is lack of uniqueness of Gibbs measure in the particular case \(q=4\).

This is a joint work with P. Ferrari and I. Armendriz.

[1] Geoffrey Grimmett; The random-cluster model, volumen 333 de Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2006

[2] Georgii, Hans-Otto and Häggström, Olle and Maes, Christian;The random geometry of equilibrium phases, volumen 18 de Phase transitions and critical phenomena, Academic Press, San Diego, 2001.

In a fascinating recent paper by Bernstein and Mettler, a new meromorphic quadratic differential P was introduced for minimal surfaces in \(\mathbb{R}^3\), using the Ricci condition. One of their basic results was the observation that prescribing two polarizations (the entropy differential P and the classical Hopf differential Q) determines a system of PDE in the conformal factor of metrics induced by minimal immersion, whose solution set is generically $3$-dimensional. The purpose of this talk is to show the existence of what might be called the entropy and Hopf quartic differentials \(\mathcal{P}\) and \(\mathcal{Q}\) on a minimal surface in \(\mathbb{R}^4\) (defined even in the absence of a Ricci condition), and then to express the entire family of minimal surfaces with a fixed double polarization (in either codimension) explicitly as certain superpositions of one surface in the family and its conjugate.