Mahler Measuring the Genetic Code of Amoebae

These are the notes of the three lectures I delivered at the mini-workshop "Knot Theory and Number Theory around the A-Polynomial" at the Instituto Superior Tecnico (IST) in Lisbon in January 2014. The goal of the lectures was to familiarize, both the author and, the audience with the A-polynomials and the connection between the Mahler measures of A-polynomials and volumes. The style of these notes is expository, written informally with the aim of giving a flavor of the subje...

We present a statistical approach for the discovery of relationships between mathematical entities that is based on linear regression and deep learning with fully connected artificial neural networks. The strategy is applied to computational knot data and empirical connections between combinatorial and hyperbolic knot invariants are revealed.

We present a geometrical approach for studying dimers. We introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial ${\bf Z}_2$ holonomy round certain curves. This holonomy has the universality property that it does not change as the number of vertices in the fundamental domain of the graph is increased. It is argued that the K-theory of the torus, with or without punctures, is the appropr...

Morphological amoebas are image-adaptive structuring elements for morphological and other local image filters introduced by Lerallut et al. Their construction is based on combining spatial distance with contrast information into an image-dependent metric. Amoeba filters show interesting parallels to image filtering methods based on partial differential equations (PDEs), which can be confirmed by asymptotic equivalence results. In computing amoebas, graph structures are genera...

Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 ...

A single-celled amoeba can solve the traveling salesman problem through its shape-changing dynamics. In this paper, we examine roles of several elements in a previously proposed computational model of the solution-search process of amoeba and three modifications towards enhancing the solution-search preformance. We find that appropriate modifications can indeed significantly improve the quality of solutions. It is also found that a condition associated with the volume conserv...

Growing interest in modelling complex systems from brains to societies to cities using networks has led to increased efforts to describe generative processes that explain those networks. Recent successes in machine learning have prompted the usage of evolutionary computation, especially genetic programming to evolve computer programs that effectively forage a multidimensional search space to iteratively find better solutions that explain network structure. Symbolic regression...

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research.

Schemata theory, Markov chains, and statistical mechanics have been used to explain how evolutionary algorithms (EAs) work. Incremental success has been achieved with all of these methods, but each has been stymied by limitations related to its less-than-global view. We show that moving the investigation into topological space improves our understanding of why EAs work.

We explore the capacity of neural networks to detect a symmetry with complex local and non-local patterns : the gauge symmetry Z 2 . This symmetry is present in physical problems from topological transitions to QCD, and controls the computational hardness of instances of spin-glasses. Here, we show how to design a neural network, and a dataset, able to learn this symmetry and to find compressed latent representations of the gauge orbits. Our method pays special attention to s...