Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.
O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.
The goal of these lectures is to give a simple and direct introduction to some of the most basic concepts and techniques in Deep Learning. We will start by reviewing the fundamentals of Linear Regression and Linear Classifiers, and from there we will find our way into Deep Dense Neural Networks (aka multi-layer perceptrons). Then, we will introduce the theoretical and practical minimum to train such neural nets to perform the classification of handwritten digits, as provided by the MNIST dataset. This will require, in particular, the efficient computation of the gradients of the loss wrt the parameters of the model, which is achieved by backpropagation. Finally, if time permits, we will briefly describe other neural network architectures, such as Convolution Networks and Transformers, and other applications of deep learning, including Physics Informed Neural Networks, which apply neural nets to find approximate solutions of Differential Equations. The lectures will be accompanied by Python code, implementing some of these basic techniques.
The goal of these lectures is to give a simple and direct introduction to some of the most basic concepts and techniques in Deep Learning. We will start by reviewing the fundamentals of Linear Regression and Linear Classifiers, and from there we will find our way into Deep Dense Neural Networks (aka multi-layer perceptrons). Then, we will introduce the theoretical and practical minimum to train such neural nets to perform the classification of handwritten digits, as provided by the MNIST dataset. This will require, in particular, the efficient computation of the gradients of the loss wrt the parameters of the model, which is achieved by backpropagation. Finally, if time permits, we will briefly describe other neural network architectures, such as Convolution Networks and Transformers, and other applications of deep learning, including Physics Informed Neural Networks, which apply neural nets to find approximate solutions of Differential Equations. The lectures will be accompanied by Python code, implementing some of these basic techniques.
Algebraic topology provides a natural framework for realizability problems, as it explores the interplay between algebraic structures and topological spaces. These questions have been around since the 1970's, with Steenrod asking when an algebra is the cohomology of a space and Kahn asking which groups are the group of self-homotopy equivalences of a simply-connected space. Addressing such questions deepens our understanding of both spaces and their associated algebraic structures, making them quite interesting.
In this talk, we present two recent realizability results concerning group actions. First, for any action of a finite group on a finitely presented abelian group, there exists a space that realizes this action as the canonical action of the group of self-homotopy equivalences on the first homology group. Second, we establish that any action of a finite group on a permutation module is the action of the group of self-homotopy equivalences of a space on its homology groups. Additionally, we show that any simplicial complex can be perturbed in a way that reduces the automorphism group to any chosen subgroup without changing the homotopy type.
(joint work with Cristina Costoya and Antonio Viruel)
In the past decade, there has been extensive research on the nonlinear Schrödinger equation (NLS) on metric graphs, driven by both the physical and mathematical communities. Metric graphs, in essence, are one-dimensional objects that can model network-like structures. The first goal of this talk, particularly for those who are new to metric graphs, is to provide an introduction to these structures and present the appropriate functional framework for studying the NLS equation on them.
It is well-established that both the metric (size) and topological (shape) properties of the graphs can impact the existence of solutions to the NLS. As a result, no general theory currently exists for analyzing the NLS equation on metric graphs. A common approach is to focus on specific classes of graphs. In this talk, we focus on two such graphs: the $\mathcal{T}$-graph and tadpole graphs. We then discuss, using techniques from the theory of ordinary differential equations (specifically, parts of the period function), how to approach questions related to the existence, uniqueness, and multiplicity of positive solutions on these graphs.
Time permitting, we will demonstrate how this careful analysis leads to a series of existence and uniqueness/multiplicity results for a class of graphs known as single-knot graphs.
This talk is based on joint work with Simão Correia and Hugo Tavares.
