# Seminar "Operator Theory and Its Applications"

## Schedule

#### Past seminars

• 10.12.2021 Chaotic composition operators on $L^p$-spaces ( Benito Frazão Pires, USP).
• 19.11.2021 Solving an abstract nonlinear eigenvalue problem by the inverse iteration method ( Grey Ercole, UFMG).

Let $\left( X,\left\Vert \cdot\right\Vert _{X}\right)$ and $\left(Y,\left\Vert \cdot\right\Vert _{Y}\right)$ be Banach spaces over $\mathbb{R}$, with $X$ uniformly convex and compactly embedded into $Y.$ We use an iteration method to solve the abstract eigenvalue problem $A(w)=\lambda\left\Vert w\right\Vert _{Y}^{p-q}B(w),$ where the maps $A:X\rightarrow X^{\star}$ and $B:Y\rightarrow Y^{\star}$ are homogeneous of degrees $p-1$ and $q-1,$ respectively. Our approach covers a large range of eigenvalue problems for quasilinear elliptic PDEs. We present examples involving the $p$-Laplacian operator with both Dirichlet and Steklov boundary conditions.Video of the talk

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• 05.11.2021 O que o espectro pode nos contar sobre os grafos? ( Renata Raposo Del-Vecchio, UFF).

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• 22.10.2021 A equação de Daugavet e suas versões modificadas ( Elisa Regina dos Santos, FAMAT-UFU).

A equação de Daugavet surgiu de um trabalho de I. K. Daugavet de 1963, onde ele provou que todo operador compacto definido no espaço $C[0,1]$ das funções contínuas de $[0,1]$ no corpo dos reais ou dos complexos satisfaz a identidade $\|Id+T\|=1+\|T\|$, onde $Id$ denota o operador identidade. Desde então, diversos pesquisadores têm se dedicado a estudar tal identidade e ela ficou conhecida como equação de Daugavet. Nesta palestra serão discutidos alguns resultados sobre tal equação e apresentadas algumas versões modificadas dela que também são amplamente estudadas. Video of the talk

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• 08.10.2021 Sobre as soluções com camadas de transição de determinados problemas elípticos ( Maicon Sônego, UNIFEI).

Nesta palestra pretendo apresentar resultados acerca das soluções que desenvolvem camadas de transição interna de alguns problemas elípticos singularmente perturbados. O objetivo principal é mostrar o papel das heterogeneidades do problema quanto a localização exata da interface de transição. Pretendo falar sobre a literatura relacionada e sobre as diversas questões abertas a respeito. Video of the talk

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• 23.07.2021 Eigenvalues and eigenvectors for nonlinear problems with Fredholm operators ( Pierluigi Benevieri, IME-USP).
• 25.06.2021 Efeito Aharonov-Bohm: algumas questões matemáticas ( César Rogério de Oliveira, UFScar).

Serão apresentadas questões matemáticas, que temos investigado nos últimos anos, a respeito do efeito Aharanov-Bohm em mecânica quântica. Por exemplo: como verificar a existência do efeito em modelos específicos? Como tratar matematicamente as idealizações de solenóide infinito e blindado? Seria possível uma situação em que a partícula (elétron) não interage com a fronteira do solenóide? Abordagens técnicas serão evitadas, com prioridade para formulação dos problemas e resultados. Video of the talk

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• 11.06.2021 Pure and Mixed States in Physics ( Ricardo Correa da Silva, IME-USP).

We will present a review on the notion of pure states and mixtures as mathematical concepts that apply for both classical and quantum physical theories, as well as for any other theory depending on statistical description. We intend to highlight the importance of several elements often forgotten when dealing with the subject. Finally, an example will be shown where a pure quantum state converges to a classical mixture of particles as Planck’s constant tends to zero. Video of the talk

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• 28.05.2021 Refined scales of decaying rates of operator semigroups on Hilbert spaces: typical behavior ( Silas Luiz de Carvalho, UFMG).

We study relations between the decaying rates of operator semigroups on Hilbert spaces and some spectral properties of their respective generators. In particular, we show that the decaying rates of orbits of semigroups which are stable but not exponentially stable, typically in Baire's sense, depend on sequences of time going to infinity. This is a joint work with Moacir Aloisio (UFAM) and César R. de Oliveira (UFSCar).Video of the talk

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• 14.05.2021 Complemented copies of $c_0(\tau)$ in tensor products of Banach spaces ( Vinícius Morelli Côrtes, IME-USP).

The injective and tensor products of $X$ and $Y$ may contain copies of unexpected spaces, even when both $X$ and $Y$ contain no copies of that space. In this talk, we will recall some classical results concerning copies of the sequence space $c_0$ in tensor products and discuss some recent results on the non-separable space $c_0(\tau)$, where $\tau$ is an uncountable cardinal. This is a joint work with Elói Medina Galego (IME-USP) and Christian Samuel (Aix-Marseille Université).Video of the talk

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• 30.04.2021 Curve shortening flow on Riemann surfaces ( Nikolaos Roidos, University of Patras).

We start by recalling some classical results concerning the problem of curve shortening flow on Riemann surfaces. Then we consider the same problem on Riemann surfaces with singular metrics. This flow is governed by a degenerate quasilinear parabolic equation. It turns out that, if we start with a curve passing from a singular point of the surface, then the evolving curve stays fixed at this point. We also discuss some collapsing and convergence results.Video of the talk

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• 16.04.2021 Pseudodifferential operators in strict deformation quantization ( Severino Toscano do Rêgo Melo, IME-USP).

I plan to report on joint work with Rodrigo Cabral and Michael Forger, in which we prove the uniqueness of the $C^*$-norm in Rieffel's approach to deformation quantization in the case of $C^*$-algebras with an action of $\mathbb{R}^d$. We rely on characterizations of pseudodifferential operators as bounded operators with smooth orbit under the action of the Heisenberg group. A good portion of the talk will be devoted to a review of old results of Cordes' and of Rieffel's. Video of the talk

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