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A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of free algebras: intuitively, this expresses that every…

The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, nervePool, for data structured as simplicial…

Network tomography, a classic research problem in the realm of network monitoring, refers to the methodology of inferring unmeasured network attributes using selected end-to-end path measurements. In the research community, network…

We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis,…

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…

Convergence is a fundamental topic in analysis that is most commonly modelled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost everywhere of a sequence of measurable…

We describe a class of systems theory based neural networks called "Network Of Recurrent neural networks" (NOR), which introduces a new structure level to RNN related models. In NOR, RNNs are viewed as the high-level neurons and are used to…

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely…

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak…

The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…

Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…

Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the…

The human brain is a complex system, and understanding its mechanisms has been a long-standing challenge in neuroscience. The study of the functional connectome, which maps the functional connections between different brain regions, has…

Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible.

We look at the internal structure of neural networks which is usually treated as a black box. The easiest and the most comprehensible thing to do is to look at a binary classification and try to understand the approach a neural network…

This thesis is a compendium of research which brings together ideas from the fields of Complex Networks and Computational Neuroscience to address two questions regarding neural systems: 1) How the activity of neurons, via synaptic changes,…

Brain function and connectivity is a pressing mystery in medicine related to many diseases. Neural connectomes have been studied as graphs with graph theory methods including topological methods. Work has started on hypergraph models and…

A primary advantage of neural networks lies in their feature learning characteristics, which is challenging to theoretically analyze due to the complexity of their training dynamics. We propose a new paradigm for studying feature learning…

Convolutional neural networks (CNN's) are powerful and widely used tools. However, their interpretability is far from ideal. One such shortcoming is the difficulty of deducing a network's ability to generalize to unseen data. We use…