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Posts
Hilbert, Schmidt and the Spectral Theorem
Published:
The Spectral Theorem comes in many flavors, one might encounter it first in it’s finite dimensional case, for $\mathbb{C}^n$. If $T:\mathbb{C}^n \to \mathbb{C}^n$ is a self-adjoint linear map then there is a basis ${x_1, …, x_n}$ of $\mathbb{C}^n$ consisting of eigenvectors of $T$ with which we have: \(Tx= \sum_{i=1}^n \lambda_i \langle x, x_i\rangle x_i.\) As its name suggests, it falls to Spectral Theory to extend this result into infinite dimensional spaces. As it’s usually the case, this brings forth considerable difficulties. One way of easing these difficulties is to retain some notion of finitude, in this case compactness.
The spectrum and bounded self-adjointness
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The spectrum $\sigma(T)$ of operators $T$ helps us them, and many of the problems spectral theorists face involve restraining this set $\sigma(T)$.
Compact operators and the unit ball
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One famous fact in functional analysis is that the closed unit ball $B_X$ of a normed space $X$ is compact if, and only if, the space $X$ is finite dimensional. As if paying homage to that, compact linear operators (operators that map bounded sets $A$ to relatively compact sets $T(A)$, where $\overline{T(A)}$ is compact), allude to both notions of compactness and finite-dimensionality.
Neumann Series
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Neumann series are series of bounded linear operators of the form
The spectrum of linear operators
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Essentially, spectral theory seeks to further the study of eigenvalues for the case of linear operators $T:X \to X$ on infinite dimensional spaces $X$.
portfolio
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publications
Bases de Riesz e séries de Fourier não harmônicas
Published in , 2023
This material is an introduction (in portuguese) to non-harmonic Fourier series.
Download here
Introdução à teoria espectral em espaços de Hilbert e aplicações em Equações Diferenciais
Published in , 2024
This material is an introduction (in portuguese) to Spectral Theory with brief applications to Differential Equations.
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O Axioma de Martin e algumas consequências
Published in , 2024
Final assignment for the course “Introduction to Set Theory” (MAT6202)
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talks
Bases de Riesz e séries de Fourier não harmônicas
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Curta apresentação no 30° SIICUSP (Simpósio Internacional de Iniciação Científica e Técnológica da USP) sobre o projeto de Iniciação Científica “Bases de Riesz e séries de Fourier não harmônicas”.
O Teorema de Hilbert-Schmidt
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Apresentação de pôster sobre o Teorema de Hilbert-Schmidt e generalizações do Teorema Espectral para o evento 1ª semana da pura.
Como treinar o seu grafão
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Seminário sobre o teorema de Ramsey finito e infinito para o grupo de seminários S4 do IME-USP.
Open for Improvement
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Apresentação de pôster sobre o Teorema de Galvin-Prikry e combinatória infinita para o XXIII Encontro Brasileiro de Topologia.
Introdução à teoria espectral e aplicações
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Apresentação de pôster sobre teoria espectral e generalizações do Teorema Espectral para a 32ª edição do SIICUSP.
O teorema de Krivine e os espaços de sequências
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Apresentação de pôster sobre o Teorema de Krivine e combinatória infinita para o 2° Encontro de Combinatória no Infinito.
teaching
Teaching Assistant: Introduction to Functional Analysis (2024)
Undergraduate course, Instituto de Matemática e Estatística, USP, 2024
In the first semester of 2024 (February-June) I acted as the Teaching Assistant for professor Vinícius Morelli in the course “Introduction to Functional Analysis” (MAT0334/MAT5721) for both undergraduates and post-graduates.