Abstracts

In this talk we discuss several aspects of the theory of global pseudo-differential operators associated to sub-Riemannian structures on arbitrary compact Lie groups, introduced recently by the author and M. Ruzhansky. The theory developed absorbs the one by M. Ruzhansky and V. Turunen in the presence of the bi-invariant Riemannian structure on a compact Lie group. Applications of the theory to partial differential equations (PDEs), spectral theory of differential operators, and harmonic analysis on Lie groups will be discussed.

Arxiv: https://arxiv.org/pdf/2008.09651.pdf

Starting with the study of the two-dimensional discrete Haar wavelet transform and its inverse, we will describe how to solve the (ill-posedness) inverse reconstruction problem, which consists of a model Af = g^Ɛ, where given an operator A, which is non-invertible, and g^Ɛ, is a blurred (or noisy image). In this talk we show how to find an approximation of f, where f is the original image without noise.

The interest in the study of quantum groups is increasing, in particular due to its relation to quantum mechanics. In this talk, we will review the main aspects concerning the quantum group Uq(sl2) which allow us to deduce the behaviour of the Chlebsch Gordan Coefficients corresponding to the quantum group SUq(2). We will discuss about the Fourier analysis elements already extended to this case.

The aim of this talk is to share with the audience the experience of the author, in collaboration with Cayo Dória, in the search for an answer to this question.

In this talk we will present necessary conditions in order to guarantee the existence of fixed points for actions of R^k on a compact surface M. The result was obtained initially by professor Elon L. Lima.

In this talk we review about homogeneous Riemannian g.o. spaces, including some recent progress by the author, jointly with L. Grama and Caio J. C. Negreiros, for real flag manifolds of classical Lie groups.

Arxiv: https://arxiv.org/pdf/2010.00542.pdf

Surfaces of general type are lovely unclassificable objects in algebraic geometry. In this talk, we want to give a historical overview of the Geography of surfaces of general type. We want to start with the very first work presented by Ulf Person. Here, as it is usual, geography refers to the problem of building surfaces for the invariants c_1^2 (self-intersection of the first Chern class) and c_2 (topological Euler characteristic). In this talk, we will present the positive results in the area, as for example, Person's theorem on the density of Chern slopes in the set [1/5, 2] for simply connected surfaces, the density results of Roulleau-Urzúa of simply connected surfaces on the interval [1,3], and the Troncoso-Urzúa density of Chern slopes of surfaces with any possible fundamental group G in the interval [1,3]. As well as, we will present the negative results or type constraint results, which show a relation between the Chern slope 1/5\leq c_1^2/c_2 < 1/2 and the fundamental group, it is a compilation of results and ideas by differents important algebraic geometers as A. Beauville, G. Xiao, R. Pardini, M. Mendes, C. Ciliberto among others.

Arxiv: https://arxiv.org/pdf/1912.07378.pdf

Arxiv: https://arxiv.org/pdf/2003.12619.pdf

In this talk we will discuss the boundedness of global pseudo-differential operators and Fourier multipliers on smooth manifolds using the nonharmonic Fourier analysis developed by Ruzhansky, Tokmagambetov and Delgado. Using the notion of a global symbol we extend a classical condition of Hörmander-Mihlin and of Hörmander to guarantee the Lp-boundedness and Lp-Lq boundedness of pseudo-differential operators respectively. As an application, we obtain the boundedness of spectral multipliers, embedding theorems and time asymptotics for heat kernels of the anharmonic oscillator.

This talk is based on my joint works with Duván Cardona (Ghent University), Marianna Chatzakou(Imperial College London), Michael Ruzhansky(Ghent University), and Niyaz Tokmagambetov (Ghent University).

Arxiv: https://arxiv.org/pdf/2005.04936.pdf

• Prof. Guillermo Restrepo Sierra (In memoriam, Universidad del Valle).

• Prof. Doris Hinestroza (In memoriam, Universidad del Valle).

• Prof. Ana María Sanabria (In memoriam, Universidad del Valle).

We study the motion of a robotic arm inside a rectangular tunnel of width 2. In this talk we show how to calculate the number of d-cubes of the configuration space of the robotic arms and his Euler characteristic using the generating functions.

Article: math.sfsu.edu/federico/Articles/robotsfpsac.pdf